CLASS MANUAL

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CLASS MANUAL
Last updated March 25, 2017

Generated by Doxygen 1.8.13

Contents

1

CLASS: Cosmic Linear Anisotropy Solving System

1

2

Where to find information and documentation on CLASS?

3

3

CLASS overview (architecture, input/output, general principles)

5

4

Data Structure Documentation

15

4.1

nonlinear Struct Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

4.1.1

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

4.1.2

Field Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.2.1

method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.2.2

k_size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.2.3

k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.2.4

tau_size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.2.5

tau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.2.6

nl_corr_density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.2.7

k_nl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.2.8

nonlinear_verbose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

4.1.2.9

error_message . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

ii

5

CONTENTS

File Documentation

19

5.1

background.c File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

5.1.1

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

5.1.2

Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

5.1.2.1

background_at_tau() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

5.1.2.2

background_tau_of_z()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

5.1.2.3

background_functions() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

5.1.2.4

background_w_fld() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

5.1.2.5

background_init()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

5.1.2.6

background_free() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

5.1.2.7

background_free_input() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

5.1.2.8

background_indices() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

5.1.2.9

background_ncdm_distribution() . . . . . . . . . . . . . . . . . . . . . . . . . .

27

5.1.2.10 background_ncdm_test_function() . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.1.2.11 background_ncdm_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.1.2.12 background_ncdm_momenta()

. . . . . . . . . . . . . . . . . . . . . . . . . .

29

5.1.2.13 background_ncdm_M_from_Omega() . . . . . . . . . . . . . . . . . . . . . . .

30

5.1.2.14 background_solve() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

5.1.2.15 background_initial_conditions() . . . . . . . . . . . . . . . . . . . . . . . . . .

31

5.1.2.16 background_output_titles() . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

5.1.2.17 background_output_data() . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

5.1.2.18 background_derivs() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

5.1.2.19 V_e_scf() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

5.1.2.20 V_p_scf() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

5.1.2.21 V_scf() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

background.h File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

5.2.1

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

5.2.2

Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

5.2.2.1

struct background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

5.2.2.2

struct background_parameters_and_workspace

41

5.2

. . . . . . . . . . . . . . . . .

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CONTENTS

iii

5.2.2.3
5.3

5.4

. . . . . . . . . . . . . . . . .

41

class.c File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

5.3.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

common.h File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

5.4.1

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

5.4.2

Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

5.4.2.1

struct precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Enumeration Type Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

5.4.3.1

evolver_type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

5.4.3.2

pk_def . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

5.4.3.3

file_format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

input.c File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

5.5.1

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

5.5.2

Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

5.5.2.1

input_init_from_arguments() . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

5.5.2.2

input_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

5.5.2.3

input_read_parameters() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

5.5.2.4

input_default_params()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

5.5.2.5

input_default_precision() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

5.5.2.6

get_machine_precision() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

5.5.2.7

class_fzero_ridder() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

5.5.2.8

input_try_unknown_parameters() . . . . . . . . . . . . . . . . . . . . . . . . .

59

5.5.2.9

input_get_guess() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

5.5.2.10 input_find_root() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

input.h File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.6.1

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5.6.2

Enumeration Type Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5.6.2.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

lensing.c File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5.7.1

64

5.4.3

5.5

5.6

5.7

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struct background_parameters_for_distributions

Detailed Description

target_names

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

CONTENTS

5.7.2

Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.7.2.1

lensing_cl_at_l() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.7.2.2

lensing_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5.7.2.3

lensing_free()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

5.7.2.4

lensing_indices() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

5.7.2.5

lensing_lensed_cl_tt() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.7.2.6

lensing_addback_cl_tt() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.7.2.7

lensing_lensed_cl_te() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.7.2.8

lensing_addback_cl_te() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.7.2.9

lensing_lensed_cl_ee_bb()

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

. . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.7.2.11 lensing_d00()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.7.2.12 lensing_d11()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.7.2.13 lensing_d1m1()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.7.2.14 lensing_d2m2()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.7.2.15 lensing_d22()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.7.2.16 lensing_d20()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.7.2.17 lensing_d31()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.7.2.18 lensing_d3m1()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.7.2.19 lensing_d3m3()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.7.2.21 lensing_d4m2()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.7.2.22 lensing_d4m4()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

lensing.h File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5.8.1

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.8.2

Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.8.2.1

struct lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

nonlinear.c File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

5.9.1

Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.9.2

Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.7.2.10 lensing_addback_cl_ee_bb()

5.7.2.20 lensing_d40()

5.8

5.9

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CONTENTS

v

5.9.2.1

nonlinear_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.9.2.2

nonlinear_halofit() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.10 nonlinear.h File Reference

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.10.2 Macro Definition Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.10.2.1 _M_EV_TOO_BIG_FOR_HALOFIT_ . . . . . . . . . . . . . . . . . . . . . . .

80

5.11 output.c File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.10.1 Detailed Description

5.11.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.11.2 Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.11.2.1 output_init()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.11.2.2 output_cl() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

5.11.2.3 output_pk() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

5.11.2.4 output_pk_nl() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

5.11.2.5 output_tk() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

5.11.2.6 output_print_data() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5.11.2.7 output_open_cl_file() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5.11.2.8 output_one_line_of_cl() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.11.2.9 output_open_pk_file() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.11.2.10 output_one_line_of_pk() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.12 output.h File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.12.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.12.2 Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.12.2.1 struct output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.12.3 Macro Definition Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.12.3.1 _Z_PK_NUM_MAX_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.13 perturbations.c File Reference

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.13.2 Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.13.2.1 perturb_sources_at_tau() . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.13.2.2 perturb_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.13.1 Detailed Description

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vi

CONTENTS

5.13.2.3 perturb_free() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.13.2.4 perturb_indices_of_perturbs() . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5.13.2.5 perturb_timesampling_for_sources()

. . . . . . . . . . . . . . . . . . . . . . .

95

5.13.2.6 perturb_get_k_list() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.13.2.7 perturb_workspace_init()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5.13.2.8 perturb_workspace_free() . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.13.2.9 perturb_solve() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.13.2.10 perturb_prepare_output() . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.13.2.11 perturb_find_approximation_number() . . . . . . . . . . . . . . . . . . . . . . . 100
5.13.2.12 perturb_find_approximation_switches() . . . . . . . . . . . . . . . . . . . . . . 100
5.13.2.13 perturb_vector_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.13.2.14 perturb_vector_free()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.13.2.15 perturb_initial_conditions() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.13.2.16 perturb_approximations() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.13.2.17 perturb_timescale() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.13.2.18 perturb_einstein() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.13.2.19 perturb_total_stress_energy() . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.13.2.20 perturb_sources() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.13.2.21 perturb_print_variables() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.13.2.22 perturb_derivs() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.13.2.23 perturb_tca_slip_and_shear() . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.14 perturbations.h File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.14.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.14.2 Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.14.2.1 struct perturbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.14.2.2 struct perturb_vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.14.2.3 struct perturb_workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.14.2.4 struct perturb_parameters_and_workspace . . . . . . . . . . . . . . . . . . . . 128
5.14.3 Macro Definition Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.14.3.1 _SELECTION_NUM_MAX_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
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5.14.3.2 _MAX_NUMBER_OF_K_FILES_ . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.14.4 Enumeration Type Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.14.4.1 tca_flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.14.4.2 tca_method

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.14.4.3 possible_gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.15 primordial.c File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.15.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.15.2 Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.15.2.1 primordial_spectrum_at_k() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.15.2.2 primordial_init()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.15.2.3 primordial_free() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.15.2.4 primordial_indices() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.15.2.5 primordial_get_lnk_list() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.15.2.6 primordial_analytic_spectrum_init() . . . . . . . . . . . . . . . . . . . . . . . . 135
5.15.2.7 primordial_analytic_spectrum() . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.15.2.8 primordial_inflation_potential() . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.15.2.9 primordial_inflation_hubble() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.15.2.10 primordial_inflation_indices()

. . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.15.2.11 primordial_inflation_solve_inflation() . . . . . . . . . . . . . . . . . . . . . . . . 137
5.15.2.12 primordial_inflation_analytic_spectra() . . . . . . . . . . . . . . . . . . . . . . . 138
5.15.2.13 primordial_inflation_spectra() . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.15.2.14 primordial_inflation_one_wavenumber() . . . . . . . . . . . . . . . . . . . . . . 139
5.15.2.15 primordial_inflation_one_k() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.15.2.16 primordial_inflation_find_attractor() . . . . . . . . . . . . . . . . . . . . . . . . 141
5.15.2.17 primordial_inflation_evolve_background() . . . . . . . . . . . . . . . . . . . . . 141
5.15.2.18 primordial_inflation_check_potential() . . . . . . . . . . . . . . . . . . . . . . . 142
5.15.2.19 primordial_inflation_check_hubble() . . . . . . . . . . . . . . . . . . . . . . . . 143
5.15.2.20 primordial_inflation_get_epsilon() . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.15.2.21 primordial_inflation_find_phi_pivot() . . . . . . . . . . . . . . . . . . . . . . . . 145
5.15.2.22 primordial_inflation_derivs() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
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5.15.2.23 primordial_external_spectrum_init() . . . . . . . . . . . . . . . . . . . . . . . . 147
5.16 primordial.h File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.16.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.16.2 Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.16.2.1 struct primordial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.16.3 Enumeration Type Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.16.3.1 primordial_spectrum_type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.16.3.2 linear_or_logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.16.3.3 potential_shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.16.3.4 target_quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.16.3.5 integration_direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.16.3.6 time_definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.16.3.7 phi_pivot_methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.16.3.8 inflation_module_behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.17 spectra.c File Reference

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.17.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.17.2 Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.17.2.1 spectra_cl_at_l() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.17.2.2 spectra_pk_at_z() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.17.2.3 spectra_pk_at_k_and_z() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.17.2.4 spectra_pk_nl_at_z() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.17.2.5 spectra_pk_nl_at_k_and_z() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.17.2.6 spectra_tk_at_z()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.17.2.7 spectra_tk_at_k_and_z() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.17.2.8 spectra_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.17.2.9 spectra_free() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.17.2.10 spectra_indices() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.17.2.11 spectra_cls() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.17.2.12 spectra_compute_cl() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.17.2.13 spectra_k_and_tau() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
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5.17.2.14 spectra_pk() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.17.2.15 spectra_sigma() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.17.2.16 spectra_matter_transfers() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.17.2.17 spectra_output_tk_data()
5.18 spectra.h File Reference

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.18.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.18.2 Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.18.2.1 struct spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.19 thermodynamics.c File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.19.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.19.2 Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.19.2.1 thermodynamics_at_z() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.19.2.2 thermodynamics_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.19.2.3 thermodynamics_free() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.19.2.4 thermodynamics_indices() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.19.2.5 thermodynamics_helium_from_bbn() . . . . . . . . . . . . . . . . . . . . . . . 180
5.19.2.6 thermodynamics_onthespot_energy_injection() . . . . . . . . . . . . . . . . . . 181
5.19.2.7 thermodynamics_energy_injection() . . . . . . . . . . . . . . . . . . . . . . . . 181
5.19.2.8 thermodynamics_reionization_function()

. . . . . . . . . . . . . . . . . . . . . 183

5.19.2.9 thermodynamics_get_xe_before_reionization() . . . . . . . . . . . . . . . . . . 184
5.19.2.10 thermodynamics_reionization() . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.19.2.11 thermodynamics_reionization_sample() . . . . . . . . . . . . . . . . . . . . . . 185
5.19.2.12 thermodynamics_recombination() . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.19.2.13 thermodynamics_recombination_with_hyrec() . . . . . . . . . . . . . . . . . . . 187
5.19.2.14 thermodynamics_recombination_with_recfast() . . . . . . . . . . . . . . . . . . 187
5.19.2.15 thermodynamics_derivs_with_recfast() . . . . . . . . . . . . . . . . . . . . . . 189
5.19.2.16 thermodynamics_merge_reco_and_reio() . . . . . . . . . . . . . . . . . . . . . 190
5.19.2.17 thermodynamics_output_titles() . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.20 thermodynamics.h File Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.20.1 Detailed Description
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5.20.2 Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.20.2.1 struct thermo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.20.2.2 struct recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.20.2.3 struct reionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.20.2.4 struct thermodynamics_parameters_and_workspace . . . . . . . . . . . . . . . 197
5.20.3 Macro Definition Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.20.3.1 f1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.20.3.2 f2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.20.3.3 _YHE_BIG_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.20.3.4 _YHE_SMALL_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.20.4 Enumeration Type Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.20.4.1 recombination_algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.20.4.2 reionization_parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.20.4.3 reionization_z_or_tau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.21 transfer.c File Reference

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

5.21.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.21.2 Function Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.21.2.1 transfer_functions_at_q()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.21.2.2 transfer_init() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.21.2.3 transfer_free() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.21.2.4 transfer_indices_of_transfers() . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.21.2.5 transfer_get_l_list()

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.21.2.6 transfer_get_q_list() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.21.2.7 transfer_get_k_list() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
5.21.2.8 transfer_get_source_correspondence() . . . . . . . . . . . . . . . . . . . . . . 206
5.21.2.9 transfer_source_tau_size() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.21.2.10 transfer_compute_for_each_q() . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.21.2.11 transfer_interpolate_sources() . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.21.2.12 transfer_sources() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.21.2.13 transfer_selection_function() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
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5.21.2.14 transfer_dNdz_analytic() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
5.21.2.15 transfer_selection_sampling() . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.21.2.16 transfer_lensing_sampling() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.21.2.17 transfer_source_resample() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.21.2.18 transfer_selection_times() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.21.2.19 transfer_selection_compute() . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.21.2.20 transfer_compute_for_each_l()

. . . . . . . . . . . . . . . . . . . . . . . . . . 214

5.21.2.21 transfer_integrate() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.21.2.22 transfer_limber() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.21.2.23 transfer_limber_interpolate() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.21.2.24 transfer_limber2() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.22 transfer.h File Reference

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5.22.1 Detailed Description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.22.2 Data Structure Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.22.2.1 struct transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.22.2.2 struct transfer_workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.22.3 Enumeration Type Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.22.3.1 radial_function_type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

6

The ‘external_Pk‘ mode

227

7

Updating the manual

231

Index

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Chapter 1

CLASS: Cosmic Linear Anisotropy Solving System
Authors: Julien Lesgourgues and Thomas Tram
with several major inputs from other people, especially Benjamin Audren, Simon Prunet, Jesus Torrado, Miguel
Zumalacarregui, Francesco Montanari, etc.
For download and information, see http://class-code.net

Compiling CLASS and getting started
(the information below can also be found on the webpage, just below the download button)
Download the code from the webpage and unpack the archive (tar -zxvf class_vx.y.z.tar.gz), or clone it from
https://github.com/lesgourg/class_public. Go to the class directory (cd class/ or class_public/ or
class_vx.y.z/) and compile (make clean; make class). You can usually speed up compilation with the option -j: make
-j class. If the first compilation attempt fails, you may need to open the Makefile and adapt the name of the compiler
(default: gcc), of the optimization flag (default: -O4 -ffast-math) and of the OpenMP flag (default: -fopenmp; this flag
is facultative, you are free to compile without OpenMP if you don't want parallel execution; note that you need the
version 4.2 or higher of gcc to be able to compile with -fopenmp). Many more details on the CLASS compilation are
given on the wiki page

https://github.com/lesgourg/class_public/wiki/Installation
(in particular, for compiling on Mac >= 10.9 despite of the clang incompatibility with OpenMP).
To check that the code runs, type:
./class explanatory.ini

The explanatory.ini file is THE reference input file, containing and explaining the use of all possible input parameters.
We recommend to read it, to keep it unchanged (for future reference), and to create for your own purposes some
shorter input files, containing only the input lines which are useful for you. Input files must have a ∗.ini extension.
If you want to play with the precision/speed of the code, you can use one of the provided precision files (e.g.
cl_permille.pre) or modify one of them, and run with two input files, for instance:
./class test.ini cl_permille.pre

The files ∗.pre are suppposed to specify the precision parameters for which you don't want to keep default values. If
you find it more convenient, you can pass these precision parameter values in your ∗.ini file instead of an additional
∗.pre file.
The automatically-generated documentation is located in
doc/manual/html/index.html
doc/manual/CLASS_manual.pdf

On top of that, if you wish to modify the code, you will find lots of comments directly in the files.

2

CLASS: Cosmic Linear Anisotropy Solving System

Python
To use CLASS from python, or ipython notebooks, or from the Monte Python parameter extraction code, you need
to compile not only the code, but also its python wrapper. This can be done by typing just 'make' instead of 'make
class' (or for speeding up: 'make -j'). More details on the wrapper and its compilation are found on the wiki page

https://github.com/lesgourg/class_public/wiki

Plotting utility
Since version 2.3, the package includes an improved plotting script called CPU.py (Class Plotting Utility), written by
Benjamin Audren and Jesus Torrado. It can plot the Cl's, the P(k) or any other CLASS output, for one or several
models, as well as their ratio or percentage difference. The syntax and list of available options is obtained by typing
'pyhton CPU.py -h'. There is a similar script for MATLAB, written by Thomas Tram. To use it, once in MATLAB, type
'help plot_CLASS_output.m'

Developing the code
If you want to develop the code, we suggest that you download it from the github webpage

https://github.com/lesgourg/class_public
rather than from class-code.net. Then you will enjoy all the feature of git repositories. You can even develop your
own branch and get it merged to the public distribution. For related instructions, check

https://github.com/lesgourg/class_public/wiki/Public-Contributing

Using the code
You can use CLASS freely, provided that in your publications, you cite at least the paper CLASS II←: Approximation schemes . Feel free to cite more C←LASS papers!

Support
To get support, please open a new issue on the

https://github.com/lesgourg/class_public
webpage!

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Chapter 2

Where to find information and documentation on
CLASS?
Author: Julien Lesgourgues

• For what the code can actually compute: all possible input parameters, all coded cosmological models,
all functionalities, all observables, etc.: read the file explanatory.ini in the main CLASS directory: it
is THE reference file where we keep track of all possible input and the definition of all input parameters. For
that reason we recommend to leave it always unchanged and to work with copies of it, or with short input files
written from scratch.
• For the structure, style, and concrete aspects of the code: this documentation, especially the CLASS
overview chapter (the extensive automatically-generated part of this documentation is more for advanced
users); plus the slides of our CLASS lectures, for instance those from Tokyo 2014 available at

http://lesgourg.github.io/class-tour-Tokyo.html
or the more recent and concise summary from the Narbonne 2016 lecture available at

http://lesgourg.github.io/class-tour/Narbonne.pdf
An updated overview of available CLASS lecture slides is always available at

http://lesgourg.github.io/courses.html
in the section Courses on numerical tools.
• For the python wrapper of CLASS: at the moment, the best are the last slides (pages 75-96) of the Narbonne 2016 lectures

http://lesgourg.github.io/class-tour/Narbonne.pdf
Later we will expand the wrapper documentation with a dedicated chapter here.
• For the physics and equations used in the code: mainly, the following papers:
– Cosmological perturbation theory in the synchronous and conformal Newtonian gauges
C. P. Ma and E. Bertschinger.

http://arxiv.org/abs/astro-ph/9506072
10.1086/176550
Astrophys. J. 455, 7 (1995)
– The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes
D. Blas, J. Lesgourgues and T. Tram.

http://arxiv.org/abs/1104.2933 [astro-ph.CO]
10.1088/1475-7516/2011/07/034
JCAP 1107, 034 (2011)

4

Where to find information and documentation on CLASS?

– The Cosmic Linear Anisotropy Solving System (CLASS) IV: efficient implementation of non-cold relics
J. Lesgourgues and T. Tram.

http://arxiv.org/abs/1104.2935 [astro-ph.CO]
10.1088/1475-7516/2011/09/032
JCAP 1109, 032 (2011)
– Optimal polarisation equations in FLRW universes
T. Tram and J. Lesgourgues.

http://arxiv.org/abs/1305.3261 [astro-ph.CO]
10.1088/1475-7516/2013/10/002
JCAP 1310, 002 (2013)
– Fast and accurate CMB computations in non-flat FLRW universes
J. Lesgourgues and T. Tram.

http://arxiv.org/abs/1312.2697 [astro-ph.CO]
10.1088/1475-7516/2014/09/032
JCAP 1409, no. 09, 032 (2014)
– The CLASSgal code for Relativistic Cosmological Large Scale Structure
E. Di Dio, F. Montanari, J. Lesgourgues and R. Durrer.

http://arxiv.org/abs/1307.1459 [astro-ph.CO]
10.1088/1475-7516/2013/11/044
JCAP 1311, 044 (2013)
plus also some latex notes on specific sectors:
– Equations for perturbed recombination
(can be turned on optionally by the user since v2.1.0)
L. Voruz.

http://lesgourg.github.io/class_public/perturbed_recombination.pdf
– PPF formalism in Newtonian and synchronous gauge
(used by default for the fluid perturbations since v2.6.0)
T. Tram.

http://lesgourg.github.io/class_public/PPF_formalism.pdf

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Chapter 3

CLASS overview (architecture, input/output, general
principles)
Author: Julien Lesgourgues

Overall architecture of class
Files and directories
After downloading CLASS, one can see the following files in the root directory contains:

• some example of input files, the most important being explanatory.ini. a reference input file containing
all possible flags, options and physical input parameters. While this documentation explains the structure and
use of the code, explanatory.ini can be seen as the physical documentation of CLASS. The other
input file are alternative parameter input files (ending with .ini) and precision input files (ending with .pre)
• the Makefile, with which you can compile the code by typing make clean; make; this will create
the executable class and some binary files in the directory build/. The Makefile contains other
compilation options that you can view inside the file.
• CPU.py is a python script designed for plotting the CLASS output; for documentation type python CP←-

U.py --help
• plot_CLASS_output.m is the counterpart of CPU.py for MatLab
• there are other input files for various applications: an example of a non-cold dark matter distribution functions (psd_FD_single.dat), and examples of evolution and selection functions for galaxy number count
observables (myevolution.dat, myselection.dat).

Other files are split between the following directories:

• source/ contains the C files for each CLASS module, i.e. each block containing some part of the physical
equations and logic of the Boltzmann code.
• tools/ contains purely numerical algorithms, applicable in any context: integrators, simple manipulation of
arrays (derivation, integration, interpolation), Bessel function calculation, quadrature algorithms, parser, etc.

6

CLASS overview (architecture, input/output, general principles)

• main/ contains the main module class.c with the main routine class(...), to be used in interactive
runs (but not necessarily when the code is interfaced with other ones).
• test/ contains alternative main routines which can be used to run only some part of the code, to test its
accuracy, to illustrate how it can be interfaced with other codes, etc.
• include/ contains all the include files with a .h suffix.
• output/ is where the output files will be written by default (this can be changed to another directory by
adjusting the input parameter root = <...>)
• python/ contains the python wrapper of CLASS, called classy (see python/README)
• cpp/ contains the C++ wrapper of CLASS, called ClassEngine (see cpp/README)
• doc/ contains the automatic documentation (manual and input files required to build it)
• external_Pk/ contains examples of external codes that can be used to generate the primordial spectrum
and be interfaced with CLASS, when one of the many options already built inside the code are not sufficient.
• bbn/ contains interpolation tables produced by BBN codes, in order to predict e.g. YHe (ωb , ∆Neff ).
• hyrec/ contains the recombination code HyRec of Yacine Ali-Haimoud and Chris Hirata, that can be used
as an alternative to the built-in Recfast (using the input parameter recombination = <...>).

The ten-module backbone

Ten tasks

The purpose of class consists in computing some background quantities, thermodynamical quantities, perturbation transfer functions, and finally 2-point statistics (power spectra) for a given set of cosmological parameters. This
task can be decomposed in few steps or modules:

1. set input parameter values.
2. compute the evolution of cosmological background quantities.
3. compute the evolution of thermodynamical quantities (ionization fractions, etc.)
4. compute the evolution of source functions S(k, τ ) (by integrating over all perturbations).
5. compute the primordial spectra.
6. eventually, compute non-linear corrections at small redshift/large wavenumber.
7. compute transfer functions in harmonic space ∆l (k) (unless one needs only Fourier spectra P (k)'s and no
harmonic spectra Cl 's).
8. compute the observable power spectra Cl 's (by convolving the primordial spectra and the harmonic transfer
functions) and/or P (k)'s (by multiplying the primordial spectra and the appropriate source functions S(k, τ )).
9. eventually, compute the lensed CMB spectra (using second-order perturbation theory)
10. write results in files (when CLASS is used interactively. The python wrapper does not go through this step,
after 1.-9. it just keeps the output stored internally).
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7

Ten structures

In class, each of these steps is associated with a structure:

1. struct precision for input precision parameters (input physical parameters are dispatched among the
other structures listed below)
2. struct background for cosmological background,
3. struct thermo for thermodynamics,
4. struct perturbs for source functions,
5. struct primordial for primordial spectra,
6. struct nonlinear for nonlinear corrections,
7. struct transfers for transfer functions,
8. struct spectra for observable spectra,
9. struct lensing for lensed CMB spectra,
10. struct output for auxiliary variable describing the output format.

A given structure contains "everything concerning one step that the subsequent steps need to know" (for instance,
struct perturbs contains everything about source functions that the transfer module needs to know). In particular, each structure contains one array of tabulated values (for struct background, background quantities
as a function of time, for struct thermo, thermodynamical quantities as a function of redshift, for struct
perturbs, sources S(k, τ ), etc.). It also contains information about the size of this array and the value of the
index of each physical quantity, so that the table can be easily read and interpolated. Finally, it contains any derived
quantity that other modules might need to know. Hence, the communication from one module A to another module
B consists in passing a pointer to the structure filled by A, and nothing else.
All "precision parameters" are grouped in the single structure struct precision. The code contains no other
arbitrary numerical coefficient.

Ten modules

Each structure is defined and filled in one of the following modules (and precisely in the order below):

1. input.c
2. background.c
3. thermodynamics.c
4. perturbations.c
5. primordial.c
6. nonlinear.c
7. transfer.c
8. spectra.c
9. lensing.c
10. output.c
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8

CLASS overview (architecture, input/output, general principles)

Each of these modules contains at least three functions:

• module_init(...)
• module_free(...)
• module_something_at_somevalue

where module is one of input, background, thermodynamics, perturb, primordial,
nonlinear, transfer, spectra, lensing, output.
The first function allocates and fills each structure. This can be done provided that the previous structures in the
hierarchy have been already allocated and filled. In summary, calling one of module_init(...) amounts in
solving entirely one of the steps 1 to 10.
The second function deallocates the fields of each structure. This can be done optionally at the end of the code (or,
when the code is embedded in a sampler, this must be done between each execution of class, and especially
before calling module_init(...) again with different input parameters).
The third function is able to interpolate the pre-computed tables. For instance, background_init() fills a
table of background quantities for discrete values of conformal time τ , but background_at_tau(tau, ∗
values) will return these values for any arbitrary τ .
Note that functions of the type module_something_at_somevalue are the only ones which are called from
another module, while functions of the type module_init(...) and module_free(...) are the only one
called by the main executable. All other functions are for internal use in each module.
When writing a C code, the ordering of the functions in the ∗.c file is in principle arbitrary. However, for the sake of
clarity, we always respected the following order in each CLASS module:

1. all functions that may be called by other modules, i.e. "external functions", usually named like module_←-

something_at_somevalue(...)
2. then, module_init(...)
3. then, module_free()
4. then, all functions used only internally by the module

The main() function(s)
The main.c file
The main executable of class is the function main() located in the file main/main.c. This function consist
only in the following lines (not including comments and error-management lines explained later):
main() {
struct precision pr;
struct background ba;
struct thermo th;
struct perturbs pt;
struct primordial pm;
struct nonlinear nl;
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9

struct transfers tr;
struct spectra sp;
struct lensing le;
struct output op;

input_init_from_arguments(argc, argv,&pr,&ba,&th,&pt,&tr,&pm,&sp,&nl,&le,&op,errmsg);
background_init(&pr,&ba);
thermodynamics_init(&pr,&ba,&th);
perturb_init(&pr,&ba,&th,&pt);
primordial_init(&pr,&pt,&pm);
nonlinear_init(&pr,&ba,&th,&pt,&pm,&nl);
transfer_init(&pr,&ba,&th,&pt,&nl,&tr);
spectra_init(&pr,&ba,&pt,&pm,&nl,&tr,&sp);
lensing_init(&pr,&pt,&sp,&nl,&le);
output_init(&ba,&th,&pt,&pm,&tr,&sp,&nl,&le,&op)

/****** done ******/

lensing_free(&le);
spectra_free(&sp);
transfer_free(&tr);
nonlinear_free(&nl);
primordial_free(&pm);
perturb_free(&pt);
thermodynamics_free(&th);
background_free(&ba);

We can come back on the role of each argument. The arguments above are all pointers to the 10 structures of the
code, excepted argc, argv which contains the input files passed by the user, and errmsg which contains the
output error message of the input module (error management will be described below).

input_init_from_arguments needs all structures, because it will set the precision parameters inside the
precision structure, and the physical parameters in some fields of the respective other structures. For instance, an input parameter relevant for the primordial spectrum calculation (like the tilt ns ) will be stored in the
primordial structure. Hence, in input_init_from_arguments, all structures can be seen as output
arguments.
Other module_init() functions typically need all previous structures, which contain the result of the previous
modules, plus its own structures, which contain some relevant input parameters before the function is called, as
well as all the result form the module when the function has been executed. Hence all passed structures can
be seen as input argument, excepted the last one which is both input and output. An example is perturb_←init(&pr,&ba,&th,&pt).
Each function module_init() does not need all previous structures, it happens that a module does not depend
on a all previous one. For instance, the primordial module does not need information on the background and thermodynamics evolution in order to compute the primordial spectra, so the dependency is reduced: primordial←_init(&pr,&pt,&pm).
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10

CLASS overview (architecture, input/output, general principles)

Each function module_init() only deallocates arrays defined in the structure of their own module, so they
need only their own structure as argument. (This is possible because all structures are self-contained, in the
sense that when the structure contains an allocated array, it also contains the size of this array). The first and last
module, input and output, have no input_free() or output_free() functions, because the structures
precision and output do not contain arrays that would need to be de-allocated after the execution of the
module.

The test_<...>.c files
For a given purpose, somebody could only be interested in the intermediate steps (only background quantities, only
the thermodynamics, only the perturbations and sources, etc.) It is then straightforward to truncate the full hierarchy
of modules 1, ... 10 at some arbitrary order. We provide several "reduced executables" achieving precisely this.
They are located in test/test_module_.c (like, for instance, test/test_perturbations.c) and they
can be complied using the Makefile, which contains the appropriate commands and definitions (for instance, you
can type make test_perturbations).
The test/ directory contains other useful example of alternative main functions, like for instance test_←loops.c which shows how to call CLASS within a loop over different parameter values. There is also a version
test/test_loops_omp.c using a double level of openMP parallelisation: one for running several CLASS
instances in parallel, one for running each CLASS instance on several cores. The comments in these files are
self-explanatory.

Input/output
Input
There are two types of input:
• "precision parameters" (controlling the precision of the output and the execution time),
• "input parameters" (cosmological parameters, flags telling to the code what it should compute, ...)
The code can be executed with a maximum of two input files, e.g.
./class explanatory.ini cl_permille.pre

The file with a .ini extension is the cosmological parameter input file, and the one with a .pre extension is the
precision file. Both files are optional: all parameters are set to default values corresponding to the "most usual
choices", and are eventually replaced by the parameters passed in the two input files. For instance, if one is happy
with default accuracy settings, it is enough to run with
./class explanatory.ini

Input files do not necessarily contain a line for each parameter, since many of them can be left to default value. The
example file explanatory.ini is very long and somewhat indigestible, since it contains all possible parameters,
together with lengthy explanations. We recommend to keep this file unchanged for reference, and to copy it in e.g.
test.ini. In the latter file, the user can erase all sections in which he/she is absolutely not interested (e.g., all
the part on isocurvature modes, or on tensors, or on non-cold species, etc.). Another option is to create an input file
from scratch, copying just the relevant lines from explanatory.ini. For the simplest applications, the user will
just need a few lines for basic cosmological parameters, one line for the output entry (where one can specifying
which power spectra must be computed), and one line for the root entry (specifying the prefix of all output files).
The syntax of the input files is explained at the beginning of explanatory.ini. Typically, lines in those files
look like:
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parameter1 = value1
free comments
parameter2 = value2 # further comments
# commented_parameter = commented_value

and parameters can be entered in arbitrary order. This is rather intuitive. The user should just be careful not to put
an "=" sign not preceded by a "#" sign inside a comment: the code would then think that one is trying to pass some
unidentified input parameter.
The syntax for the cosmological and precision parameters is the same. It is clearer to split these parameters in
the two files .ini and .pre, but there is no strict rule about which parameter goes into which file: in principle,
precision parameters could be passed in the .ini, and vice-versa. The only important thing is not to pass the
same parameter twice: the code would then complain and not run.
The CLASS input files are also user-friendly in the sense that many different cosmological parameter bases can
be used. This is made possible by the fact that the code does not only read parameters, it "interprets them" with
the level of logic which has been coded in the input.c module. For instance, the Hubble parameter, the photon
density, the baryon density and the ultra-relativistic neutrino density can be entered as:
h = 0.7
T_cmb = 2.726

# Kelvin units

omega_b = 0.02
N_eff = 3.04

(in arbitrary order), or as

H0 = 70
omega_g = 2.5e-5

# g is the label for photons

Omega_b = 0.04
omega_ur = 1.7e-5

# ur is the label for ultra-relativistic species

or any combination of the two. The code knows that for the photon density, one should pass one (but not more
than one) parameter out of T_cmb, omega_g, Omega_g (where small omega's refer to ωi ≡ Ωi h2 ). It searches
for one of these values, and if needed, it converts it into one of the other two parameters, using also other input
parameters. For instance, omega_g will be converted into Omega_g even if h is written later in the file than
omega_g: the order makes no difference. Lots of alternatives have been defined. If the code finds that not enough
parameters have been passed for making consistent deductions, it will complete the missing information with in-built
default values. On the contrary, if it finds that there is too much information and no unique solution, it will complain
and return an error.
In summary, the input syntax has been defined in such way that the user does not need to think too much, and can
pass his preferred set of parameters in a nearly informal way.
Let us mention a two useful parameters defined at the end of explanatory.ini, that we recommend setting to

yes in order to run the code in a safe way:
write parameters = [yes or no] (default: no)
When set to yes, all input/precision parameters which have been read are written in a file parameters.←ini, to keep track all the details of this execution; this file can also be re-used as a new input file. Also, with this
option, all parameters that have been passed and that the code did not read (because the syntax was wrong,
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12

CLASS overview (architecture, input/output, general principles)

or because the parameter was not relevant in the context of the run) are written in a file unused_←parameters. When you have doubts about your input or your results, you can check what is in there.

write warnings = [yes or no] (default: no)
When set to yes, the parameters that have been passed and that the code did not read (because the syntax was
wrong, or because the parameter was not relevant in the context of the run) are written in the standard output as

[Warning:]....
There is also a list of "verbose" parameters at the end of explanatory.ini. They can be used to control the
level of information passed to the standard output (0 means silent; 1 means normal, e.g. information on age of the
universe, etc.; 2 is useful for instance when you want to check on how many cores the run is parallelised; 3 and
more are intended for debugging).

CLASS comes with a list of precision parameter files ending by .pre. Honestly we have not been updating all these
files recently, and we need to do a bit of cleaning there. However you can trust cl_ref.pre. We have derived
this file by studying both the convergence of the CMB output with respect to all CLASS precision parameters, and
the agreement with CAMB. We consider that this file generates good reference CMB spectra, accurate up to the
hundredth of per cent level, as explained in the CLASS IV paper and re-checked since then. You can try it with e.g.

./class explanatory.ini cl_ref.pre

but the run will be extremely long. This is an occasion to run a many-core machine with a lot of RAM. It may work
also on your laptop, but in half an hour or so.
If you want a reference matter power spectrum P(k), also accurate up to the hundredth of percent level, we recommend using the file pk_ref.pre, identical to cl_ref.pre excepted that the truncation of the neutrino hierarchy
has been pushed to l_max_ur=150.
In order to increase moderately the precision to a tenth of percent, without prohibitive computing time, we recommend using cl_permille.pre.

Output

The input file may contain a line

root = 

where  is a path of your choice, e.g. output/test_. Then all output files will start like this, e.←g. output/test_cl.dat, output/test_cl_lensed.dat, etc. Of course the number of output files
depends on your settings in the input file. There can be input files for CMB, LSS, background, thermodynamics,
transfer functions, primordial spectra, etc. All this is documented in explanatory.ini.
If you do not pass explicitly a root = , the code will name the output in its own way, by concatenating
output/, the name of the input parameter file, and the first available integer number, e.g.

output/explanatory03_cl.dat, etc.
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13

General principles
Error management
Error management is based on the fact that all functions are defined as integers returning either _SUCCESS_ or
_FAILURE_. Before returning _FAILURE_, they write an error message in the structure of the module to which
they belong. The calling function will read this message, append it to its own error message, and return a _FAIL←URE_; and so on and so forth, until the main routine is reached. This error management allows the user to see the
whole nested structure of error messages when an error has been met. The structure associated to each module
contains a field for writing error messages, called structure_i.error_message, where structure_i
could be one of background, thermo, perturbs, etc. So, when a function from a module i is called within
module j and returns an error, the goal is to write in structure_j.error_message a local error message,
and to append to it the error message in structure_i.error_message. These steps are implemented in a
macro class_call(), used for calling whatever function:

class_call(module_i_function(...,structure_i),
structure_i.error_message,
structure_j.error_message);

So, the first argument of call_call() is the function we want to call; the second argument is the location of
the error message returned by this function; and the third one is the location of the error message which should
be returned to the higher level. Usually, in the bulk of the code, we use pointer to structures rather than structure
themselves; then the syntax is
class_call(module_i_function(...,pi),
pi->error_message,
pj->error_message);‘

where in this generic example, pi and pj are assumed to be pointers towards the structures structure_i and
structure_j.
The user will find in include/common.h a list of additional macros, all starting by class_...(), which are all
based on this logic. For instance, the macro class_test() offers a generic way to return an error in a standard
format if a condition is not fulfilled. A typical error message from CLASS looks like:

Error in module_j_function1
module_j_function1 (L:340) :

error in module_i_function2(...)

module_i_function2 (L:275) :

error in module_k_function3(...)

...
=> module_x_functionN (L:735) : your choice of input parameter blabla=30
is not consistent with the constraint blabla<1
where the L's refer to line numbers in each file. These error messages are very informative, and are built almost
entirely automatically by the macros. For instance, in the above example, it was only necessary to write inside the
function module_x_functionN() a test like:

class_test(blabla >= 1,
px->error_message,
"your choice of input parameter blabla=%e
is not consistent with the constraint blabla<%e",
blabla,blablamax);

All the rest was added step by step by the various class_call() macros.
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CLASS overview (architecture, input/output, general principles)

Dynamical allocation of indices
On might be tempted to decide that in a given array, matrix or vector, a given quantity is associated with an explicit
index value. However, when modifying the code, extra entries will be needed and will mess up the initial scheme;
the user will need to study which index is associated to which quantity, and possibly make an error. All this can be
avoided by using systematically a dynamical index allocation. This means that all indices remain under a symbolic
form, and in each, run the code attributes automatically a value to each index. The user never needs to know this
value.
Dynamical indexing is implemented in a very generic way in CLASS, the same rules apply everywhere. They are
explained in these lecture slides:

https://www.dropbox.com/sh/ma5muh76sggwk8k/AABl_DDUBEzAjjdywMjeTya2a?dl=0
in the folder CLASS_Lecture_slides/lecture5_index_and_error.pdf.

No hard coding
Any feature or equation which could be true in one cosmology and not in another one should not be written explicitly
in the code, and should not be taken as granted in several other places. Discretization and integration steps are
usually defined automatically by the code for each cosmology, instead of being set to something which might be
optimal for minimal models, and not sufficient for other ones. You will find many example of this in the code.
As a consequence, in the list of precision parameter, you rarely find actual stepsize. You find rather parameters
representing the ratio between a stepsize and a physical quantity computed for each cosmology.

Modifying the code
Implementing a new idea completly from scratch would be rather intimidating, even for the main developpers of C←LASS. Fortunately, we never have to work from scratch. Usually we want to code a new species, a new observable,
a new approximation scheme, etc. The trick is to think of another species, observable, approximation scheme, etc.,
looking as close as possible to the new one.
Then, playing with the grep command and the search command of your editor, search for all occurences of this
nearest-as-possible other feature. This is usually easy thanks to our naming scheme. For each species, observable,
approximation scheme, etc., we usually use the same sequence of few letters everywhere (fo instance, fld for the
fluid usually representing Dark Energy). Grep for fld and you'll get all the lines related to the fluid. There is another
way: we use everywhere some conditional jumps related to a given feature. For instance, the lines related to the fluid
are always in between if (pba->has_fld == _TRUE_) { ... } and the lines related to the cosmic
shear observables are always in between if (ppt->has_lensing_potential == _TRUE_) { ...
}. Locating these flags and conditional jumps shows you all the parts related to a given feature/ingredient.
Once you have localised your nearest-as-possible other feature, you can copy/paste these lines and adapt them to
the case of your new feature! You are then sure that you didn't miss any step, even the smallest technical steps
(definition of indices, etc.)

Units
Internally, the code uses almost everywhere units of Mpc to some power, excepted in the inflation module, where
many quantities are in natural units (wrt the true Planck mass).

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Chapter 4

Data Structure Documentation
4.1

nonlinear Struct Reference

#include 

Data Fields
- input parameters initialized by user in input module
(all other quantitites are computed in this module, given these parameters and the content of the 'precision',
'background', 'thermo', 'primordial' and 'spectra' structures)
• enum non_linear_method method

- table non-linear corrections for matter density, sqrt(P_NL(k,z)/P_NL(k,z))
•
•
•
•
•
•

int k_size
double ∗ k
int tau_size
double ∗ tau
double ∗ nl_corr_density
double ∗ k_nl

- technical parameters
• short nonlinear_verbose
• ErrorMsg error_message

4.1.1

Detailed Description

Structure containing all information on non-linear spectra.
Once initialized by nonlinear_init(), contains a table for all two points correlation functions and for all the ai,bj functions (containing the three points correlation functions), for each time and wave-number.
Structure containing all information on non-linear spectra.
Once initialised by nonlinear_init(), contains a table for all two points correlation functions and for all the ai,bj functions (containing the three points correlation functions), for each time and wave-number.

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Data Structure Documentation

4.1.2

Field Documentation

4.1.2.1

method

enum non_linear_method nonlinear::method

method for computing non-linear corrections (none, Halogit, etc.)

4.1.2.2

k_size

int nonlinear::k_size

k_size = total number of k values

4.1.2.3

k

double ∗ nonlinear::k

k[index_k] = list of k values

4.1.2.4

tau_size

int nonlinear::tau_size

tau_size = number of values

4.1.2.5

tau

double ∗ nonlinear::tau

tau[index_tau] = list of time values

4.1.2.6

nl_corr_density

double ∗ nonlinear::nl_corr_density

nl_corr_density[index_tau ∗ ppt->k_size + index_k]

4.1.2.7

k_nl

double ∗ nonlinear::k_nl

wavenumber at which non-linear corrections become important, defined differently by different non_linear_method's

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4.1 nonlinear Struct Reference

4.1.2.8

nonlinear_verbose

short nonlinear::nonlinear_verbose

amount of information written in standard output

4.1.2.9

error_message

ErrorMsg nonlinear::error_message

zone for writing error messages
The documentation for this struct was generated from the following files:

• nonlinear.h
• nonlinear_exp.h
• nonlinear_test.h

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18

Data Structure Documentation

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Chapter 5

File Documentation

5.1

background.c File Reference

#include "background.h"
Include dependency graph for background.c:

background.c

background.h

quadrature.h

growTable.h

arrays.h

dei_rkck.h

parser.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int background_at_tau (struct background ∗pba, double tau, short return_format, short intermode, int ∗last←_index, double ∗pvecback)
• int background_tau_of_z (struct background ∗pba, double z, double ∗tau)
• int background_functions (struct background ∗pba, double ∗pvecback_B, short return_format, double
∗pvecback)
• int background_w_fld (struct background ∗pba, double a, double ∗w_fld, double ∗dw_over_da_fld, double
∗integral_fld)
• int background_init (struct precision ∗ppr, struct background ∗pba)
• int background_free (struct background ∗pba)
• int background_free_input (struct background ∗pba)
• int background_indices (struct background ∗pba)

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•
•
•
•
•
•
•
•
•
•
•
•
•

5.1.1

int background_ncdm_distribution (void ∗pbadist, double q, double ∗f0)
int background_ncdm_test_function (void ∗pbadist, double q, double ∗test)
int background_ncdm_init (struct precision ∗ppr, struct background ∗pba)
int background_ncdm_momenta (double ∗qvec, double ∗wvec, int qsize, double M, double factor, double z,
double ∗n, double ∗rho, double ∗p, double ∗drho_dM, double ∗pseudo_p)
int background_ncdm_M_from_Omega (struct precision ∗ppr, struct background ∗pba, int n_ncdm)
int background_solve (struct precision ∗ppr, struct background ∗pba)
int background_initial_conditions (struct precision ∗ppr, struct background ∗pba, double ∗pvecback, double
∗pvecback_integration)
int background_output_titles (struct background ∗pba, char titles[_MAXTITLESTRINGLENGTH_])
int background_output_data (struct background ∗pba, int number_of_titles, double ∗data)
int background_derivs (double tau, double ∗y, double ∗dy, void ∗parameters_and_workspace, ErrorMsg
error_message)
double V_e_scf (struct background ∗pba, double phi)
double V_p_scf (struct background ∗pba, double phi)
double V_scf (struct background ∗pba, double phi)

Detailed Description

Documented background module

• Julien Lesgourgues, 17.04.2011
• routines related to ncdm written by T. Tram in 2011

Deals with the cosmological background evolution. This module has two purposes:

• at the beginning, to initialize the background, i.e. to integrate the background equations, and store all background quantities as a function of conformal time inside an interpolation table.
• to provide routines which allow other modules to evaluate any background quantity for a given value of the
conformal time (by interpolating within the interpolation table), or to find the correspondence between redshift
and conformal time.

The overall logic in this module is the following:

1. most background parameters that we will call {A} (e.g. rho_gamma, ..) can be expressed as simple analytical
functions of a few variables that we will call {B} (in simplest models, of the scale factor 'a'; in extended
cosmologies, of 'a' plus e.g. (phi, phidot) for quintessence, or some temperature for exotic particles, etc...).
2. in turn, quantities {B} can be found as a function of conformal time by integrating the background equations.
3. some other quantities that we will call {C} (like e.g. the sound horizon or proper time) also require an integration with respect to time, that cannot be inferred analytically from parameters {B}.

So, we define the following routines:

• background_functions() returns all background quantities {A} as a function of quantities {B}.
• background_solve() integrates the quantities {B} and {C} with respect to conformal time; this integration requires many calls to background_functions().
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5.1 background.c File Reference

21

• the result is stored in the form of a big table in the background structure. There is one column for conformal
time 'tau'; one or more for quantities {B}; then several columns for quantities {A} and {C}.

Later in the code, if we know the variables {B} and need some quantity {A}, the quickest and most precise way is to
call directly background_functions() (for instance, in simple models, if we want H at a given value of the scale factor).
If we know 'tau' and want any other quantity, we can call background_at_tau(), which interpolates in the table and
returns all values. Finally it can be useful to get 'tau' for a given redshift 'z': this can be done with background_←tau_of_z(). So if we are somewhere in the code, knowing z and willing to get background quantities, we should call
first background_tau_of_z() and then background_at_tau().
In order to save time, background_at_tau() can be called in three modes: short_info, normal_info, long_info (returning only essential quantities, or useful quantities, or rarely useful quantities). Each line in the interpolation table is a
vector whose first few elements correspond to the short_info format; a larger fraction contribute to the normal format;
and the full vector corresponds to the long format. The guideline is that short_info returns only geometric quantities like a, H, H'; normal format returns quantities strictly needed at each step in the integration of perturbations;
long_info returns quantities needed only occasionally.
In summary, the following functions can be called from other modules:

1. background_init() at the beginning
2. background_at_tau(), background_tau_of_z() at any later time
3. background_free() at the end, when no more calls to the previous functions are needed

5.1.2

Function Documentation

5.1.2.1

background_at_tau()

int background_at_tau (
struct background ∗ pba,
double tau,
short return_format,
short intermode,
int ∗ last_index,
double ∗ pvecback )

Background quantities at given conformal time tau.
Evaluates all background quantities at a given value of conformal time by reading the pre-computed table and
interpolating.
Parameters
pba

Input: pointer to background structure (containing pre-computed table)

tau

Input: value of conformal time

return_format

Input: format of output vector (short, normal, long)

intermode

Input: interpolation mode (normal or closeby)

last_index

Input/Output: index of the previous/current point in the interpolation array (input only for
closeby mode, output for both)

pvecback

Output: vector (assumed to be already allocated)

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Returns
the error status

Summary:

• define local variables
• check that tau is in the pre-computed range
• deduce length of returned vector from format mode
• interpolate from pre-computed table with array_interpolate() or array_interpolate_growing_closeby() (depending on interpolation mode)

5.1.2.2

background_tau_of_z()

int background_tau_of_z (
struct background ∗ pba,
double z,
double ∗ tau )

Conformal time at given redshift.
Returns tau(z) by interpolation from pre-computed table.
Parameters
pba

Input: pointer to background structure

z

Input: redshift

tau

Output: conformal time

Returns
the error status

Summary:

• define local variables
• check that z is in the pre-computed range
• interpolate from pre-computed table with array_interpolate()

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5.1 background.c File Reference

5.1.2.3

23

background_functions()

int background_functions (
struct background ∗ pba,
double ∗ pvecback_B,
short return_format,
double ∗ pvecback )

Background quantities at given a.
Function evaluating all background quantities which can be computed analytically as a function of {B} parameters
such as the scale factor 'a' (see discussion at the beginning of this file). In extended cosmological models, the
pvecback_B vector contains other input parameters than just 'a', e.g. (phi, phidot) for quintessence, some temperature of exotic relics, etc...
Parameters
pba

Input: pointer to background structure

pvecback_B

Input: vector containing all {B} type quantities (scale factor, ...)

return_format

Input: format of output vector

pvecback

Output: vector of background quantities (assumed to be already allocated)

Returns
the error status

Summary:

• define local variables
• initialize local variables
• pass value of a to output
• compute each component's density and pressure
• compute expansion rate H from Friedmann equation: this is the only place where the Friedmann equation is
assumed. Remember that densities are all expressed in units of [3c2 /8πG], ie ρclass = [8πGρphysical /3c2 ]
• compute derivative of H with respect to conformal time
• compute relativistic density to total density ratio
• compute other quantities in the exhaustive, redundant format
• compute critical density
• compute Omega_m

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5.1.2.4

File Documentation

background_w_fld()

int background_w_fld (
struct
double
double
double
double

background ∗ pba,
a,
∗ w_fld,
∗ dw_over_da_fld,
∗ integral_fld )

Single place where the fluid equation of state is defined. Parameters of the function are passed through the background structure. Generalisation to arbitrary functions should be simple.

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25

Parameters
pba

Input: pointer to background structure

a

Input: current value of scale factor

w_fld

Output: equation of state parameter w_fld(a)

dw_over_da_fld

Output: function dw_fld/da

integral_fld

Output: function

R a0
a

da3(1 + wf ld )/a

Returns
the error status

• first, define the function w(a)
• then, give the corresponding analytic derivative dw/da (used by perturbation equations; we could compute it
numerically, but with a loss of precision; as long as there is a simple analytic expression of the derivative of
the previous function, let's use it!

R a0

• finally, give the analytic solution of the following integral: a da3(1 + wf ld )/a. This is used in only one
place, in the initial conditions for the background, and with a=a_ini. If your w(a) does not lead to a simple
analytic solution of this integral, no worry: instead of writing something here, the best would then be to leave
it equal to zero, and then in background_initial_conditions() you should implement a numerical calculation of
this integral only for a=a_ini, using for instance Romberg integration. It should be fast, simple, and accurate
enough.

note: of course you can generalise these formulas to anything, defining new parameters pba->w..._fld. Just remember that so far, HyRec explicitely assumes that w(a)= w0 + wa (1-a/a0); but Recfast does not assume anything

5.1.2.5

background_init()

int background_init (
struct precision ∗ ppr,
struct background ∗ pba )

Initialize the background structure, and in particular the background interpolation table.
Parameters
ppr

Input: pointer to precision structure

pba

Input/Output: pointer to initialized background structure

Returns
the error status

Summary:

• define local variables
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File Documentation

• in verbose mode, provide some information
• if shooting failed during input, catch the error here
• assign values to all indices in vectors of background quantities with background_indices()
• control that cosmological parameter values make sense
• this function integrates the background over time, allocates and fills the background table

5.1.2.6

background_free()

int background_free (
struct background ∗ pba )

Free all memory space allocated by background_init().
Parameters
pba

Input: pointer to background structure (to be freed)

Returns
the error status

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5.1.2.7

background_free_input

background_free_input()

int background_free_input (
struct background ∗ pba )

Free pointers inside background structure which were allocated in input_read_parameters()
Parameters
pba

Input: pointer to background structure

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Returns
the error status

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5.1.2.8

background_free

background_indices()

int background_indices (
struct background ∗ pba )

Assign value to each relevant index in vectors of background quantities.
Parameters
pba

Input: pointer to background structure

Returns
the error status
Summary:
• define local variables
• initialize all flags: which species are present?
• initialize all indices

5.1.2.9

background_ncdm_distribution()

int background_ncdm_distribution (
void ∗ pbadist,
double q,
double ∗ f0 )

This is the routine where the distribution function f0(q) of each ncdm species is specified (it is the only place to
modify if you need a partlar f0(q))

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File Documentation

Parameters
pbadist

Input: structure containing all parameters defining f0(q)

q

Input: momentum

f0

Output: phase-space distribution

• extract from the input structure pbadist all the relevant information
• shall we interpolate in file, or shall we use analytical formula below?
• a) deal first with the case of interpolating in files
• b) deal now with case of reading analytical function

Next enter your analytic expression(s) for the p.s.d.'s. If you need different p.s.d.'s for different species, put each
p.s.d inside a condition, like for instance: if (n_ncdm==2) {∗f0=...}. Remember that n_ncdm = 0 refers to the first
species.
This form is only appropriate for approximate studies, since in reality the chemical potentials are associated with
flavor eigenstates, not mass eigenstates. It is easy to take this into account by introducing the mixing angles. In the
later part (not read by the code) we illustrate how to do this.

5.1.2.10

background_ncdm_test_function()

int background_ncdm_test_function (
void ∗ pbadist,
double q,
double ∗ test )

This function is only used for the purpose of finding optimal quadrature weights. The logic is: if we can accurately
convolve f0(q) with this function, then we can convolve it accurately with any other relevant function.
Parameters
pbadist

Input: structure containing all background parameters

q

Input: momentum

test

Output: value of the test function test(q)

Using a + bq creates problems for otherwise acceptable distributions which diverges as 1/r or 1/r 2 for r → 0

5.1.2.11

background_ncdm_init()

int background_ncdm_init (
struct precision ∗ ppr,
struct background ∗ pba )

This function finds optimal quadrature weights for each ncdm species
Parameters
ppr

Input: precision structure

pba

Input/Output: background structure

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• in verbose mode, inform user of number of sampled momenta for background quantities

5.1.2.12

background_ncdm_momenta()

int background_ncdm_momenta (
double ∗ qvec,
double ∗ wvec,
int qsize,
double M,
double factor,
double z,
double
double
double
double
double

∗
∗
∗
∗
∗

n,
rho,
p,
drho_dM,
pseudo_p )

For a given ncdm species: given the quadrature weights, the mass and the redshift, find background quantities by
a quick weighted sum over. Input parameters passed as NULL pointers are not evaluated for speed-up
Parameters
qvec

Input: sampled momenta

wvec

Input: quadrature weights

qsize

Input: number of momenta/weights

M

Input: mass

factor

Input: normalization factor for the p.s.d.

z

Input: redshift

n

Output: number density

rho

Output: energy density

p

Output: pressure

drho_dM

Output: derivative used in next function

pseudo←_p

Output: pseudo-pressure used in perturbation module for fluid approx

Summary:

• rescale normalization at given redshift
• initialize quantities
• loop over momenta
• adjust normalization
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File Documentation

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5.1.2.13

background_ncdm_M_from
_Omega

background_ncdm_M_from_Omega()

int background_ncdm_M_from_Omega (
struct precision ∗ ppr,
struct background ∗ pba,
int n_ncdm )

When the user passed the density fraction Omega_ncdm or omega_ncdm in input but not the mass, infer the mass
with Newton iteration method.
Parameters
ppr

Input: precision structure

pba

Input/Output: background structure

n_ncdm

Input: index of ncdm species

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5.1.2.14

background_ncdm_momenta

background_solve()

int background_solve (
struct precision ∗ ppr,
struct background ∗ pba )

This function integrates the background over time, allocates and fills the background table

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Parameters
ppr

Input: precision structure

pba

Input/Output: background structure

Summary:

• define local variables
• allocate vector of quantities to be integrated
• initialize generic integrator with initialize_generic_integrator()
• impose initial conditions with background_initial_conditions()
• create a growTable with gt_init()
• loop over integration steps: call background_functions(), find step size, save data in growTable with gt_add(),
perform one step with generic_integrator(), store new value of tau
• save last data in growTable with gt_add()
• clean up generic integrator with cleanup_generic_integrator()
• retrieve data stored in the growTable with gt_getPtr()
• interpolate to get quantities precisely today with array_interpolate()
• deduce age of the Universe
• allocate background tables
• In a loop over lines, fill background table using the result of the integration plus background_functions()
• free the growTable with gt_free()
• fill tables of second derivatives (in view of spline interpolation)
• compute remaining "related parameters"
– so-called "effective neutrino number", computed at earliest time in interpolation table. This should be
seen as a definition: Neff is the equivalent number of instantaneously-decoupled neutrinos accounting
for the radiation density, beyond photons
• done

5.1.2.15

background_initial_conditions()

int background_initial_conditions (
struct
struct
double
double

precision ∗ ppr,
background ∗ pba,
∗ pvecback,
∗ pvecback_integration )

Assign initial values to background integrated variables.

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File Documentation

Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pvecback

Input: vector of background quantities used as workspace

pvecback_integration

Output: vector of background quantities to be integrated, returned with proper initial values

Returns
the error status

Summary:

• define local variables
• fix initial value of a
If we have ncdm species, perhaps we need to start earlier than the standard value for the species to be
relativistic. This could happen for some WDM models.
• We must add the relativistic contribution from NCDM species
– f is the critical density fraction of DR. The exact solution is:

f = -Omega_rad+pow(pow(Omega_rad,3./2.)+0.5∗pow(a/pba->a_today,6)∗pvecback←_integration[pba->index_bi_rho_dcdm]∗pba->Gamma_dcdm/pow(pba->H0,3),2./3.);
but it is not numerically stable for very small f which is always the case. Instead we use the Taylor expansion of this
equation, which is equivalent to ignoring f(a) in the Hubble rate.
There is also a space reserved for a future case where dr is not sourced by dcdm
• Fix initial value of φ, φ0 set directly in the radiation attractor => fixes the units in terms of rho_ur

TODO:
• There seems to be some small oscillation when it starts.
• Check equations and signs. Sign of phi_prime?
• is rho_ur all there is early on?
• –> If there is no attractor solution for scf_lambda, assign some value. Otherwise would give a nan.
• –> If no attractor initial conditions are assigned, gets the provided ones.
• compute initial proper time, assuming radiation-dominated universe since Big Bang and therefore t = 1/(2H)
(good approximation for most purposes)
• compute initial conformal time, assuming radiation-dominated universe since Big Bang and therefore τ =
1/(aH) (good approximation for most purposes)

√

• compute initial sound horizon, assuming cs = 1/ 3 initially
• set initial value of D and D' in RD. D will be renormalised later, but D' must be correct.

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5.1.2.16

33

background_output_titles()

int background_output_titles (
struct background ∗ pba,
char titles[_MAXTITLESTRINGLENGTH_] )

Subroutine for formatting background output

• Length of the column title should be less than OUTPUTPRECISION+6 to be indented correctly, but it can be
as long as .

5.1.2.17

background_output_data()

int background_output_data (
struct background ∗ pba,
int number_of_titles,
double ∗ data )

Stores quantities

5.1.2.18

background_derivs()

int background_derivs (
double tau,
double ∗ y,
double ∗ dy,
void ∗ parameters_and_workspace,
ErrorMsg error_message )

Subroutine evaluating the derivative with respect to conformal time of quantities which are integrated (a, t, etc).
This is one of the few functions in the code which is passed to the generic_integrator() routine. Since generic_←integrator() should work with functions passed from various modules, the format of the arguments is a bit special:

• fixed input parameters and workspaces are passed through a generic pointer. Here, this is just a pointer to
the background structure and to a background vector, but generic_integrator() doesn't know its fine structure.
• the error management is a bit special: errors are not written as usual to pba->error_message, but to a generic
error_message passed in the list of arguments.

Parameters
tau

Input: conformal time

y

Input: vector of variable

dy

Output: its derivative (already allocated)

parameters_and_workspace

Input: pointer to fixed parameters (e.g. indices)

error_message

Output: error message

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Summary:

• define local variables
• calculate functions of a with background_functions()
• Short hand notation
• calculate a0 = a2 H
• calculate t0 = a
• calculate rs0 = cs
• solve second order growth equation [D 00 (τ ) = −aHD 0 (τ ) + 3/2a2 ρM D(τ )
• compute dcdm density ρ0 = −3aHρ − aΓρ
• Compute dr density ρ0 = −4aHρ − aΓρ
• Compute fld density ρ0 = −3aH(1 + wf ld (a))ρ
• Scalar field equation: φ00 + 2aHφ0 + a2 dV = 0 (note H is wrt cosmic time)

5.1.2.19

V_e_scf()

double V_e_scf (
struct background ∗ pba,
double phi )

Scalar field potential and its derivatives with respect to the field _scf For Albrecht & Skordis model: 9908085

• V = Vpscf ∗ Vescf
• Ve = exp(−λφ) (exponential)
• Vp = (φ − B)α + A (polynomial bump)

TODO:

• Add some functionality to include different models/potentials (tuning would be difficult, though)
• Generalize to Kessence/Horndeski/PPF and/or couplings
• A default module to numerically compute the derivatives when no analytic functions are given should be
added.
• Numerical derivatives may further serve as a consistency check.

The units of phi, tau in the derivatives and the potential V are the following:

• phi is given in units of the reduced Planck mass mpl = (8πG)(−1/2)
• tau in the derivative is given in units of Mpc.
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• the potential V (φ) is given in units of m2pl /M pc2 .

(8πG)/3ρ

physical

=

1/(3m2pl )ρphysical

With this convention, we have ρclass

=
= 1/3 ∗ [1/(2a )(φ0 )2 + V (φ)] and ρclass has the proper
2

dimension M pc− 2.

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5.1.2.20

V_scf

V_p_scf()

double V_p_scf (
struct background ∗ pba,
double phi )

parameters and functions for the polynomial coefficient Vp = (φ − B)α + A(polynomial bump)

double scf_alpha = 2;

double scf_B = 34.8;

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5.1.2.21

File Documentation

V_scf()

double V_scf (
struct background ∗ pba,
double phi )

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5.2

background.h File Reference

#include
#include
#include
#include
#include
#include

"common.h"
"quadrature.h"
"growTable.h"
"arrays.h"
"dei_rkck.h"
"parser.h"

Include dependency graph for background.h:

background.h

quadrature.h

growTable.h

arrays.h

dei_rkck.h

parser.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

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37

This graph shows which files directly or indirectly include this file:

background.h

thermodynamics.h

thermodynamics.c

background.c

perturbations.h

primordial.h

nonlinear_exp.h

nonlinear_test.h

perturbations.c

primordial.c

nonlinear.h

transfer.h

transfer.c

spectra.h

lensing.h

lensing.c

input.h

nonlinear.c

spectra.c

output.h

output.c

input.c

class.h

class.c

Data Structures
• struct background
• struct background_parameters_and_workspace
• struct background_parameters_for_distributions

5.2.1

Detailed Description

Documented includes for background module

5.2.2

Data Structure Documentation

5.2.2.1

struct background

All background parameters and evolution that other modules need to know.
Once initialized by the backgound_init(), contains all necessary information on the background evolution (except
thermodynamics), and in particular, a table of all background quantities as a function of time and scale factor, used
for interpolation in other modules.
Data Fields
double

H0

H0 : Hubble parameter (in fact, [ H0 /c]) in M pc−1

double

Omega0_g

Ω0γ : photons

double

T_cmb

double

Omega0_b

Tcmb : current CMB temperature in Kelvins
Ω0b : baryons

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Data Fields

Ω0cdm : cold dark matter
Ω0Λ : cosmological constant

double

Omega0_cdm

double

Omega0_lambda

double

Omega0_fld

double

w0_fld

double

wa_fld

Ω0de : fluid
w0DE : current fluid equation of state parameter
waDE : fluid equation of state parameter derivative

double

cs2_fld

c2s DE : sound speed of the fluid in the frame comoving with the
fluid (so, this is not [delta p/delta rho] in the synchronous or
newtonian gauge!!!)

short

use_ppf

flag switching on PPF perturbation equations instead of true fluid
equations for perturbations. It could have been defined inside
perturbation structure, but we leave it here in such way to have all
fld parameters grouped.

double

c_gamma_over_c_fld

ppf parameter defined in eq. (16) of 0808.3125 [astro-ph]

double

Omega0_ur

double

Omega0_dcdmdr

Ω0νr : ultra-relativistic neutrinos
Ω0dcdm + Ω0dr : decaying cold dark matter (dcdm) decaying to

double

Gamma_dcdm

double

Omega_ini_dcdm

dark radiation (dr)

Γdcdm : decay constant for decaying cold dark matter
Ωini,dcdm : rescaled initial value for dcdm density (see 1407.2418
for definitions)

double
short

Omega0_scf

Ω0scf : scalar field

attractor_ic_scf

whether the scalar field has attractor initial conditions

double

phi_ini_scf

φ(t0 ): scalar field initial value

double

phi_prime_ini_scf

dφ(t0 )/dτ : scalar field initial derivative wrt conformal time

scf_parameters

list of parameters describing the scalar field potential

int

scf_parameters_size

size of scf_parameters

int

scf_tuning_index

index in scf_parameters used for tuning

Omega0_k

Ω0k : curvature contribution

int

N_ncdm

Number of distinguishable ncdm species

double ∗

M_ncdm

vector of masses of non-cold relic: dimensionless ratios
m_ncdm/T_ncdm

double ∗

Omega0_ncdm

double ∗

double

double
double ∗
double

double ∗
double
double ∗
double
double ∗
int ∗

Omega0_ncdm_tot

Omega0_ncdm for each species and for the total Omega0_ncdm

deg_ncdm
deg_ncdm_default

vector of degeneracy parameters in factor of p-s-d: 1 for one
family of neutrinos (= one neutrino plus its anti-neutrino, total
g∗=1+1=2, so deg = 0.5 g∗); and its default value

T_ncdm
T_ncdm_default

list of 1st parameters in p-s-d of non-cold relics: relative
temperature T_ncdm1/T_gamma; and its default value

ksi_ncdm
ksi_ncdm_default

list of 2nd parameters in p-s-d of non-cold relics: relative chemical
potential ksi_ncdm1/T_ncdm1; and its default value

ncdm_psd_parameters

list of parameters for specifying/modifying ncdm p.s.d.'s, to be
customized for given model (could be e.g. mixing angles)

got_files

char ∗

ncdm_psd_files

list of flags for each species, set to true if p-s-d is passed through
file
list of filenames for tabulated p-s-d

double

h

reduced Hubble parameter

double

age

age in Gyears

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39

Data Fields
double

conformal_age

conformal age in Mpc

double

K

K : Curvature parameter K = −Ω0k ∗ a2today ∗ H02 ;

sgnK

K/|K|: -1, 0 or 1

m_ncdm_in_eV

list of ncdm masses in eV (inferred from M_ncdm and other
parameters above)

double

Neff

so-called "effective neutrino number", computed at earliest time in
interpolation table

double

Omega0_dcdm

double

Omega0_dr

Ω0dcdm : decaying cold dark matter
Ω0dr : decay radiation

double

a_today

scale factor today (arbitrary and irrelevant for most purposes)

int

index_bg_a

scale factor

int

index_bg_H

Hubble parameter in M pc−1

int

index_bg_H_prime

its derivative w.r.t. conformal time

int

index_bg_rho_g

photon density

int

index_bg_rho_b

baryon density

int

index_bg_rho_cdm

cdm density

int

index_bg_rho_lambda

cosmological constant density

int

index_bg_rho_fld

fluid density

int

index_bg_w_fld

fluid equation of state

int

index_bg_rho_ur

relativistic neutrinos/relics density

int

index_bg_rho_dcdm

dcdm density

int

index_bg_rho_dr

dr density

int

index_bg_phi_scf

scalar field value

int

index_bg_phi_prime_scf

scalar field derivative wrt conformal time

int

index_bg_V_scf

scalar field potential V

int

index_bg_dV_scf

scalar field potential derivative V'

int

index_bg_ddV_scf

scalar field potential second derivative V''

int

index_bg_rho_scf

scalar field energy density

int

index_bg_p_scf

scalar field pressure

int

index_bg_rho_ncdm1

density of first ncdm species (others contiguous)

int

index_bg_p_ncdm1

pressure of first ncdm species (others contiguous)

int

index_bg_pseudo_p_ncdm1

another statistical momentum useful in ncdma approximation

int

index_bg_Omega_r

relativistic density fraction ( Ωγ + Ωνr )

int

index_bg_rho_crit

critical density

int

index_bg_Omega_m

non-relativistic density fraction ( Ωb + Ωc dm + Ωνnr )

int

index_bg_conf_distance

conformal distance (from us) in Mpc

int

index_bg_ang_distance

angular diameter distance in Mpc

int

index_bg_lum_distance

luminosity distance in Mpc

int

index_bg_time

proper (cosmological) time in Mpc

int

index_bg_rs

comoving sound horizon in Mpc

int

index_bg_D

scale independent growth factor D(a) for CDM perturbations

int

index_bg_f

corresponding velocity growth factor [dlnD]/[dln a]

int

bg_size_short

size of background vector in the "short format"

int

bg_size_normal

size of background vector in the "normal format"

int

bg_size

size of background vector in the "long format"

int

bt_size

number of lines (i.e. time-steps) in the array

int
double ∗

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Data Fields
double ∗

tau_table

vector tau_table[index_tau] with values of τ (conformal time)

double ∗

z_table

vector z_table[index_tau] with values of z (redshift)

double ∗

background_table

table
background_table[index_tau∗pba->bg_size+pba->index_bg] with
all other quantities (array of size bg_size∗bt_size)

double ∗

d2tau_dz2_table

vector d2tau_dz2_table[index_tau] with values of d2 τ /dz 2
(conformal time)

double ∗

d2background_dtau2_table

table d2background_dtau2_table[index_tau∗pba->bg_size+pba>index_bg] with values of d2 bi /dτ 2 (conformal
time)

int

index_bi_a

{B} scale factor

int

index_bi_rho_dcdm

{B} dcdm density

int

index_bi_rho_dr

{B} dr density

int

index_bi_rho_fld

{B} fluid density

int

index_bi_phi_scf

{B} scalar field value

int

index_bi_phi_prime_scf

{B} scalar field derivative wrt conformal time

int

index_bi_time

{C} proper (cosmological) time in Mpc

int

index_bi_rs

{C} sound horizon

int

index_bi_tau

{C} conformal time in Mpc

int

index_bi_D

{C} scale independent growth factor D(a) for CDM perturbations.

int

index_bi_D_prime

{C} D satisfies [D 00 (τ ) = −aHD 0 (τ ) + 3/2a2 ρM D(τ )

int

bi_B_size

Number of {B} parameters

int

bi_size

Number of {B}+{C} parameters

short

has_cdm

presence of cold dark matter?

short

has_dcdm

presence of decaying cold dark matter?

short

has_dr

presence of relativistic decay radiation?

short

has_scf

presence of a scalar field?

short

has_ncdm

presence of non-cold dark matter?

short

has_lambda

presence of cosmological constant?

short

has_fld

presence of fluid with constant w and cs2?

short

has_ur

presence of ultra-relativistic neutrinos/relics?

short

has_curvature

presence of global spatial curvature?

double ∗∗

q_ncdm_bg

Pointers to vectors of background sampling in q

double ∗∗

w_ncdm_bg

Pointers to vectors of corresponding quadrature weights w

double ∗∗

q_ncdm

Pointers to vectors of perturbation sampling in q

double ∗∗

w_ncdm

Pointers to vectors of corresponding quadrature weights w

double ∗∗

dlnf0_dlnq_ncdm

Pointers to vectors of logarithmic derivatives of p-s-d

int ∗

q_size_ncdm_bg

Size of the q_ncdm_bg arrays

int ∗

q_size_ncdm

Size of the q_ncdm arrays

double ∗

factor_ncdm

List of normalization factors for calculating energy density etc.

short

short_info

flag for calling background_at_eta and return little information

short

normal_info

flag for calling background_at_eta and return medium information

short

long_info

flag for calling background_at_eta and return all information

short

inter_normal

flag for calling background_at_eta and find position in
interpolation table normally

short

inter_closeby

flag for calling background_at_eta and find position in
interpolation table starting from previous position in previous call

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41

Data Fields
short

shooting_failed

flag is set to true if shooting failed.

ErrorMsg

shooting_error

Error message from shooting failed.

background_verbose

flag regulating the amount of information sent to standard output
(none if set to zero)

error_message

zone for writing error messages

short
ErrorMsg

5.2.2.2

struct background_parameters_and_workspace

temporary parameters and workspace passed to the background_derivs function

5.2.2.3

struct background_parameters_for_distributions

temporary parameters and workspace passed to phase space distribution function

5.3

class.c File Reference

#include "class.h"
Include dependency graph for class.c:

class.c

class.h

input.h

output.h

lensing.h

spectra.h

transfer.h

nonlinear.h

sys/stat.h

primordial.h

errno.h

sys/shm.h

hyperspherical.h

perturbations.h

evolver_rkck.h

thermodynamics.h

evolver_ndf15.h

background.h

sparse.h

growTable.h

arrays.h

dei_rkck.h

parser.h

quadrature.h

common.h

stdio.h

5.3.1

Detailed Description

Julien Lesgourgues, 17.04.2011
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stdlib.h

math.h

stdarg.h

svnversion.h

string.h

float.h

42

5.4

File Documentation

common.h File Reference

#include
#include
#include
#include
#include
#include
#include

"stdio.h"
"stdlib.h"
"math.h"
"string.h"
"float.h"
"svnversion.h"


Include dependency graph for common.h:

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

This graph shows which files directly or indirectly include this file:

common.h

parser.h

quadrature.h

sparse.h

background.h

background.c

dei_rkck.h

evolver_ndf15.h

arrays.h

evolver_rkck.h

thermodynamics.h

thermodynamics.c

hyperspherical.h

perturbations.h

primordial.h

nonlinear.h

perturbations.c

nonlinear_exp.h

nonlinear.c

nonlinear_test.h

primordial.c

transfer.h

spectra.h

spectra.c

transfer.c

lensing.h

lensing.c

input.h

input.c

output.h

output.c

class.h

class.c

Data Structures
• struct precision

Enumerations
• enum evolver_type
• enum pk_def { delta_m_squared, delta_tot_squared, delta_bc_squared, delta_tot_from_poisson_squared }
• enum file_format
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5.4 common.h File Reference

5.4.1

43

Detailed Description

Generic libraries, parameters and functions used in the whole code.

5.4.2

Data Structure Documentation

5.4.2.1

struct precision

All precision parameters.
Includes integrations steps, flags telling how the computation is to be performed, etc.
Data Fields
double

a_ini_over_a_today_default

default initial value of scale factor in
background integration, in units of scale
factor today

double

back_integration_stepsize

default step d tau in background
integration, in units of conformal Hubble
time ( dτ = back_integration_stepsize / aH
)

double

tol_background_integration

parameter controlling precision of
background integration

double

tol_initial_Omega_r

double

tol_M_ncdm

parameter controlling how deep inside
radiation domination must the initial time
be chosen
parameter controlling relative precision of
ncdm mass for given ncdm current density

double

tol_ncdm_newtonian

parameter controlling relative precision of
integrals over ncdm phase-space
distribution during perturbation
calculation: value to be applied in
Newtonian gauge

double

tol_ncdm_synchronous

parameter controlling relative precision of
integrals over ncdm phase-space
distribution during perturbation
calculation: value to be applied in
synchronous gauge

double

tol_ncdm

parameter controlling relative precision of
integrals over ncdm phase-space
distribution during perturbation
calculation: value actually applied in
chosen gauge

double

tol_ncdm_bg

parameter controlling relative precision of
integrals over ncdm phase-space
distribution during background evolution

double

tol_ncdm_initial_w

double

safe_phi_scf

parameter controlling how relativistic must
non-cold relics be at initial time
parameter controlling the initial scalar field
in background functions

double

recfast_z_initial

initial redshift in recfast

recfast_Nz0

number of integration steps

tol_thermo_integration

precision of each integration step

int
double
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Data Fields
int

recfast_Heswitch

recfast 1.4 parameter

recfast_fudge_He

recfast 1.4 parameter

recfast_Hswitch

recfast 1.5 switching parameter

double

recfast_fudge_H

H fudge factor when recfast_Hswitch set to
false (v1.4 fudging)

double

recfast_delta_fudge_H

correction to H fudge factor in v1.5

double

recfast_AGauss1

Amplitude of 1st Gaussian

double

recfast_AGauss2

Amplitude of 2nd Gaussian

double

recfast_zGauss1

ln(1+z) of 1st Gaussian

double

recfast_zGauss2

ln(1+z) of 2nd Gaussian

double

recfast_wGauss1

Width of 1st Gaussian

double

recfast_wGauss2

Width of 2nd Gaussian

double

recfast_z_He_1

down to which redshift Helium fully ionized

double

recfast_delta_z_He_1

z range over which transition is smoothed

double

recfast_z_He_2

down to which redshift first Helium
recombination not complete

double

recfast_delta_z_He_2

z range over which transition is smoothed

double

recfast_z_He_3

double

recfast_delta_z_He_3

down to which redshift Helium singly
ionized
z range over which transition is smoothed

double

recfast_x_He0_trigger

value below which recfast uses the full
equation for Helium

double

recfast_x_He0_trigger2

double

recfast_x_He0_trigger_delta

double

recfast_x_H0_trigger

a second threshold used in derivative
routine
x_He range over which transition is
smoothed
value below which recfast uses the full
equation for Hydrogen

double

recfast_x_H0_trigger2

double

recfast_x_H0_trigger_delta

double

recfast_H_frac

double

reionization_z_start_max

double

reionization_sampling

control stepsize in z during reionization

double

reionization_optical_depth_tol

fractional error on optical_depth

double

reionization_start_factor

parameter for CAMB-like parametrization

thermo_rate_smoothing_radius

plays a minor (almost aesthetic) role in the
definition of the variation rate of
thermodynamical quantities

evolver

which type of evolver for integrating
perturbations (Runge-Kutta? Stiff?...)

k_min_tau0

number defining k_min for the computation
of Cl's and P(k)'s (dimensionless): (k_min
tau_0), usually chosen much smaller than
one

double
int

int

enum evolver_type
double

a second threshold used in derivative
routine
x_H range over which transition is
smoothed
governs time at which full equation of
evolution for Tmat is used
maximum redshift at which reionization
should start. If not, return an error.

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45

Data Fields
double

k_max_tau0_over_l_max

double

k_step_sub

double

k_step_super

step in k space, in units of one period of
acoustic oscillation at decoupling, for
scales above sound horizon at decoupling

double

k_step_transition

dimensionless number regulating the
transition from 'sub' steps to 'super' steps.
Decrease for more precision.

double

k_step_super_reduction

the step k_step_super is reduced by this
amount in the k–>0 limit (below scale of
Hubble and/or curvature radius)

double

k_per_decade_for_pk

if values needed between kmax inferred
from k_oscillations and k_kmax_for_pk,
this gives the number of k per decade
outside the BAO region

double

k_per_decade_for_bao

if values needed between kmax inferred
from k_oscillations and k_kmax_for_pk,
this gives the number of k per decade
inside the BAO region (for finer sampling)

double

k_bao_center

in ln(k) space, the central value of the BAO
region where sampling is finer is defined
as k_rec times this number
(recommended: 3, i.e. finest sampling
near 3rd BAO peak)

double

k_bao_width

in ln(k) space, width of the BAO region
where sampling is finer: this number gives
roughly the number of BAO oscillations
well resolved on both sides of the central
value (recommended: 4, i.e. finest
sampling from before first up to 3+4=7th
peak)

double

start_small_k_at_tau_c_over_tau_h

double

start_large_k_at_tau_h_over_tau_k

largest wavelengths start being sampled
when universe is sufficiently opaque. This
is quantified in terms of the ratio of thermo
to hubble time scales, τc /τH . Start when
start_largek_at_tau_c_over_tau_h equals
this ratio. Decrease this value to start
integrating the wavenumbers earlier in
time.
largest wavelengths start being sampled
when mode is sufficiently outside Hubble
scale. This is quantified in terms of the
ratio of hubble time scale to wavenumber
time scale, τh /τk which is roughly equal to
(k∗tau). Start when this ratio equals
start_large_k_at_tau_k_over_tau_h.
Decrease this value to start integrating the
wavenumbers earlier in time.

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number defining k_max for the
computation of Cl's (dimensionless):
(k_max tau_0)/l_max, usually chosen
around two
step in k space, in units of one period of
acoustic oscillation at decoupling, for
scales inside sound horizon at decoupling

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Data Fields
double

tight_coupling_trigger_tau_c_over_tau_h

double

tight_coupling_trigger_tau_c_over_tau_k

when to switch off tight-coupling
approximation: first condition: τc /τH >
tight_coupling_trigger_tau_c_over_tau_h.
Decrease this value to switch off earlier in
time. If this number is larger than
start_sources_at_tau_c_over_tau_h, the
code returns an error, because the source
computation requires tight-coupling to be
switched off.
when to switch off tight-coupling
approximation: second condition:

τc /τk ≡ kτc <

double

start_sources_at_tau_c_over_tau_h

tight_coupling_trigger_tau_c_over_tau_k.
Decrease this value to switch off earlier in
time.
sources start being sampled when
universe is sufficiently opaque. This is
quantified in terms of the ratio of thermo to
hubble time scales, τc /τH . Start when
start_sources_at_tau_c_over_tau_h
equals this ratio. Decrease this value to
start sampling the sources earlier in time.

int

tight_coupling_approximation

method for tight coupling approximation

int

l_max_g

int

l_max_pol_g

int

l_max_dr

number of momenta in Boltzmann
hierarchy for photon temperature (scalar),
at least 4
number of momenta in Boltzmann
hierarchy for photon polarization (scalar),
at least 4
number of momenta in Boltzmann
hierarchy for decay radiation, at least 4

int

l_max_ur

number of momenta in Boltzmann
hierarchy for relativistic neutrino/relics
(scalar), at least 4

int

l_max_ncdm

number of momenta in Boltzmann
hierarchy for relativistic neutrino/relics
(scalar), at least 4

int

l_max_g_ten

int

l_max_pol_g_ten

number of momenta in Boltzmann
hierarchy for photon temperature (tensor),
at least 4
number of momenta in Boltzmann
hierarchy for photon polarization (tensor),
at least 4
initial condition for curvature for adiabatic

double

curvature_ini

double

entropy_ini

double

gw_ini

double

perturb_integration_stepsize

initial condition for entropy perturbation for
isocurvature
initial condition for tensor metric
perturbation h
default step dτ in perturbation integration,
in units of the timescale involved in the
equations (usually, the min of 1/k , 1/aH ,
1/κ̇)

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Data Fields
double

perturb_sampling_stepsize

default step dτ for sampling the source
function, in units of the timescale involved
in the sources: (κ̇ − κ̈/κ̇)−1

double

tol_perturb_integration

control parameter for the precision of the
perturbation integration

double

tol_tau_approx

precision with which the code should
determine (by bisection) the times at
which sources start being sampled, and at
which approximations must be switched
on/off (units of Mpc)

radiation_streaming_approximation

method for switching off photon
perturbations

int
double

radiation_streaming_trigger_tau_over_tau_k when to switch off photon perturbations, ie
when to switch on photon free-streaming
approximation (keep density and thtau, set
shear and higher momenta to zero): first
condition: kτ > radiation_streaming_←trigger_tau_h_over_tau_k

double

radiation_streaming_trigger_tau_c_over_tau when to switch off photon perturbations, ie
when to switch on photon free-streaming
approximation (keep density and theta, set
shear and higher momenta to zero):
second condition:
ur_fluid_approximation
method for ultra relativistic fluid
approximation

int
double

ur_fluid_trigger_tau_over_tau_k

when to switch off ur (massless neutrinos /
ultra-relativistic relics) fluid approximation

ncdm_fluid_approximation

method for non-cold dark matter fluid
approximation

double

ncdm_fluid_trigger_tau_over_tau_k

when to switch off ncdm (massive
neutrinos / non-cold relics) fluid
approximation

double

neglect_CMB_sources_below_visibility

whether CMB source functions can be
approximated as zero when visibility
function g(tau) is tiny

double

k_per_decade_primordial

logarithmic sampling for primordial spectra
(number of points per decade in k space)

double

primordial_inflation_ratio_min

for each k, start following wavenumber
when aH = k/primordial_inflation_ratio_min

double

primordial_inflation_ratio_max

for each k, stop following wavenumber, at
the latest, when aH =
k/primordial_inflation_ratio_max

primordial_inflation_phi_ini_maxit

maximum number of iteration when
searching a suitable initial field value
phi_ini (value reached when no
long-enough slow-roll period before the
pivot scale)

double

primordial_inflation_pt_stepsize

controls the integration timestep for
inflaton perturbations

double

primordial_inflation_bg_stepsize

controls the integration timestep for
inflaton background

double

primordial_inflation_tol_integration

controls the precision of the ODE
integration during inflation

int

int

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Data Fields
double

primordial_inflation_attractor_precision_pivot targeted precision when searching
attractor solution near phi_pivot

double

primordial_inflation_attractor_precision_initial targeted precision when searching
attractor solution near phi_ini

int

primordial_inflation_attractor_maxit

maximum number of iteration when
searching attractor solution

double

primordial_inflation_tol_curvature

for each k, stop following wavenumber, at
the latest, when curvature perturbation R
is stable up to to this tolerance

double

primordial_inflation_aH_ini_target

double

primordial_inflation_end_dphi

control the step size in the search for a
suitable initial field value
first bracketing width, when trying to
bracket the value phi_end at which
inflation ends naturally

double

primordial_inflation_end_logstep

logarithmic step for updating the
bracketing width, when trying to bracket
the value phi_end at which inflation ends
naturally

double

primordial_inflation_small_epsilon

value of slow-roll parameter epsilon used
to define a field value phi_end close to the
end of inflation (doesn't need to be exactly
at the end):
epsilon(phi_end)=small_epsilon (should
be smaller than one)

double

primordial_inflation_small_epsilon_tol

tolerance in the search for phi_end

double

primordial_inflation_extra_efolds

a small number of efolds, irrelevant at the
end, used in the search for the pivot scale
(backward from the end of inflation)

l_linstep

factor for logarithmic spacing of values of l
over which bessel and transfer functions
are sampled

double

l_logstep

maximum spacing of values of l over which
Bessel and transfer functions are sampled
(so, spacing becomes linear instead of
logarithmic at some point)

double

hyper_x_min

flat case: lower bound on the smallest
value of x at which we sample Φνl (x) or

int

jl (x)
double

hyper_sampling_flat

flat case: number of sampled points x per
approximate wavelength 2π

double

hyper_sampling_curved_low_nu

open/closed cases: number of sampled
points x per approximate wavelength
2π/ν , when ν smaller than
hyper_nu_sampling_step

double

hyper_sampling_curved_high_nu

open/closed cases: number of sampled
points x per approximate wavelength
2π/ν , when ν greater than
hyper_nu_sampling_step

double

hyper_nu_sampling_step

open/closed cases: value of nu at which
sampling changes

double

hyper_phi_min_abs

small value of Bessel function used in
calculation of first point x ( Φνl (x) equals
hyper_phi_min_abs)

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5.4 common.h File Reference

49

Data Fields
double

hyper_x_tol

double

hyper_flat_approximation_nu

double

q_linstep

double

q_logstep_spline

initial logarithmic sampling step in q space,
in units of 2π/ra (τrec ) (comoving angular
diameter distance to recombination)

double

q_logstep_open

in open models, the value of
q_logstep_spline must be decreased
according to curvature. Increasing this
number will make the calculation more
accurate for large positive Omega_k

double

q_logstep_trapzd

initial logarithmic sampling step in q space,
in units of 2π/ra (τrec ) (comoving angular
diameter distance to recombination), in the
case of small q's in the closed case, for
which one must used trapezoidal
integration instead of spline (the number of
q's for which this is the case decreases
with curvature and vanishes in the flat
limit)

double

q_numstep_transition

number of steps for the transition from
q_logstep_trapzd steps to
q_logstep_spline steps (transition must be
smooth for spline)

double

transfer_neglect_delta_k_S_t0

for temperature source function T0 of
scalar mode, range of k values (in 1/Mpc)
taken into account in transfer function: for l
< (k-delta_k)∗tau0, ie for k > (l/tau0 +
delta_k), the transfer function is set to zero

double

transfer_neglect_delta_k_S_t1

double

transfer_neglect_delta_k_S_t2

double

transfer_neglect_delta_k_S_e

double

transfer_neglect_delta_k_V_t1

double

transfer_neglect_delta_k_V_t2

double

transfer_neglect_delta_k_V_e

double

transfer_neglect_delta_k_V_b

double

transfer_neglect_delta_k_T_t2

double

transfer_neglect_delta_k_T_e

same for temperature source function T1
of scalar mode
same for temperature source function T2
of scalar mode
same for polarization source function E of
scalar mode
same for temperature source function T1
of vector mode
same for temperature source function T2
of vector mode
same for polarization source function E of
vector mode
same for polarization source function B of
vector mode
same for temperature source function T2
of tensor mode
same for polarization source function E of
tensor mode

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tolerance parameter used to determine
first value of x
value of nu below which the flat
approximation is used to compute Bessel
function
asymptotic linear sampling step in q
space, in units of 2π/ra (τr ec) (comoving
angular diameter distance to
recombination)

50

File Documentation

Data Fields
double

transfer_neglect_delta_k_T_b

same for polarization source function B of
tensor mode
value of l below which the CMB source
functions can be neglected at late time,
excepted when there is a Late ISW
contribution
when to use the Limber approximation for
project gravitational potential cl's

double

transfer_neglect_late_source

double

l_switch_limber

double

l_switch_limber_for_nc_local_over_z

when to use the Limber approximation for
local number count contributions to cl's
(relative to central redshift of each bin)

double

l_switch_limber_for_nc_los_over_z

when to use the Limber approximation for
number count contributions to cl's
integrated along the line-of-sight (relative
to central redshift of each bin)

double

selection_cut_at_sigma

double

selection_sampling

in sigma units, where to cut gaussian
selection functions
controls sampling of integral over time
when selection functions vary quicker than
Bessel functions. Increase for better
sampling.

double

selection_sampling_bessel

controls sampling of integral over time
when selection functions vary slower than
Bessel functions. Increase for better
sampling

double

selection_sampling_bessel_los

controls sampling of integral over time
when selection functions vary slower than
Bessel functions. This parameter is
specific to number counts contributions to
Cl integrated along the line of sight.
Increase for better sampling

double

selection_tophat_edge

controls how smooth are the edge of
top-hat window function (<<1 for very
sharp, 0.1 for sharp)

double

halofit_dz

parameters relevant for HALOFIT
computation spacing in redshift space
defining values of z at which HALOFIT will
be used. Intermediate values will be
obtained by interpolation. Decrease for
more precise interpolations, at the
expense of increasing time spent in
nonlinear_init()

double

halofit_min_k_nonlinear

value of k in 1/Mpc above which non-linear
corrections will be computed

double

halofit_sigma_precision

a smaller value will lead to a more precise
halofit result at the highest requested
redshift, at the expense of requiring a
larger k_max

double

halofit_min_k_max

when halofit is used, k_max must be at
least equal to this value (otherwise halofit
could not find the scale of non-linearity)

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5.4 common.h File Reference

51

Data Fields
double

halofit_k_per_decade

halofit needs to evalute integrals (linear
power spectrum times some kernels).
They are sampled using this logarithmic
step size.

int

accurate_lensing

switch between Gauss-Legendre
quadrature integration and simple
quadrature on a subdomain of angles

int

num_mu_minus_lmax

difference between num_mu and l_max,
increase for more precision

int

delta_l_max

difference between l_max in unlensed and
lensed spectra

double

tol_gauss_legendre

tolerance with which quadrature points are
found: must be very small for an accurate
integration (if not entered manually, set
automatically to match machine precision)

double

smallest_allowed_variation

machine-dependent, assigned
automatically by the code

error_message

zone for writing error messages

ErrorMsg

5.4.3

Enumeration Type Documentation

5.4.3.1

evolver_type

enum evolver_type

parameters related to the precision of the code and to the method of calculation list of evolver types for integrating
perturbations over time

5.4.3.2

pk_def

enum pk_def

List of ways in which matter power spectrum P(k) can be defined. The standard definition is the first one (delta_←m_squared) but alternative definitions can be useful in some projects.
Enumerator
delta_m_squared

normal definition (delta_m includes all non-relativistic species at late times)

delta_tot_squared

delta_tot includes all species contributions to (delta rho), and only
non-relativistic contributions to rho
delta_bc includes contribution of baryons and cdm only to (delta rho) and
to rho
use delta_tot inferred from gravitational potential through Poisson equation

delta_bc_squared
delta_tot_from_poisson_squared

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52

File Documentation

5.4.3.3

file_format

enum file_format

Different ways to present output files

5.5

input.c File Reference

#include "input.h"
Include dependency graph for input.c:

input.c

input.h

output.h

lensing.h

spectra.h

transfer.h

nonlinear.h

sys/shm.h

sys/stat.h

primordial.h

errno.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

parser.h

quadrature.h

arrays.h

evolver_ndf15.h

growTable.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int input_init_from_arguments (int argc, char ∗∗argv, struct precision ∗ppr, struct background ∗pba, struct
thermo ∗pth, struct perturbs ∗ppt, struct transfers ∗ptr, struct primordial ∗ppm, struct spectra ∗psp, struct
nonlinear ∗pnl, struct lensing ∗ple, struct output ∗pop, ErrorMsg errmsg)
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5.5 input.c File Reference

53

• int input_init (struct file_content ∗pfc, struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct
perturbs ∗ppt, struct transfers ∗ptr, struct primordial ∗ppm, struct spectra ∗psp, struct nonlinear ∗pnl, struct
lensing ∗ple, struct output ∗pop, ErrorMsg errmsg)
• int input_read_parameters (struct file_content ∗pfc, struct precision ∗ppr, struct background ∗pba, struct
thermo ∗pth, struct perturbs ∗ppt, struct transfers ∗ptr, struct primordial ∗ppm, struct spectra ∗psp, struct
nonlinear ∗pnl, struct lensing ∗ple, struct output ∗pop, ErrorMsg errmsg)
• int input_default_params (struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt, struct transfers
∗ptr, struct primordial ∗ppm, struct spectra ∗psp, struct nonlinear ∗pnl, struct lensing ∗ple, struct output ∗pop)
• int input_default_precision (struct precision ∗ppr)
• int get_machine_precision (double ∗smallest_allowed_variation)
• int class_fzero_ridder (int(∗func)(double x, void ∗param, double ∗y, ErrorMsg error_message), double x1,
double x2, double xtol, void ∗param, double ∗Fx1, double ∗Fx2, double ∗xzero, int ∗fevals, ErrorMsg error←_message)
• int input_try_unknown_parameters (double ∗unknown_parameter, int unknown_parameters_size, void
∗voidpfzw, double ∗output, ErrorMsg errmsg)
• int input_get_guess (double ∗xguess, double ∗dxdy, struct fzerofun_workspace ∗pfzw, ErrorMsg errmsg)
• int input_find_root (double ∗xzero, int ∗fevals, struct fzerofun_workspace ∗pfzw, ErrorMsg errmsg)

5.5.1

Detailed Description

Documented input module.
Julien Lesgourgues, 27.08.2010

5.5.2

Function Documentation

5.5.2.1

input_init_from_arguments()

int input_init_from_arguments (
int argc,
char ∗∗ argv,
struct precision ∗ ppr,
struct background ∗ pba,
struct
struct
struct
struct
struct
struct

thermo ∗ pth,
perturbs ∗ ppt,
transfers ∗ ptr,
primordial ∗ ppm,
spectra ∗ psp,
nonlinear ∗ pnl,

struct lensing ∗ ple,
struct output ∗ pop,
ErrorMsg errmsg )

Use this routine to extract initial parameters from files 'xxx.ini' and/or 'xxx.pre'. They can be the arguments of the
main() routine.
If class is embedded into another code, you will probably prefer to call directly input_init() in order to pass input
parameters through a 'file_content' structure. Summary:

• define local variables
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54

File Documentation

• –> the final structure with all parameters
• –> a temporary structure with all input parameters
• –> a temporary structure with all precision parameters
• –> a temporary structure with only the root name
• –> sum of fc_inoput and fc_root
• –> a pointer to either fc_root or fc_inputroot
• Initialize the two file_content structures (for input parameters and precision parameters) to some null content.
If no arguments are passed, they will remain null and inform init_params() that all parameters take default
values.
• If some arguments are passed, identify eventually some 'xxx.ini' and 'xxx.pre' files, and store their name.
• if there is an 'xxx.ini' file, read it and store its content.
• check whether a root name has been set
• if root has not been set, use root=output/inputfilennameN_
• if there is an 'xxx.pre' file, read it and store its content.
• if one or two files were read, merge their contents in a single 'file_content' structure.
• Finally, initialize all parameters given the input 'file_content' structure. If its size is null, all parameters take
their default values.

5.5.2.2

input_init()

int input_init (
struct file_content ∗ pfc,
struct precision ∗ ppr,
struct
struct
struct
struct
struct
struct

background ∗ pba,
thermo ∗ pth,
perturbs ∗ ppt,
transfers ∗ ptr,
primordial ∗ ppm,
spectra ∗ psp,

struct nonlinear ∗ pnl,
struct lensing ∗ ple,
struct output ∗ pop,
ErrorMsg errmsg )

Initialize each parameter, first to its default values, and then from what can be interpreted from the values passed
in the input 'file_content' structure. If its size is null, all parameters keep their default values. These two arrays must
contain the strings of names to be searched for and the corresponding new parameter

• Do we need to fix unknown parameters?
• –> input_auxillary_target_conditions() takes care of the case where for instance Omega_dcdmdr is set to
0.0.
• case with unknown parameters
• –> go through all cases with unknown parameters:
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5.5 input.c File Reference

55

• –> Read all parameters from tuned pfc
• –> Set status of shooting
• –> Free arrays allocated
• case with no unknown parameters
• –> just read all parameters from input pfc:
• eventually write all the read parameters in a file, unread parameters in another file, and warnings about unread
parameters

5.5.2.3

input_read_parameters()

int input_read_parameters (
struct file_content ∗ pfc,
struct precision ∗ ppr,
struct
struct
struct
struct
struct

background ∗ pba,
thermo ∗ pth,
perturbs ∗ ppt,
transfers ∗ ptr,
primordial ∗ ppm,

struct spectra ∗ psp,
struct nonlinear ∗ pnl,
struct lensing ∗ ple,
struct output ∗ pop,
ErrorMsg errmsg )

Summary:

• define local variables
• set all parameters (input and precision) to default values
• if entries passed in file_content structure, carefully read and interpret each of them, and tune the relevant
input parameters accordingly

Knowing the gauge from the very beginning is useful (even if this could be a run not requiring perturbations at all:
even in that case, knowing the gauge is important e.g. for fixing the sampling in momentum space for non-cold dark
matter)
(a) background parameters

• scale factor today (arbitrary)
• h (dimensionless) and [ H0 /c] in M pc−1 = h/2997.9... = h ∗ 105 /c
• Omega_0_g (photons) and T_cmb
• Omega0_g = rho_g / rho_c0, each of them expressed in Kg/m/s2
• rho_g = (4 sigma_B / c) T 4
• rho_c0 = 3c2 H02 /(8πG)
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File Documentation

• Omega_0_b (baryons)
• Omega_0_ur (ultra-relativistic species / massless neutrino)
• Omega_0_cdm (CDM)
• Omega_0_dcdmdr (DCDM)
• Read Omega_ini_dcdm or omega_ini_dcdm
• Read Gamma in same units as H0, i.e. km/(s Mpc)
• non-cold relics (ncdm)
• Omega_0_k (effective fractional density of curvature)
• Set curvature parameter K
• Set curvature sign
• Omega_0_lambda (cosmological constant), Omega0_fld (dark energy fluid), Omega0_scf (scalar field)
• –> (flag3 == FALSE) || (param3 >= 0.) explained: it means that either we have not read Omega_scf
so we are ignoring it (unlike lambda and fld!) OR we have read it, but it had a positive value and should
not be used for filling. We now proceed in two steps: 1) set each Omega0 and add to the total for each
specified component. 2) go through the components in order {lambda, fld, scf} and fill using first unspecified
component.
• Test that the user have not specified Omega_scf = -1 but left either Omega_lambda or Omega_fld
unspecified:
• Read parameters describing scalar field potential
• Assign shooting parameter

(b) assign values to thermodynamics cosmological parameters

• primordial helium fraction
• recombination parameters
• reionization parametrization
• reionization parameters if reio_parametrization=reio_camb
• reionization parameters if reio_parametrization=reio_bins_tanh
• reionization parameters if reio_parametrization=reio_many_tanh
• reionization parameters if reio_parametrization=reio_many_tanh
• energy injection parameters from CDM annihilation/decay

(c) define which perturbations and sources should be computed, and down to which scale
(d) define the primordial spectrum
(e) parameters for final spectra
(f) parameter related to the non-linear spectra computation
(g) amount of information sent to standard output (none if all set to zero)
(h) all precision parameters
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5.5 input.c File Reference

57

• (h.1.) parameters related to the background
• (h.2.) parameters related to the thermodynamics
• (h.3.) parameters related to the perturbations
• —> Include ur and ncdm shear in tensor computation?
• —> derivatives of baryon sound speed only computed if some non-minimal tight-coupling schemes is requested
• (h.4.) parameter related to the primordial spectra
• (h.5.) parameter related to the transfer functions
• (h.6.) parameters related to nonlinear calculations
• (h.7.) parameter related to lensing

(i) Write values in file

• (i.1.) shall we write background quantities in a file?
• (i.2.) shall we write thermodynamics quantities in a file?
• (i.3.) shall we write perturbation quantities in files?
• (i.4.) shall we write primordial spectra in a file?

5.5.2.4

input_default_params()

int input_default_params (
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt,
struct
struct
struct
struct
struct
struct

transfers ∗ ptr,
primordial ∗ ppm,
spectra ∗ psp,
nonlinear ∗ pnl,
lensing ∗ ple,
output ∗ pop )

All default parameter values (for input parameters)
Parameters
pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfer structure

ppm

Input: pointer to primordial structure

psp

Input: pointer to spectra structure

pnl

Input: pointer to nonlinear structure

ple

Input: pointer to lensing structure

pop

Input: pointer to output structure

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58

File Documentation

Returns
the error status

Define all default parameter values (for input parameters) for each structure:

• background structure
• thermodynamics structure
• perturbation structure
• primordial structure
• transfer structure
• output structure
• spectra structure
• nonlinear structure
• lensing structure
• nonlinear structure
• all verbose parameters

5.5.2.5

input_default_precision()

int input_default_precision (
struct precision ∗ ppr )

Initialize the precision parameter structure.
All precision parameters used in the other modules are listed here and assigned here a default value.
Parameters
ppr

Input/Output: a precision_params structure pointer

Returns
the error status

Initialize presicion parameters for different structures:

• parameters related to the background
• parameters related to the thermodynamics
• parameters related to the perturbations
• parameter related to the primordial spectra
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5.5 input.c File Reference

59

• parameter related to the transfer functions
• parameters related to spectra module
• parameters related to nonlinear module
• parameter related to lensing
• automatic estimate of machine precision

5.5.2.6

get_machine_precision()

int get_machine_precision (
double ∗ smallest_allowed_variation )

Automatically computes the machine precision.
Parameters
smallest_allowed_variation

a pointer to the smallest allowed variation

Returns the smallest allowed variation (minimum epsilon ∗ TOLVAR)

5.5.2.7

class_fzero_ridder()

int class_fzero_ridder (
int(∗)(double x, void ∗param, double ∗y, ErrorMsg error_message) func,
double
double
double
void ∗
double
double

x1,
x2,
xtol,
param,
∗ Fx1,
∗ Fx2,

double ∗ xzero,
int ∗ fevals,
ErrorMsg error_message )

Using Ridders' method, return the root of a function func known to lie between x1 and x2. The root, returned as
zriddr, will be found to an approximate accuracy xtol.

5.5.2.8

input_try_unknown_parameters()

int input_try_unknown_parameters (
double ∗ unknown_parameter,
int unknown_parameters_size,
void ∗ voidpfzw,
double ∗ output,
ErrorMsg errmsg )

Summary:
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60

File Documentation

• Call the structures
• Do computations
• In case scalar field is used to fill, pba->Omega0_scf is not equal to pfzw->target_value[i].
• Free structures
• Set filecontent to unread

5.5.2.9

input_get_guess()

int input_get_guess (
double ∗ xguess,
double ∗ dxdy,
struct fzerofun_workspace ∗ pfzw,
ErrorMsg errmsg )

Summary:
• Here we should write reasonable guesses for the unknown parameters. Also estimate dxdy, i.e. how the
unknown parameter responds to the known. This can simply be estimated as the derivative of the guess
formula.
• Update pb to reflect guess
– This guess is arbitrary, something nice using WKB should be implemented.
• Version 2: use a fit: xguess[index_guess] = 1.77835∗pow(ba.Omega0_scf,-2./7.);
dxdy[index_guess] = -0.5081∗pow(ba.Omega0_scf,-9./7.);
• Version 3: use attractor solution
• This works since correspondence is Omega_ini_dcdm -> Omega_dcdmdr and omega_ini_dcdm -> omega←_dcdmdr
• Deallocate everything allocated by input_read_parameters

5.5.2.10

input_find_root()

int input_find_root (
double ∗ xzero,
int ∗ fevals,
struct fzerofun_workspace ∗ pfzw,
ErrorMsg errmsg )

Summary:
• Fisrt we do our guess
• Do linear hunt for boundaries
• root has been bracketed
• Find root using Ridders method. (Exchange for bisection if you are old-school.)
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5.6 input.h File Reference

5.6

61

input.h File Reference

#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include

"common.h"
"parser.h"
"quadrature.h"
"background.h"
"thermodynamics.h"
"perturbations.h"
"transfer.h"
"primordial.h"
"spectra.h"
"nonlinear.h"
"lensing.h"
"output.h"

Include dependency graph for input.h:

input.h

output.h

lensing.h

spectra.h

transfer.h

nonlinear.h

sys/shm.h

sys/stat.h

primordial.h

errno.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

parser.h

quadrature.h

arrays.h

evolver_ndf15.h

growTable.h

dei_rkck.h

sparse.h

common.h

stdio.h

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stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

62

File Documentation

This graph shows which files directly or indirectly include this file:

input.h

class.h

input.c

class.c

Enumerations

• enum target_names

5.6.1

Detailed Description

Documented includes for input module

5.6.2

Enumeration Type Documentation

5.6.2.1

target_names

enum target_names

temporary parameters for background fzero function
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5.7 lensing.c File Reference

5.7

63

lensing.c File Reference

#include "lensing.h"
#include 
Include dependency graph for lensing.c:

lensing.c

lensing.h

time.h

spectra.h

transfer.h

nonlinear.h

sys/shm.h

primordial.h

sys/stat.h

errno.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int lensing_cl_at_l (struct lensing ∗ple, int l, double ∗cl_lensed)
• int lensing_init (struct precision ∗ppr, struct perturbs ∗ppt, struct spectra ∗psp, struct nonlinear ∗pnl, struct
lensing ∗ple)
• int lensing_free (struct lensing ∗ple)
• int lensing_indices (struct precision ∗ppr, struct spectra ∗psp, struct lensing ∗ple)
• int lensing_lensed_cl_tt (double ∗ksi, double ∗∗d00, double ∗w8, int nmu, struct lensing ∗ple)
• int lensing_addback_cl_tt (struct lensing ∗ple, double ∗cl_tt)
• int lensing_lensed_cl_te (double ∗ksiX, double ∗∗d20, double ∗w8, int nmu, struct lensing ∗ple)
• int lensing_addback_cl_te (struct lensing ∗ple, double ∗cl_te)
• int lensing_lensed_cl_ee_bb (double ∗ksip, double ∗ksim, double ∗∗d22, double ∗∗d2m2, double ∗w8, int
nmu, struct lensing ∗ple)
• int lensing_addback_cl_ee_bb (struct lensing ∗ple, double ∗cl_ee, double ∗cl_bb)
• int lensing_d00 (double ∗mu, int num_mu, int lmax, double ∗∗d00)
• int lensing_d11 (double ∗mu, int num_mu, int lmax, double ∗∗d11)
• int lensing_d1m1 (double ∗mu, int num_mu, int lmax, double ∗∗d1m1)
• int lensing_d2m2 (double ∗mu, int num_mu, int lmax, double ∗∗d2m2)
• int lensing_d22 (double ∗mu, int num_mu, int lmax, double ∗∗d22)
• int lensing_d20 (double ∗mu, int num_mu, int lmax, double ∗∗d20)
• int lensing_d31 (double ∗mu, int num_mu, int lmax, double ∗∗d31)
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File Documentation

•
•
•
•
•

5.7.1

int lensing_d3m1 (double ∗mu, int num_mu, int lmax, double ∗∗d3m1)
int lensing_d3m3 (double ∗mu, int num_mu, int lmax, double ∗∗d3m3)
int lensing_d40 (double ∗mu, int num_mu, int lmax, double ∗∗d40)
int lensing_d4m2 (double ∗mu, int num_mu, int lmax, double ∗∗d4m2)
int lensing_d4m4 (double ∗mu, int num_mu, int lmax, double ∗∗d4m4)

Detailed Description

Documented lensing module
Simon Prunet and Julien Lesgourgues, 6.12.2010
This module computes the lensed temperature and polarization anisotropy power spectra ClX , P (k), ...'s given the
unlensed temperature, polarization and lensing potential spectra.
Follows Challinor and Lewis full-sky method, astro-ph/0502425
The following functions can be called from other modules:

1. lensing_init() at the beginning (but after spectra_init())
2. lensing_cl_at_l() at any time for computing Cl_lensed at any l
3. lensing_free() at the end

5.7.2

Function Documentation

5.7.2.1

lensing_cl_at_l()

int lensing_cl_at_l (
struct lensing ∗ ple,
int l,
double ∗ cl_lensed )

Anisotropy power spectra Cl 's for all types, modes and initial conditions. SO FAR: ONLY SCALAR
This routine evaluates all the lensed Cl 's at a given value of l by picking it in the pre-computed table. When relevant,
it also sums over all initial conditions for each mode, and over all modes.
This function can be called from whatever module at whatever time, provided that lensing_init() has been called
before, and lensing_free() has not been called yet.
Parameters
ple

Input: pointer to lensing structure

l

Input: multipole number

cl_lensed

Output: lensed Cl 's for all types (TT, TE, EE, etc..)

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5.7 lensing.c File Reference

65

Returns
the error status

5.7.2.2

lensing_init()

int lensing_init (
struct
struct
struct
struct
struct

precision ∗ ppr,
perturbs ∗ ppt,
spectra ∗ psp,
nonlinear ∗ pnl,
lensing ∗ ple )

This routine initializes the lensing structure (in particular, computes table of lensed anisotropy spectra ClX )
Parameters
ppr

Input: pointer to precision structure

ppt

Input: pointer to perturbation structure (just in case, not used in current version...)

psp

Input: pointer to spectra structure

pnl

Input: pointer to nonlinear structure

ple

Output: pointer to initialized lensing structure

Returns
the error status

Summary:

• Define local variables
• check that we really want to compute at least one spectrum
• initialize indices and allocate some of the arrays in the lensing structure
• put all precision variables hare; will be stored later in precision structure
• Last element in µ will be for µ = 1, needed for sigma2. The rest will be chosen as roots of a Gauss-Legendre
quadrature
• allocate array of µ values, as well as quadrature weights
• Compute dlmm0 (µ)
• Allocate main contiguous buffer
• compute Cgl(µ), Cgl2(µ) and sigma2( µ)
• Locally store unlensed temperature cltt and potential clpp spectra
• Compute sigma2 (µ) and Cgl2( µ)
• compute ksi, ksi+, ksi-, ksiX
• –> ksi is for TT
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File Documentation

• –> ksiX is for TE
• –> ksip, ksim for EE, BB
• compute lensed Cl 's by integration
• spline computed Cl 's in view of interpolation
• Free lots of stuff
• Exit

5.7.2.3

lensing_free()

int lensing_free (
struct lensing ∗ ple )

This routine frees all the memory space allocated by lensing_init().
To be called at the end of each run, only when no further calls to lensing_cl_at_l() are needed.
Parameters
ple

Input: pointer to lensing structure (which fields must be freed)

Returns
the error status

5.7.2.4

lensing_indices()

int lensing_indices
struct
struct
struct

(
precision ∗ ppr,
spectra ∗ psp,
lensing ∗ ple )

This routine defines indices and allocates tables in the lensing structure
Parameters
ppr

Input: pointer to precision structure

psp

Input: pointer to spectra structure

ple

Input/output: pointer to lensing structure

Returns
the error status
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5.7 lensing.c File Reference

5.7.2.5

67

lensing_lensed_cl_tt()

int lensing_lensed_cl_tt (
double ∗ ksi,
double ∗∗ d00,
double ∗ w8,
int nmu,
struct lensing ∗ ple )

This routine computes the lensed power spectra by Gaussian quadrature
Parameters
ksi

Input: Lensed correlation function (ksi[index_mu])

d00

Input: Legendre polynomials ( dl00 [l][index_mu])

w8

Input: Legendre quadrature weights (w8[index_mu])

nmu

Input: Number of quadrature points (0<=index_mu<=nmu)

ple

Input/output: Pointer to the lensing structure

Returns
the error status

Integration by Gauss-Legendre quadrature.

5.7.2.6

lensing_addback_cl_tt()

int lensing_addback_cl_tt (
struct lensing ∗ ple,
double ∗ cl_tt )

This routine adds back the unlensed cltt power spectrum Used in case of fast (and BB inaccurate) integration of
correlation functions.
Parameters
ple

Input/output: Pointer to the lensing structure

cl←_tt

Input: Array of unlensed power spectrum

Returns
the error status

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File Documentation

5.7.2.7

lensing_lensed_cl_te()

int lensing_lensed_cl_te (
double ∗ ksiX,
double ∗∗ d20,
double ∗ w8,
int nmu,
struct lensing ∗ ple )

This routine computes the lensed power spectra by Gaussian quadrature
Parameters
ksiX

Input: Lensed correlation function (ksiX[index_mu])

d20

Input: Wigner d-function ( dl20 [l][index_mu])

w8

Input: Legendre quadrature weights (w8[index_mu])

nmu

Input: Number of quadrature points (0<=index_mu<=nmu)

ple

Input/output: Pointer to the lensing structure

Returns
the error status

Integration by Gauss-Legendre quadrature.

5.7.2.8

lensing_addback_cl_te()

int lensing_addback_cl_te (
struct lensing ∗ ple,
double ∗ cl_te )

This routine adds back the unlensed clte power spectrum Used in case of fast (and BB inaccurate) integration of
correlation functions.
Parameters
ple

Input/output: Pointer to the lensing structure

cl←_te

Input: Array of unlensed power spectrum

Returns
the error status

5.7.2.9

lensing_lensed_cl_ee_bb()

int lensing_lensed_cl_ee_bb (
double ∗ ksip,
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5.7 lensing.c File Reference

69

double ∗ ksim,
double ∗∗ d22,
double ∗∗ d2m2,
double ∗ w8,
int nmu,
struct lensing ∗ ple )

This routine computes the lensed power spectra by Gaussian quadrature
Parameters
ksip

Input: Lensed correlation function (ksi+[index_mu])

ksim

Input: Lensed correlation function (ksi-[index_mu])

d22

Input: Wigner d-function ( dl22 [l][index_mu])

d2m2

Input: Wigner d-function ( dl2−2 [l][index_mu])

w8

Input: Legendre quadrature weights (w8[index_mu])

nmu

Input: Number of quadrature points (0<=index_mu<=nmu)

ple

Input/output: Pointer to the lensing structure

Returns
the error status

Integration by Gauss-Legendre quadrature.

5.7.2.10

lensing_addback_cl_ee_bb()

int lensing_addback_cl_ee_bb (
struct lensing ∗ ple,
double ∗ cl_ee,
double ∗ cl_bb )

This routine adds back the unlensed clee , clbb power spectra Used in case of fast (and BB inaccurate) integration of
correlation functions.
Parameters
ple

Input/output: Pointer to the lensing structure

cl_ee

Input: Array of unlensed power spectrum

cl_bb

Input: Array of unlensed power spectrum

Returns
the error status

5.7.2.11

lensing_d00()

int lensing_d00 (
double ∗ mu,
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File Documentation

int num_mu,
int lmax,
double ∗∗ d00 )

This routine computes the d00 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d00

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.12

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d11()

int lensing_d11 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d11 )

This routine computes the d11 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d11

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.13

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d1m1()

int lensing_d1m1 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d1m1 )

This routine computes the d1m1 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d1m1

Input/output: Result is stored here

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5.7 lensing.c File Reference

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.14

71

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d2m2()

int lensing_d2m2 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d2m2 )

This routine computes the d2m2 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d2m2

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.15

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d22()

int lensing_d22 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d22 )

This routine computes the d22 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d22

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.16

lensing_d20()

int lensing_d20 (
double ∗ mu,
int num_mu,
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p
(2l + 1)/2dlmm0 for stability Formulae from

72

File Documentation

int lmax,
double ∗∗ d20 )

This routine computes the d20 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d20

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.17

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d31()

int lensing_d31 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d31 )

This routine computes the d31 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d31

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.18

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d3m1()

int lensing_d3m1 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d3m1 )

This routine computes the d3m1 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d3m1

Input/output: Result is stored here
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5.7 lensing.c File Reference

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.19

73

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d3m3()

int lensing_d3m3 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d3m3 )

This routine computes the d3m3 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d3m3

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.20

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d40()

int lensing_d40 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d40 )

This routine computes the d40 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d40

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.21

lensing_d4m2()

int lensing_d4m2 (
double ∗ mu,
int num_mu,
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p
(2l + 1)/2dlmm0 for stability Formulae from

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File Documentation

int lmax,
double ∗∗ d4m2 )

This routine computes the d4m2 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d4m2

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.7.2.22

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing_d4m4()

int lensing_d4m4 (
double ∗ mu,
int num_mu,
int lmax,
double ∗∗ d4m4 )

This routine computes the d4m4 term
Parameters
mu

Input: Vector of cos(beta) values

num_mu

Input: Number of cos(beta) values

lmax

Input: maximum multipole

d4m4

Input/output: Result is stored here

Wigner d-functions, computed by recurrence actual recurrence on
Kostelec & Rockmore 2003

5.8

p
(2l + 1)/2dlmm0 for stability Formulae from

lensing.h File Reference

#include "spectra.h"
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5.8 lensing.h File Reference

75

Include dependency graph for lensing.h:

lensing.h

spectra.h

transfer.h

nonlinear.h

sys/shm.h

primordial.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

This graph shows which files directly or indirectly include this file:

lensing.h

output.h

output.c

class.h

class.c

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lensing.c

input.h

input.c

sys/stat.h

errno.h

76

File Documentation

Data Structures
• struct lensing

5.8.1

Detailed Description

Documented includes for spectra module

5.8.2

Data Structure Documentation

5.8.2.1

struct lensing

Structure containing everything about lensed spectra that other modules need to know.
Once initialized by lensing_init(), contains a table of all lensed Cl 's for the all modes (scalar/tensor), all types (TT,
TE...), and all pairs of initial conditions (adiabatic, isocurvatures...). FOR THE MOMENT, ASSUME ONLY SCALAR
& ADIABATIC
Data Fields
has_lensed_cls

do we need to compute lensed Cl 's at all ?

int

has_tt

do we want lensed ClT T ? (T = temperature)

int

has_ee

do we want lensed ClEE ? (E = E-polarization)

int

has_te

do we want lensed ClT E ?

int

has_bb

do we want ClBB ? (B = B-polarization)

int

has_pp

do we want Cl ? ( φ = CMB lensing potential)

int

has_tp

do we want Cl

int

has_dd

do we want Cldd ? (d = matter density)

int

has_td

do we want ClT d ?

int

has_ll

do we want Clll ? (l = lensing potential)

int

has_tl

do we want ClT l ?

int

index_lt_tt

index for type ClT T

int

index_lt_ee

index for type ClEE

int

index_lt_te

index for type ClT E

int

index_lt_bb

index for type ClBB

int

index_lt_pp

index for type Cl

int

index_lt_tp

index for type Cl

int

index_lt_dd

index for type Cldd

int

index_lt_td

index for type ClT d

int

index_lt_ll

index for type Cldd

int

index_lt_tl

index for type ClT d

int

lt_size

number of Cl types requested

int

l_unlensed_max

last multipole in all calculations (same as in spectra module)

int

l_lensed_max

last multipole at which lensed spectra are computed

short

φφ

Tφ

?

φφ
Tφ

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5.9 nonlinear.c File Reference

77

Data Fields
int

l_size

number of l values

l_max_lt

last multipole (given as an input) at which we want to output Cl 's for a given
mode and type

double ∗

l

table of multipole values l[index_l]

double ∗

cl_lens

table of anisotropy spectra for each multipole and types, cl[index_l ∗
ple->lt_size + index_lt]

double ∗

ddcl_lens

second derivatives for interpolation

lensing_verbose

flag regulating the amount of information sent to standard output (none if set to
zero)

error_message

zone for writing error messages

int ∗

short
ErrorMsg

5.9

nonlinear.c File Reference

#include "nonlinear.h"
Include dependency graph for nonlinear.c:

nonlinear.c

nonlinear.h

primordial.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int nonlinear_init (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt,
struct primordial ∗ppm, struct nonlinear ∗pnl)
• int nonlinear_halofit (struct precision ∗ppr, struct background ∗pba, struct primordial ∗ppm, struct nonlinear
∗pnl, double tau, double ∗pk_l, double ∗pk_nl, double ∗lnk_l, double ∗lnpk_l, double ∗ddlnpk_l, double ∗k_nl)
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78

File Documentation

5.9.1

Detailed Description

Documented nonlinear module
Julien Lesgourgues, 6.03.2014
New module replacing an older one present up to version 2.0 The new module is located in a better place in the
main, allowing it to compute non-linear correction to Cl 's and not just P (k). It will also be easier to generalize to
new methods. The old implementation of one-loop calculations and TRG calculations has been dropped from this
version, they can still be found in older versions.

5.9.2

Function Documentation

5.9.2.1

nonlinear_init()

int nonlinear_init (
struct precision ∗ ppr,
struct background ∗ pba,
struct
struct
struct
struct

thermo ∗ pth,
perturbs ∗ ppt,
primordial ∗ ppm,
nonlinear ∗ pnl )

Summary
(a) First deal with the case where non non-linear corrections requested
(b) Compute for HALOFIT non-linear spectrum

• copy list of (k,tau) from perturbation module
• loop over time

5.9.2.2

nonlinear_halofit()

int nonlinear_halofit (
struct precision ∗ ppr,
struct
struct
struct
double
double

background ∗ pba,
primordial ∗ ppm,
nonlinear ∗ pnl,
tau,
∗ pk_l,

double
double
double
double
double

∗
∗
∗
∗
∗

pk_nl,
lnk_l,
lnpk_l,
ddlnpk_l,
k_nl )

Determine non linear ratios (from pk)
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5.10 nonlinear.h File Reference

5.10

79

nonlinear.h File Reference

#include "primordial.h"
Include dependency graph for nonlinear.h:

nonlinear.h

primordial.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

This graph shows which files directly or indirectly include this file:

nonlinear.h

transfer.h

spectra.h

spectra.c

output.h

input.h

input.c

transfer.c

lensing.h

lensing.c

output.c

class.h

class.c

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nonlinear.c

80

File Documentation

Data Structures
• struct nonlinear

Macros
• #define _M_EV_TOO_BIG_FOR_HALOFIT_ 10.

5.10.1

Detailed Description

Documented includes for trg module

5.10.2

Macro Definition Documentation

5.10.2.1

_M_EV_TOO_BIG_FOR_HALOFIT_

#define _M_EV_TOO_BIG_FOR_HALOFIT_ 10.

above which value of non-CDM mass (in eV) do we stop trusting halofit?

5.11

output.c File Reference

#include "output.h"
Include dependency graph for output.c:

output.c

output.h

lensing.h

spectra.h

transfer.h

nonlinear.h

sys/shm.h

primordial.h

sys/stat.h

errno.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

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5.11 output.c File Reference

81

Functions
• int output_init (struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt, struct primordial ∗ppm, struct
transfers ∗ptr, struct spectra ∗psp, struct nonlinear ∗pnl, struct lensing ∗ple, struct output ∗pop)
• int output_cl (struct background ∗pba, struct perturbs ∗ppt, struct spectra ∗psp, struct lensing ∗ple, struct
output ∗pop)
• int output_pk (struct background ∗pba, struct perturbs ∗ppt, struct spectra ∗psp, struct output ∗pop)
• int output_pk_nl (struct background ∗pba, struct perturbs ∗ppt, struct spectra ∗psp, struct output ∗pop)
• int output_tk (struct background ∗pba, struct perturbs ∗ppt, struct spectra ∗psp, struct output ∗pop)
• int output_print_data (FILE ∗out, char titles[_MAXTITLESTRINGLENGTH_], double ∗dataptr, int size_←dataptr)
• int output_open_cl_file (struct spectra ∗psp, struct output ∗pop, FILE ∗∗clfile, FileName filename, char ∗first←_line, int lmax)
• int output_one_line_of_cl (struct background ∗pba, struct spectra ∗psp, struct output ∗pop, FILE ∗clfile, double l, double ∗cl, int ct_size)
• int output_open_pk_file (struct background ∗pba, struct spectra ∗psp, struct output ∗pop, FILE ∗∗pkfile, File←Name filename, char ∗first_line, double z)
• int output_one_line_of_pk (FILE ∗pkfile, double one_k, double one_pk)

5.11.1

Detailed Description

Documented output module
Julien Lesgourgues, 26.08.2010
This module writes the output in files.
The following functions can be called from other modules or from the main:
1. output_init() (must be called after spectra_init())
2. output_total_cl_at_l() (can be called even before output_init())
No memory needs to be deallocated after that, hence there is no output_free() routine like in other modules.

5.11.2

Function Documentation

5.11.2.1

output_init()

int output_init (
struct
struct
struct
struct
struct
struct

background ∗ pba,
thermo ∗ pth,
perturbs ∗ ppt,
primordial ∗ ppm,
transfers ∗ ptr,
spectra ∗ psp,

struct nonlinear ∗ pnl,
struct lensing ∗ ple,
struct output ∗ pop )

This routine writes the output in files.

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82

File Documentation

Parameters
pba

Input: pointer to background structure (needed for calling spectra_pk_at_z())

pth

Input: pointer to thermodynamics structure

ppt

Input: pointer perturbation structure

ppm

Input: pointer to primordial structure

ptr

Input: pointer to transfer structure

psp

Input: pointer to spectra structure

pnl

Input: pointer to nonlinear structure

ple

Input: pointer to lensing structure

pop

Input: pointer to output structure

Summary:

• check that we really want to output at least one file
• deal with all anisotropy power spectra Cl 's
• deal with all Fourier matter power spectra P(k)'s
• deal with density and matter power spectra
• deal with background quantities
• deal with thermodynamics quantities
• deal with perturbation quantities
• deal with primordial spectra

5.11.2.2

output_cl()

int output_cl (
struct
struct
struct
struct

background ∗ pba,
perturbs ∗ ppt,
spectra ∗ psp,
lensing ∗ ple,

struct output ∗ pop )

This routines writes the output in files for anisotropy power spectra Cl 's.
Parameters
pba

Input: pointer to background structure (needed for Tcmb )

ppt

Input: pointer perturbation structure

psp

Input: pointer to spectra structure

ple

Input: pointer to lensing structure

pop

Input: pointer to output structure

Summary:
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5.11 output.c File Reference

83

• define local variables
• first, allocate all arrays of files and Cl 's
• second, open only the relevant files, and write a heading in each of them
• third, perform loop over l. For each multipole, get all Cl 's by calling spectra_cl_at_l() and distribute the results
to relevant files
• finally, close files and free arrays of files and Cl 's

5.11.2.3

output_pk()

int output_pk (
struct
struct
struct
struct

background ∗ pba,
perturbs ∗ ppt,
spectra ∗ psp,
output ∗ pop )

This routines writes the output in files for Fourier matter power spectra P(k)'s.
Parameters
pba

Input: pointer to background structure (needed for calling spectra_pk_at_z())

ppt

Input: pointer perturbation structure

psp

Input: pointer to spectra structure

pop

Input: pointer to output structure

Summary:
• define local variables
• first, check that requested redshift z_pk is consistent
• second, open only the relevant files and write a heading in each of them
• third, compute P(k) for each k (if several ic's, compute it for each ic and compute also the total); if z_pk = 0,
this is done by directly reading inside the pre-computed table; if not, this is done by interpolating the table at
the correct value of tau.
• fourth, write in files
• fifth, free memory and close files

5.11.2.4

output_pk_nl()

int output_pk_nl (
struct
struct
struct
struct

background ∗ pba,
perturbs ∗ ppt,
spectra ∗ psp,
output ∗ pop )

This routines writes the output in files for Fourier non-linear matter power spectra P(k)'s.

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Parameters
pba

Input: pointer to background structure (needed for calling spectra_pk_at_z())

ppt

Input: pointer perturbation structure

psp

Input: pointer to spectra structure

pop

Input: pointer to output structure

Summary:

• define local variables
• first, check that requested redshift z_pk is consistent
• second, open only the relevant files, and write a heading in each of them
• third, compute P(k) for each k (if several ic's, compute it for each ic and compute also the total); if z_pk = 0,
this is done by directly reading inside the pre-computed table; if not, this is done by interpolating the table at
the correct value of tau.
• fourth, write in files
• fifth, free memory and close files

5.11.2.5

output_tk()

int output_tk (
struct background ∗ pba,
struct perturbs ∗ ppt,
struct spectra ∗ psp,
struct output ∗ pop )

This routines writes the output in files for matter transfer functions Ti (k)'s.
Parameters
pba

Input: pointer to background structure (needed for calling spectra_pk_at_z())

ppt

Input: pointer perturbation structure

psp

Input: pointer to spectra structure

pop

Input: pointer to output structure

Summary:
• define local variables
• first, check that requested redshift z_pk is consistent
• second, open only the relevant files, and write a heading in each of them
• free memory and close files

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5.11 output.c File Reference

5.11.2.6

85

output_print_data()

int output_print_data (
FILE ∗ out,
char titles[_MAXTITLESTRINGLENGTH_],
double ∗ dataptr,
int size_dataptr )

Summary

• First we print the titles
• Then we print the data

5.11.2.7

output_open_cl_file()

int output_open_cl_file (
struct spectra ∗ psp,
struct output ∗ pop,
FILE ∗∗ clfile,
FileName filename,
char ∗ first_line,
int lmax )

This routine opens one file where some Cl 's will be written, and writes a heading with some general information
concerning its content.
Parameters
psp

Input: pointer to spectra structure

pop

Input: pointer to output structure

clfile

Output: returned pointer to file pointer

filename

Input: name of the file

first_line

Input: text describing the content (mode, initial condition..)

lmax

Input: last multipole in the file (the first one is assumed to be 2)

Returns
the error status

Summary

• First we deal with the entries that are dependent of format type
• Next deal with entries that are independent of format type

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5.11.2.8

output_one_line_of_cl()

int output_one_line_of_cl (
struct
struct
struct
FILE ∗
double

background ∗ pba,
spectra ∗ psp,
output ∗ pop,
clfile,
l,

double ∗ cl,
int ct_size )

This routine write one line with l and all Cl 's for all types (TT, TE...)
Parameters
pba

Input: pointer to background structure (needed for Tcmb )

psp

Input: pointer to spectra structure

pop

Input: pointer to output structure

clfile

Input: file pointer

l

Input: multipole

cl

Input: Cl 's for all types

ct_size

Input: number of types

Returns
the error status

5.11.2.9

output_open_pk_file()

int output_open_pk_file (
struct background ∗ pba,
struct spectra ∗ psp,
struct output ∗ pop,
FILE ∗∗ pkfile,
FileName filename,
char ∗ first_line,
double z )

This routine opens one file where some P(k)'s will be written, and writes a heading with some general information
concerning its content.
Parameters
pba

Input: pointer to background structure (needed for h)

psp

Input: pointer to spectra structure

pop

Input: pointer to output structure

pkfile

Output: returned pointer to file pointer

filename

Input: name of the file

first_line

Input: text describing the content (initial conditions, ...)

z

Input: redshift of the output
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5.12 output.h File Reference
Returns
the error status

5.11.2.10

output_one_line_of_pk()

int output_one_line_of_pk (
FILE ∗ pkfile,
double one_k,
double one_pk )

This routine writes one line with k and P(k)
Parameters
pkfile

Input: file pointer

one_k

Input: wavenumber

one_pk

Input: matter power spectrum

Returns
the error status

5.12

output.h File Reference

#include "common.h"
#include "lensing.h"
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88

File Documentation

Include dependency graph for output.h:

output.h

lensing.h

spectra.h

transfer.h

nonlinear.h

sys/shm.h

primordial.h

sys/stat.h

errno.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

This graph shows which files directly or indirectly include this file:

output.h

input.h

class.h

output.c

input.c

class.c

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89

Data Structures
• struct output

Macros
• #define _Z_PK_NUM_MAX_ 100

5.12.1

Detailed Description

Documented includes for output module

5.12.2

Data Structure Documentation

5.12.2.1

struct output

Structure containing various informations on the output format, all of them initialized by user in input module.
Data Fields
FileName
int
double
short
enum file_format

root
z_pk_num
z_pk[_Z_PK_NUM_MAX_]
write_header

root for all file names
number of redshift at which P(k,z) and T_i(k,z) should be
written
value(s) of redshift at which P(k,z) and T_i(k,z) should be
written
flag stating whether we should write a header in output files

output_format

which format for output files (definitions, order of columns,
etc.)

short

write_background

flag for outputing background evolution in file

short

write_thermodynamics

flag for outputing thermodynamical evolution in file

short

write_perturbations

flag for outputing perturbations of selected wavenumber(s) in
file(s)

short

write_primordial

flag for outputing scalar/tensor primordial spectra in files

short

output_verbose

flag regulating the amount of information sent to standard
output (none if set to zero)

ErrorMsg

error_message

zone for writing error messages

5.12.3

Macro Definition Documentation

5.12.3.1

_Z_PK_NUM_MAX_

#define _Z_PK_NUM_MAX_ 100

Maximum number of values of redshift at which the spectra will be written in output files
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5.13

perturbations.c File Reference

#include "perturbations.h"
Include dependency graph for perturbations.c:

perturbations.c

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int perturb_sources_at_tau (struct perturbs ∗ppt, int index_md, int index_ic, int index_type, double tau, double
∗psource)
• int perturb_init (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt)
• int perturb_free (struct perturbs ∗ppt)
• int perturb_indices_of_perturbs (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct
perturbs ∗ppt)
• int perturb_timesampling_for_sources (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth,
struct perturbs ∗ppt)
• int perturb_get_k_list (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt)
• int perturb_workspace_init (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs
∗ppt, int index_md, struct perturb_workspace ∗ppw)
• int perturb_workspace_free (struct perturbs ∗ppt, int index_md, struct perturb_workspace ∗ppw)
• int perturb_solve (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt, int
index_md, int index_ic, int index_k, struct perturb_workspace ∗ppw)
• int perturb_prepare_output (struct background ∗pba, struct perturbs ∗ppt)
• int perturb_find_approximation_number (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth,
struct perturbs ∗ppt, int index_md, double k, struct perturb_workspace ∗ppw, double tau_ini, double tau_end,
int ∗interval_number, int ∗interval_number_of)
• int perturb_find_approximation_switches (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth,
struct perturbs ∗ppt, int index_md, double k, struct perturb_workspace ∗ppw, double tau_ini, double tau_end,
double precision, int interval_number, int ∗interval_number_of, double ∗interval_limit, int ∗∗interval_approx)
• int perturb_vector_init (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt,
int index_md, int index_ic, double k, double tau, struct perturb_workspace ∗ppw, int ∗pa_old)
• int perturb_vector_free (struct perturb_vector ∗pv)
• int perturb_initial_conditions (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, int index←_md, int index_ic, double k, double tau, struct perturb_workspace ∗ppw)
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• int perturb_approximations (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs
∗ppt, int index_md, double k, double tau, struct perturb_workspace ∗ppw)
• int perturb_timescale (double tau, void ∗parameters_and_workspace, double ∗timescale, ErrorMsg error_←message)
• int perturb_einstein (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt,
int index_md, double k, double tau, double ∗y, struct perturb_workspace ∗ppw)
• int perturb_total_stress_energy (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct
perturbs ∗ppt, int index_md, double k, double ∗y, struct perturb_workspace ∗ppw)
• int perturb_sources (double tau, double ∗y, double ∗dy, int index_tau, void ∗parameters_and_workspace,
ErrorMsg error_message)
• int perturb_print_variables (double tau, double ∗y, double ∗dy, void ∗parameters_and_workspace, ErrorMsg
error_message)
• int perturb_derivs (double tau, double ∗y, double ∗dy, void ∗parameters_and_workspace, ErrorMsg error_←message)
• int perturb_tca_slip_and_shear (double ∗y, void ∗parameters_and_workspace, ErrorMsg error_message)

5.13.1

Detailed Description

Documented perturbation module
Julien Lesgourgues, 23.09.2010
Deals with the perturbation evolution. This module has two purposes:
• at the beginning; to initialize the perturbations, i.e. to integrate the perturbation equations, and store temporarily the terms contributing to the source functions as a function of conformal time. Then, to perform a few
manipulations of these terms in order to infer the actual source functions S X (k, τ ), and to store them as a
function of conformal time inside an interpolation table.
• at any time in the code; to evaluate the source functions at a given conformal time (by interpolating within the
interpolation table).
Hence the following functions can be called from other modules:
1. perturb_init() at the beginning (but after background_init() and thermodynamics_init())
2. perturb_sources_at_tau() at any later time
3. perturb_free() at the end, when no more calls to perturb_sources_at_tau() are needed

5.13.2

Function Documentation

5.13.2.1

perturb_sources_at_tau()

int perturb_sources_at_tau (
struct perturbs ∗ ppt,
int index_md,
int index_ic,
int index_type,
double tau,
double ∗ psource )

Source function S X (k, τ ) at a given conformal time tau.
Evaluate source functions at given conformal time tau by reading the pre-computed table and interpolating.

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Parameters
ppt

Input: pointer to perturbation structure containing interpolation tables

index_md

Input: index of requested mode

index_ic

Input: index of requested initial condition

index_type

Input: index of requested source function type

tau

Input: any value of conformal time

psource

Output: vector (already allocated) of source function as a function of k

Returns
the error status

Summary:

• interpolate in pre-computed table contained in ppt

5.13.2.2

perturb_init()

int perturb_init (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt )

Initialize the perturbs structure, and in particular the table of source functions.
Main steps:

• given the values of the flags describing which kind of perturbations should be considered (modes←: scalar/vector/tensor, initial conditions, type of source functions needed...), initialize indices and wavenumber
list
• define the time sampling for the output source functions
• for each mode (scalar/vector/tensor): initialize the indices of relevant perturbations, integrate the differential
system, compute and store the source functions.

Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

ppt

Output: Initialized perturbation structure

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Returns
the error status

Summary:

• define local variables
• perform preliminary checks
• initialize all indices and lists in perturbs structure using perturb_indices_of_perturbs()
• define the common time sampling for all sources using perturb_timesampling_for_sources()
• if we want to store perturbations, write titles and allocate storage
• create an array of workspaces in multi-thread case
• loop over modes (scalar, tensors, etc). For each mode:
• –> (a) create a workspace (one per thread in multi-thread case)
• –> (b) initialize indices of vectors of perturbations with perturb_indices_of_current_vectors()
• –> (c) loop over initial conditions and wavenumbers; for each of them, evolve perturbations and compute
source functions with perturb_solve()

5.13.2.3

perturb_free()

int perturb_free (
struct perturbs ∗ ppt )

Free all memory space allocated by perturb_init().
To be called at the end of each run, only when no further calls to perturb_sources_at_tau() are needed.
Parameters
ppt

Input: perturbation structure to be freed

Returns
the error status

Stuff related to perturbations output:

• Free non-NULL pointers

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5.13.2.4

perturb_indices_of_perturbs()

int perturb_indices_of_perturbs (
struct
struct
struct
struct

precision ∗ ppr,
background ∗ pba,
thermo ∗ pth,
perturbs ∗ ppt )

Initialize all indices and allocate most arrays in perturbs structure.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

ppt

Input/Output: Initialized perturbation structure

Returns
the error status
Summary:
• define local variables
• count modes (scalar, vector, tensor) and assign corresponding indices
• allocate array of number of types for each mode, ppt->tp_size[index_md]
• allocate array of number of initial conditions for each mode, ppt->ic_size[index_md]
• allocate array of arrays of source functions for each mode, ppt->source[index_md]
• initialization of all flags to false (will eventually be set to true later)
• source flags and indices, for sources that all modes have in common (temperature, polarization, ...). For
temperature, the term t2 is always non-zero, while other terms are non-zero only for scalars and vectors. For
polarization, the term e is always non-zero, while the term b is only for vectors and tensors.
• define k values with perturb_get_k_list()
• loop over modes. Initialize flags and indices which are specific to each mode.
• (a) scalars
• –> source flags and indices, for sources that are specific to scalars
• –> count scalar initial conditions (for scalars: ad, cdi, nid, niv; for tensors: only one) and assign corresponding
indices
• (b) vectors
• –> source flags and indices, for sources that are specific to vectors
• –> initial conditions for vectors
• (c) tensors
• –> source flags and indices, for sources that are specific to tensors
• –> only one initial condition for tensors
• (d) for each mode, allocate array of arrays of source functions for each initial conditions and wavenumber,
(ppt->source[index_md])[index_ic][index_type]

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5.13.2.5

95

perturb_timesampling_for_sources()

int perturb_timesampling_for_sources (
struct
struct
struct
struct

precision ∗ ppr,
background ∗ pba,
thermo ∗ pth,
perturbs ∗ ppt )

Define time sampling for source functions.
For each type, compute the list of values of tau at which sources will be sampled. Knowing the number of tau values,
allocate all arrays of source functions.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

ppt

Input/Output: Initialized perturbation structure

Returns
the error status
Summary:
• define local variables
• allocate background/thermodynamics vectors
• first, just count the number of sampling points in order to allocate the array containing all values
• (a) if CMB requested, first sampling point = when the universe stops being opaque; otherwise, start sampling
gravitational potential at recombination [however, if perturbed recombination is requested, we also need to
start the system before recombination. Otherwise, the initial conditions for gas temperature and ionization
fraction perturbations (delta_T = 1/3 delta_b, delta_x_e) are not valid].
• (b) next sampling point = previous + ppr->perturb_sampling_stepsize ∗ timescale_source, where:
• –> if CMB requested: timescale_source1 = |g/ġ| = |κ̇ − κ̈/κ̇|−1 ; timescale_source2 = |2ä/a − (ȧ/a)2 |−1/2
(to sample correctly the late ISW effect; and timescale_source=1/(1/timescale_source1+1/timescale_←source2); repeat till today.
• –> if CMB not requested: timescale_source = 1/aH; repeat till today.
• –> infer total number of time steps, ppt->tau_size
• –> allocate array of time steps, ppt->tau_sampling[index_tau]
• –> repeat the same steps, now filling the array with each tau value:
• –> (b.1.) first sampling point = when the universe stops being opaque
• –> (b.2.) next sampling point = previous + ppr->perturb_sampling_stepsize ∗ timescale_source, where
timescale_source1 = |g/ġ| = |κ̇ − κ̈/κ̇|−1 ; timescale_source2 = |2ä/a − (ȧ/a)2 |−1/2 (to sample correctly
the late ISW effect; and timescale_source=1/(1/timescale_source1+1/timescale_source2); repeat till today. If
CMB not requested: timescale_source = 1/aH; repeat till today.
• last sampling point = exactly today
• loop over modes, initial conditions and types. For each of them, allocate array of source functions.

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5.13.2.6

perturb_get_k_list()

int perturb_get_k_list (
struct
struct
struct
struct

precision ∗ ppr,
background ∗ pba,
thermo ∗ pth,
perturbs ∗ ppt )

Define the number of comoving wavenumbers using the information passed in the precision structure.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

ppt

Input: pointer to perturbation structure

Returns
the error status

Summary:

• allocate arrays related to k list for each mode
• scalar modes
• –> find k_max (as well as k_max_cmb[ppt->index_md_scalars], k_max_cl[ppt->index_md_scalars])
• –> test that result for k_min, k_max make sense
• vector modes
• –> find k_max (as well as k_max_cmb[ppt->index_md_vectors], k_max_cl[ppt->index_md_vectors])
• –> test that result for k_min, k_max make sense
• tensor modes
• –> find k_max (as well as k_max_cmb[ppt->index_md_tensors], k_max_cl[ppt->index_md_tensors])
• –> test that result for k_min, k_max make sense
• If user asked for k_output_values, add those to all k lists:
• –> Find indices in ppt->k[index_md] corresponding to 'k_output_values'. We are assuming that ppt->k is
sorted and growing, and we have made sure that ppt->k_output_values is also sorted and growing.
• –> Decide if we should add k_output_value now. This has to be this complicated, since we can only compare
the k-values when both indices are in range.
• –> The two MIN statements are here because in a normal run, the cl and cmb arrays contain a single k value
larger than their respective k_max. We are mimicking this behavior.
• finally, find the global k_min and k_max for the ensemble of all modes 9scalars, vectors, tensors)

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5.13.2.7

97

perturb_workspace_init()

int perturb_workspace_init (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt,
int index_md,
struct perturb_workspace ∗ ppw )

Initialize a perturb_workspace structure. All fields are allocated here, with the exception of the perturb_vector '–>pv'
field, which is allocated separately in perturb_vector_init. We allocate one such perturb_workspace structure per
thread and per mode (scalar/../tensor). Then, for each thread, all initial conditions and wavenumbers will use the
same workspace.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to the thermodynamics structure

ppt

Input: pointer to the perturbation structure

index_md

Input: index of mode under consideration (scalar/.../tensor)

ppw

Input/Output: pointer to perturb_workspace structure which fields are allocated or filled here

Returns
the error status

Summary:

• define local variables
• Compute maximum l_max for any multipole
• Allocate sl [ ] array for freestreaming of multipoles (see arXiv:1305.3261) and initialize to 1.0, which is the K=0
value.
• define indices of metric perturbations obeying constraint equations (this can be done once and for all, because the vector of metric perturbations is the same whatever the approximation scheme, unlike the vector of
quantities to be integrated, which is allocated separately in perturb_vector_init)
• allocate some workspace in which we will store temporarily the values of background, thermodynamics, metric
and source quantities at a given time
• count number of approximations, initialize their indices, and allocate their flags
• For definiteness, initialize approximation flags to arbitrary values (correct values are overwritten in pertub_←find_approximation_switches)
• allocate fields where some of the perturbations are stored

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5.13.2.8

perturb_workspace_free()

int perturb_workspace_free (
struct perturbs ∗ ppt,
int index_md,
struct perturb_workspace ∗ ppw )

Free the perturb_workspace structure (with the exception of the perturb_vector '–>pv' field, which is freed separately in perturb_vector_free).
Parameters
ppt

Input: pointer to the perturbation structure

index_md

Input: index of mode under consideration (scalar/.../tensor)

ppw

Input: pointer to perturb_workspace structure to be freed

Returns
the error status

5.13.2.9

perturb_solve()

int perturb_solve (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt,
int index_md,
int index_ic,
int index_k,
struct perturb_workspace ∗ ppw )

Solve the perturbation evolution for a given mode, initial condition and wavenumber, and compute the corresponding
source functions.
For a given mode, initial condition and wavenumber, this function finds the time ranges over which the perturbations
can be described within a given approximation. For each such range, it initializes (or redistributes) perturbations
using perturb_vector_init(), and integrates over time. Whenever a "source sampling time" is passed, the source
terms are computed and stored in the source table using perturb_sources().
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to the thermodynamics structure

ppt

Input/Output: pointer to the perturbation structure (output source functions S(k,tau) written here)

index_md

Input: index of mode under consideration (scalar/.../tensor)

index_ic

Input: index of initial condition under consideration (ad, iso...)

index_k

Input: index of wavenumber

ppw

Input: pointer to perturb_workspace structure containing index values and workspaces
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Returns
the error status

Summary:

• define local variables
• initialize indices relevant for back/thermo tables search
• get wavenumber value
• If non-zero curvature, update array of free-streaming coefficients ppw->s_l
• maximum value of tau for which sources are calculated for this wavenumber
• using bisection, compute minimum value of tau for which this wavenumber is integrated
• find the number of intervals over which approximation scheme is constant
• fill the structure containing all fixed parameters, indices and workspaces needed by perturb_derivs
• check whether we need to print perturbations to a file for this wavenumber
• loop over intervals over which approximation scheme is uniform. For each interval:
• –> (a) fix the approximation scheme
• –> (b) get the previous approximation scheme. If the current interval starts from the initial time tau_ini,
the previous approximation is set to be a NULL pointer, so that the function perturb_vector_init() knows that
perturbations must be initialized
• –> (c) define the vector of perturbations to be integrated over. If the current interval starts from the initial time
tau_ini, fill the vector with initial conditions for each mode. If it starts from an approximation switching point,
redistribute correctly the perturbations from the previous to the new vector of perturbations.
• –> (d) integrate the perturbations over the current interval.
• if perturbations were printed in a file, close the file
• fill the source terms array with zeros for all times between the last integrated time tau_max and tau_today.
• free quantities allocated at the beginning of the routine

5.13.2.10

perturb_prepare_output()

int perturb_prepare_output (
struct background ∗ pba,
struct perturbs ∗ ppt )

Write titles for all perturbations that we would like to print/store.
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5.13.2.11

perturb_find_approximation_number()

int perturb_find_approximation_number (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt,
int index_md,
double k,
struct perturb_workspace ∗ ppw,
double tau_ini,
double tau_end,
int ∗ interval_number,
int ∗ interval_number_of )

For a given mode and wavenumber, find the number of intervals of time between tau_ini and tau_end such that the
approximation scheme (and the number of perturbation equations) is uniform.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to the thermodynamics structure

ppt

Input: pointer to the perturbation structure

index_md

Input: index of mode under consideration (scalar/.../tensor)

k

Input: index of wavenumber

ppw

Input: pointer to perturb_workspace structure containing index values and workspaces

tau_ini

Input: initial time of the perturbation integration

tau_end

Input: final time of the perturbation integration

interval_number

Output: total number of intervals

interval_number←_of

Output: number of intervals with respect to each particular approximation

Returns
the error status

Summary:

• fix default number of intervals to one (if no approximation switch)
• loop over each approximation and add the number of approximation switching times

5.13.2.12

perturb_find_approximation_switches()

int perturb_find_approximation_switches (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
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struct perturbs ∗ ppt,
int index_md,
double k,
struct perturb_workspace ∗ ppw,
double tau_ini,
double tau_end,
double precision,
int interval_number,
int ∗ interval_number_of,
double ∗ interval_limit,
int ∗∗ interval_approx )

For a given mode and wavenumber, find the values of time at which the approximation changes.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to the thermodynamics structure

ppt

Input: pointer to the perturbation structure

index_md

Input: index of mode under consideration (scalar/.../tensor)

k

Input: index of wavenumber

ppw

Input: pointer to perturb_workspace structure containing index values and workspaces

tau_ini

Input: initial time of the perturbation integration

tau_end

Input: final time of the perturbation integration

precision

Input: tolerance on output values

interval_number

Input: total number of intervals

interval_number←_of

Input: number of intervals with respect to each particular approximation

interval_limit

Output: value of time at the boundary of the intervals: tau_ini, tau_switch1, ..., tau_end

interval_approx

Output: value of approximations in each interval

Returns
the error status

Summary:

• write in output arrays the initial time and approximation
• if there are no approximation switches, just write final time and return
• if there are switches, consider approximations one after each other. Find switching time by bisection. Store
all switches in arbitrary order in array unsorted_tau_switch[ ]
• now sort interval limits in correct order
• store each approximation in chronological order

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5.13.2.13

perturb_vector_init()

int perturb_vector_init (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt,
int index_md,
int index_ic,
double k,
double tau,
struct perturb_workspace ∗ ppw,
int ∗ pa_old )

Initialize the field '–>pv' of a perturb_workspace structure, which is a perturb_vector structure. This structure contains indices and values of all quantities which need to be integrated with respect to time (and only them: quantities
fixed analytically or obeying constraint equations are NOT included in this vector). This routine distinguishes between two cases:
–> the input pa_old is set to the NULL pointer:
This happens when we start integrating over a new wavenumber and we want to set initial conditions for the perturbations. Then, it is assumed that ppw–>pv is not yet allocated. This routine allocates it, defines all indices, and
then fills the vector ppw–>pv–>y with the initial conditions defined in perturb_initial_conditions.
–> the input pa_old is not set to the NULL pointer and describes some set of approximations:
This happens when we need to change approximation scheme while integrating over a given wavenumber. The
new approximation described by ppw–>pa is then different from pa_old. Then, this routine allocates a new vector
with a new size and new index values; it fills this vector with initial conditions taken from the previous vector passed
as an input in ppw–>pv, and eventually with some analytic approximations for the new variables appearing at this
time; then the new vector comes in replacement of the old one, which is freed.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to the thermodynamics structure

ppt

Input: pointer to the perturbation structure

index_md

Input: index of mode under consideration (scalar/.../tensor)

index_ic

Input: index of initial condition under consideration (ad, iso...)

k

Input: wavenumber

tau

Input: conformal time

ppw

Input/Output: workspace containing in input the approximation scheme, the
background/thermodynamics/metric quantities, and eventually the previous vector y; and in output
the new vector y.

pa_old

Input: NULL is we need to set y to initial conditions for a new wavenumber; points towards a
perturb_approximations if we want to switch of approximation.

Returns
the error status

Summary:
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• define local variables
• allocate a new perturb_vector structure to which ppw–>pv will point at the end of the routine
• initialize pointers to NULL (they will be allocated later if needed), relevant for perturb_vector_free()
• define all indices in this new vector (depends on approximation scheme, described by the input structure
ppw–>pa)
• (a) metric perturbations V or hv depending on gauge
• (b) metric perturbation h is a propagating degree of freedom, so h and hdot are included in the vector of
ordinary perturbations, no in that of metric perturbations
• allocate vectors for storing the values of all these quantities and their time-derivatives at a given time
• specify which perturbations are needed in the evaluation of source terms
• case of setting initial conditions for a new wavenumber
• –> (a) check that current approximation scheme is consistent with initial conditions
• –> (b) let ppw–>pv points towards the perturb_vector structure that we just created
• –> (c) fill the vector ppw–>pv–>y with appropriate initial conditions
• case of switching approximation while a wavenumber is being integrated
• –> (a) for the scalar mode:
• —> (a.1.) check that the change of approximation scheme makes sense (note: before calling this routine
there is already a check that we wish to change only one approximation flag at a time)
• —> (a.2.) some variables (b, cdm, fld, ...) are not affected by any approximation. They need to be reconducted whatever the approximation switching is. We treat them here. Below we will treat other variables case
by case.
• –> (b) for the vector mode
• —> (b.1.) check that the change of approximation scheme makes sense (note: before calling this routine
there is already a check that we wish to change only one approximation flag at a time)
• —> (b.2.) some variables (gw, gwdot, ...) are not affected by any approximation. They need to be reconducted whatever the approximation switching is. We treat them here. Below we will treat other variables case
by case.
• –> (c) for the tensor mode
• —> (c.1.) check that the change of approximation scheme makes sense (note: before calling this routine
there is already a check that we wish to change only one approximation flag at a time)
• —> (c.2.) some variables (gw, gwdot, ...) are not affected by any approximation. They need to be reconducted whatever the approximation switching is. We treat them here. Below we will treat other variables case
by case.
• –> (d) free the previous vector of perturbations
• –> (e) let ppw–>pv points towards the perturb_vector structure that we just created

5.13.2.14

perturb_vector_free()

int perturb_vector_free (
struct perturb_vector ∗ pv )

Free the perturb_vector structure.

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Parameters
pv

Input: pointer to perturb_vector structure to be freed

Returns
the error status

5.13.2.15

perturb_initial_conditions()

int perturb_initial_conditions (
struct precision ∗ ppr,
struct background ∗ pba,
struct perturbs ∗ ppt,
int index_md,
int index_ic,
double k,
double tau,
struct perturb_workspace ∗ ppw )

For each mode, wavenumber and initial condition, this function initializes in the vector all values of perturbed variables (in a given gauge). It is assumed here that all values have previously been set to zero, only non-zero values
are set here.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

ppt

Input: pointer to the perturbation structure

index_md

Input: index of mode under consideration (scalar/.../tensor)

index_ic

Input: index of initial condition under consideration (ad, iso...)

k

Input: wavenumber

tau

Input: conformal time

ppw

Input/Output: workspace containing in input the approximation scheme, the
background/thermodynamics/metric quantities, and eventually the previous vector y; and in output
the new vector y.

Returns
the error status

Summary:
–> Declare local variables
–> For scalars

• (a) compute relevant background quantities: compute rho_r, rho_m, rho_nu (= all relativistic except photons),
and their ratio.
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• (b) starts by setting everything in synchronous gauge. If another gauge is needed, we will perform a gauge
transformation below.
• –> (b.1.) adiabatic
• —> Canonical field (solving for the perturbations): initial perturbations set to zero, they should reach the
attractor soon enough.
• —> TODO: Incorporate the attractor IC from 1004.5509. delta_phi = −(a/k)2 /φ0 (ρ + p)θ , delta_phi_prime
= a2 /φ0 (delta_rho_phi + V'delta_phi), and assume theta, delta_rho as for perfect fluid with c2s = 1 and w =
1/3 (ASSUMES radiation TRACKING)
• –> (b.2.) Cold dark matter Isocurvature
• –> (b.3.) Baryon Isocurvature
• –> (b.4.) Neutrino density Isocurvature
• –> (b.5.) Neutrino velocity Isocurvature
• (c) If the needed gauge is really the synchronous gauge, we need to affect the previously computed value of
eta to the actual variable eta
• (d) If the needed gauge is the newtonian gauge, we must compute alpha and then perform a gauge transformation for each variable
• (e) In any gauge, we should now implement the relativistic initial conditions in ur and ncdm variables

–> For tensors
tensor initial conditions take into account the fact that scalar (resp. tensor) Cl 's are related to the real space power
spectrum of curvature (resp. of the tensor part of metric perturbations)

< R(x)R(x) >

X

< hij (x)hij (x) >

ij

In momentum space it is conventional to use the modes R(k) and h(k) where the quantity h obeying to the equation
of propagation:

h00 +

2a0
h + [k2 + 2K]h = 12πGa2(ρ + p)σ = 8πGa2pπ
a

and the power spectra in real space and momentum space are related through:


 Z
dk k 3
dk
∗
< R(x)R(x) >=
< R(k)R(k) > =
PR (k)
k 2π 2
k

 Z
 2
 2
X
k
dk
k
dk k 3
∗
< hij (x)hij (x) >=
F
<
h(k)h(k)
>
=
F
Ph (k)
2
k
2π
K
k
K
ij
Z

where PR and Ph are the dimensionless spectrum of curvature R, and F is a function of k2/K, where K is the curvature parameter. F is equal to one in flat space (K=0), and coming from the contraction of the laplacian eigentensor
Qij with itself. We will give F explicitly below.
Similarly the scalar (S) and tensor (T) Cl 's are given by
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Z

dk S
[∆l (q)]2 PR (k)
k
 2
Z
k
dk T
ClT = 4π
[∆l (q)]2 F
Ph (k)
k
K
ClS

= 4π

The usual convention for the tensor-to-scalar ratio r = At /As at pivot scale = 16 epsilon in single-field inflation is
such that for constant PR (k) and Ph (k),

r=6

Ph (k)
PR (k)

so

Ph (k) =

PR (k)r
As r
At
=
=
6
6
6

A priori it would make sense to say that for a power-law primordial spectrum there is an extra factor (k/kpivot )nt
(and eventually running and so on and so forth...)
However it has been shown that the minimal models of inflation in a negatively curved bubble lead to Ph (k) =
tanh(π ∗ ν/2). In open models it is customary to define the tensor tilt in a non-flat universe as a deviation from this
behavior rather than from true scale-invariance in the above sense.
Hence we should have

Ph (k) =

ν
At
[tanh(π ∗ )](k/kpivot )(nt +...)
6
2

where the brackets

[...]
mean "if K<0"
Then

ClT

Z
= 4π

dk T
[∆l (q)]2 F
k



k2
K



At
ν
[tanh(π ∗ )](k/kpivot )(nt +...)
6
2

In the code, it is then a matter of choice to write:

• In the primordial module: Ph (k) =

At
6

tanh (π ∗ ν2 )(k/k ∗ )nT

• In the perturbation initial conditions: h = 1
• In the spectra module: ClT =

4
π

R

dk
T
2
k [∆l (q)] F



k2
K



Ph (k)

or:

• In the primordial module: Ph (k) = At (k/k ∗ )nT
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• In the perturbation initial conditions: h =
• In the spectra module: ClT =

4
π

R

q

k2
K

[F



/6] tanh (π ∗ ν2 )

dk
T
2
k [∆l (q)] Ph (k)

We choose this last option, such that the primordial and spectra module differ minimally in flat and non-flat space.
Then we must impose

s
h=

F
6


tanh (π ∗

ν
)
2

The factor F is found to be given by:

X

Z

ij

< hij (x)h (x) >=

ij

k2(k2 − K)
dk
Ph (k)
k (k2 + 3K)(k2 + 2K)

Introducing as usual q2 = k2 − 3K and using qdq = kdk this gives

X

< hij (x)hij (x) >=

dk (q2 − 3K)(q2 − 4K)
Ph (k)
k
q2(q2 − K)

Z

ij

Using qdq = kdk this is equivalent to

X

< hij (x)hij (x) >=

ij

p

Finally, introducing ν = q/

Z

dq q2 − 4K
Ph (k(q))
q q2 − K

|K| and sgnK=SIGN(k) = ±1, this could also be written

X
ij

ij

< hij (x)h (x) >=

Z

dν (ν2 − 4sgnK)
Ph (k(ν))
ν (ν2 − sgnK)

Equation (43,44) of Hu, Seljak, White, Zaldarriaga is equivalent to absorbing the above factor (ν2 − 4sgnK)/(ν2 −
sgnK) in the definition of the primordial spectrum. Since the initial condition should be written in terms of k rather
than nu, they should read

r
h=

[k2(k2 − K)]/[(k2 + 3K)(k2 + 2K)]/6 ∗ tanh (π ∗

ν
)
2

We leave the freedom to multiply by an arbitrary number ppr->gw_ini. The standard convention corresponding to
standard definitions of r, AT , nT is however ppr->gw_ini=1.
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5.13.2.16

perturb_approximations()

int perturb_approximations (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt,
int index_md,
double k,
double tau,
struct perturb_workspace ∗ ppw )

Evaluate background/thermodynamics at τ , infer useful flags / time scales for integrating perturbations.
Evaluate background quantities at τ , as well as thermodynamics for scalar mode; infer useful flags and time scales
for integrating the perturbations:

• check whether tight-coupling approximation is needed.
• check whether radiation (photons, massless neutrinos...) perturbations are needed.
• choose step of integration: step = ppr->perturb_integration_stepsize ∗ min_time_scale, where min_time_←scale = smallest time scale involved in the equations. There are three time scales to compare:
1. that of recombination, τc = 1/κ0
2. Hubble time scale, τh = a/a0
3. Fourier mode, τk = 1/k

So, in general, min_time_scale = min(τc , τb , τh , τk ).
However, if τc  τh and τc  τk , we can use the tight-coupling regime for photons and write equations in such
way that the time scale τc becomes irrelevant (no effective mass term in 1/τc ). Then, the smallest scale in the
equations is only min(τh , τk ). In practise, it is sufficient to use only the condition τc  τh .
Also, if ρmatter  ρradiation and k  aH , we can switch off radiation perturbations (i.e. switch on the freestreaming approximation) and then the smallest scale is simply τh .
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

ppt

Input: pointer to the perturbation structure

index_md

Input: index of mode under consideration (scalar/.../tensor)

k

Input: wavenumber

tau

Input: conformal time

ppw

Input/Output: in output contains the approximation to be used at this time

Returns
the error status

Summary:
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• define local variables
• compute Fourier mode time scale = τk = 1/k
• evaluate background quantities with background_at_tau() and Hubble time scale τh = a/a0
• for scalar modes:
• –> (a) evaluate thermodynamical quantities with thermodynamics_at_z()
• —> (b.1.) if κ0 = 0, recombination is finished; tight-coupling approximation must be off
• —> (b.2.) if κ0 6= 0, recombination is not finished: check tight-coupling approximation
• -—> (b.2.a) compute recombination time scale for photons, τγ = 1/κ0
• -—> (b.2.b) check whether tight-coupling approximation should be on
• –> (c) free-streaming approximations
• for tensor modes:
• –> (a) evaluate thermodynamical quantities with thermodynamics_at_z()
• —> (b.1.) if κ0 = 0, recombination is finished; tight-coupling approximation must be off
• —> (b.2.) if κ0 6= 0, recombination is not finished: check tight-coupling approximation
• -—> (b.2.a) compute recombination time scale for photons, τγ = 1/κ0
• -—> (b.2.b) check whether tight-coupling approximation should be on

5.13.2.17

perturb_timescale()

int perturb_timescale (
double tau,
void ∗ parameters_and_workspace,
double ∗ timescale,
ErrorMsg error_message )

Compute typical timescale over which the perturbation equations vary. Some integrators (e.g. Runge-Kunta) benefit
from calling this routine at each step in order to adapt the next step.
This is one of the few functions in the code which is passed to the generic_integrator() routine. Since generic_←integrator() should work with functions passed from various modules, the format of the arguments is a bit special:

• fixed parameters and workspaces are passed through a generic pointer. generic_integrator() doesn't know
the content of this pointer.
• the error management is a bit special: errors are not written as usual to pth->error_message, but to a generic
error_message passed in the list of arguments.

Parameters
tau

Input: conformal time

parameters_and_workspace

Input: fixed parameters (e.g. indices), workspace, approximation used, etc.

timescale

Output: perturbation variation timescale (given the approximation used)

error_message

Output: error message

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Summary:

• define local variables
• extract the fields of the parameter_and_workspace input structure
• compute Fourier mode time scale = τk = 1/k
• evaluate background quantities with background_at_tau() and Hubble time scale τh = a/a0
• for scalars modes:
• –> compute recombination time scale for photons, τγ = 1/κ0
• for vector modes:
• –> compute recombination time scale for photons, τγ = 1/κ0
• for tensor modes:
• –> compute recombination time scale for photons, τγ = 1/κ0

5.13.2.18

perturb_einstein()

int perturb_einstein (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt,
int index_md,
double k,
double tau,
double ∗ y,
struct perturb_workspace ∗ ppw )

Compute metric perturbations (those not integrated over time) using Einstein equations
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

ppt

Input: pointer to the perturbation structure

index_md

Input: index of mode under consideration (scalar/.../tensor)

k

Input: wavenumber

tau

Input: conformal time

y

Input: vector of perturbations (those integrated over time) (already allocated)

ppw

Input/Output: in output contains the updated metric perturbations

Returns
the error status
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Summary:
• define local variables
• define wavenumber and scale factor related quantities
• sum up perturbations from all species
• for scalar modes:
• –> infer metric perturbations from Einstein equations
• for vector modes
• for tensor modes

5.13.2.19

perturb_total_stress_energy()

int perturb_total_stress_energy (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct perturbs ∗ ppt,
int index_md,
double k,
double ∗ y,
struct perturb_workspace ∗ ppw )

Summary:
• define local variables
• wavenumber and scale factor related quantities
• for scalar modes
• –> (a) deal with approximation schemes
• —> (a.1.) photons
• -—> (a.1.1.) no approximation
• -—> (a.1.2.) radiation streaming approximation
• -—> (a.1.3.) tight coupling approximation
• —> (a.2.) ur
• –> (b) compute the total density, velocity and shear perturbations
• for vector modes
• –> photon contribution to vector sources:
• –> baryons
• for tensor modes
• –> photon contribution to gravitational wave source:
• –> ur contribution to gravitational wave source:
• –> ncdm contribution to gravitational wave source:

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5.13.2.20

perturb_sources()

int perturb_sources (
double tau,
double ∗ y,
double ∗ dy,
int index_tau,
void ∗ parameters_and_workspace,
ErrorMsg error_message )

Compute the source functions (three terms for temperature, one for E or B modes, etc.)
This is one of the few functions in the code which is passed to the generic_integrator() routine. Since generic_←integrator() should work with functions passed from various modules, the format of the arguments is a bit special:

• fixed parameters and workspaces are passed through a generic pointer. generic_integrator() doesn't know
the content of this pointer.
• the error management is a bit special: errors are not written as usual to pth->error_message, but to a generic
error_message passed in the list of arguments.

Parameters
tau

Input: conformal time

y

Input: vector of perturbations

dy

Input: vector of time derivative of perturbations

index_tau

Input: index in the array tau_sampling

parameters_and_workspace

Input/Output: in input, all parameters needed by perturb_derivs, in output,
source terms
Output: error message

error_message

Returns
the error status
Summary:
• define local variables
• rename structure fields (just to avoid heavy notations)
• get background/thermo quantities in this point
• for scalars
• –> compute metric perturbations
• –> compute quantities depending on approximation schemes
• –> for each type, compute source terms
• for tensors
• –> compute quantities depending on approximation schemes

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5.13.2.21

113

perturb_print_variables()

int perturb_print_variables (
double tau,
double ∗ y,
double ∗ dy,
void ∗ parameters_and_workspace,
ErrorMsg error_message )

When testing the code or a cosmological model, it can be useful to output perturbations at each step of integration
(and not just the delta's at each source sampling point, which is achieved simply by asking for matter transfer
functions). Then this function can be passed to the generic_evolver routine.
By default, instead of passing this function to generic_evolver, one passes a null pointer. Then this function is just
not used.
Parameters
tau

Input: conformal time

y

Input: vector of perturbations

dy

Input: vector of its derivatives (already allocated)

parameters_and_workspace

Input: fixed parameters (e.g. indices)

error_message

Output: error message

Summary:

• define local variables
• ncdm sector begins
• ncdm sector ends
• rename structure fields (just to avoid heavy notations)
• calculate perturbed recombination
• for scalar modes
• –> Get delta, deltaP/rho, theta, shear and store in array
• –> Do gauge transformation of delta, deltaP/rho (?) and theta using -= 3aH(1+w_ncdm) alpha for delta.
• –> Handle (re-)allocation
• for tensor modes:
• –> Handle (re-)allocation

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5.13.2.22

perturb_derivs()

int perturb_derivs (
double tau,
double ∗ y,
double ∗ dy,
void ∗ parameters_and_workspace,
ErrorMsg error_message )

Compute derivative of all perturbations to be integrated
For each mode (scalar/vector/tensor) and each wavenumber k, this function computes the derivative of all values in
the vector of perturbed variables to be integrated.
This is one of the few functions in the code which is passed to the generic_integrator() routine. Since generic_←integrator() should work with functions passed from various modules, the format of the arguments is a bit special:

• fixed parameters and workspaces are passed through a generic pointer. generic_integrator() doesn't know
what the content of this pointer is.
• errors are not written as usual in pth->error_message, but in a generic error_message passed in the list of
arguments.

Parameters
tau

Input: conformal time

y

Input: vector of perturbations

dy

Output: vector of its derivatives (already allocated)

parameters_and_workspace

Input/Output: in input, fixed parameters (e.g. indices); in output, background
and thermo quantities evaluated at tau.

error_message

Output: error message

Summary:

• define local variables
• rename the fields of the input structure (just to avoid heavy notations)
• get background/thermo quantities in this point
• get metric perturbations with perturb_einstein()
• compute related background quantities
• Compute 'generalised cotK function of argument
arXiv:1305.3261):

p

|K| ∗ τ , for closing hierarchy. (see equation 2.34 in

• for scalar modes:
• –> (a) define short-cut notations for the scalar perturbations
• –> (b) perturbed recombination
• –> (c) compute metric-related quantities (depending on gauge; additional gauges can be coded below)
– Each continuity equation contains a term in (theta+metric_continuity) with metric_continuity = (h_←prime/2) in synchronous gauge, (-3 phi_prime) in newtonian gauge
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– Each Euler equation contains a source term metric_euler with metric_euler = 0 in synchronous gauge,
(k2 psi) in newtonian gauge
– Each shear derivative equation contains a source term metric_shear equal to metric_shear = (h_←prime+6eta_prime)/2 in synchronous gauge, 0 in newtonian gauge
– metric_shear_prime is the derivative of metric_shear
– In the ufa_class approximation, the leading-order source term is (h_prime/2) in synchronous gauge, (-3
(phi_prime+psi_prime)) in newtonian gauge: we approximate the later by (-6 phi_prime)
• –> (d) if some approximation schemes are turned on, enforce a few y[] values computed in perturb_einstein
• –> (e) BEGINNING OF ACTUAL SYSTEM OF EQUATIONS OF EVOLUTION
• —> photon temperature density
• —> baryon density
• —> baryon velocity (depends on tight-coupling approximation=tca)
• -—> perturbed recombination has an impact
• —> photon temperature higher momenta and photon polarization (depend on tight-coupling approximation)
• -—> if photon tight-coupling is off
• —–> define Π = Gγ0 + Gγ2 + Fγ2
• —–> photon temperature velocity
• —–> photon temperature shear
• —–> photon temperature l=3
• —–> photon temperature l>3
• —–> photon temperature lmax
• —–> photon polarization l=0
• —–> photon polarization l=1
• —–> photon polarization l=2
• —–> photon polarization l>2
• —–> photon polarization lmax_pol
• -—> if photon tight-coupling is on:
• —–> in that case, only need photon velocity
• —> cdm
• -—> newtonian gauge: cdm density and velocity
• -—> synchronous gauge: cdm density only (velocity set to zero by definition of the gauge)
• —> dcdm and dr
• -—> dcdm
• —> dr
• -—> dr F0
• -—> dr F1
• -—> exact dr F2
• -—> exact dr l=3
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• -—> exact dr l>3
• -—> exact dr lmax_dr
• —> fluid (fld)
• -—> factors w, w_prime, adiabatic sound speed ca2 (all three background-related), plus actual sound speed
in the fluid rest frame cs2
• -—> fluid density
• -—> fluid velocity
• —> scalar field (scf)
• -—> field value
• -—> Klein Gordon equation
• —> ultra-relativistic neutrino/relics (ur)
• -—> if radiation streaming approximation is off
• —–> ur density
• —–> ur velocity
• —–> exact ur shear
• —–> exact ur l=3
• —–> exact ur l>3
• —–> exact ur lmax_ur
• —–> in fluid approximation (ufa): only ur shear needed
• —> non-cold dark matter (ncdm): massive neutrinos, WDM, etc.
• -—> first case: use a fluid approximation (ncdmfa)
• —–> loop over species
• —–> define intermediate quantitites
• —–> exact continuity equation
• —–> exact euler equation
• —–> different ansatz for approximate shear derivative
• —–> jump to next species
• -—> second case: use exact equation (Boltzmann hierarchy on momentum grid)
• —–> loop over species
• —–> loop over momentum
• —–> define intermediate quantities
• —–> ncdm density for given momentum bin
• —–> ncdm velocity for given momentum bin
• —–> ncdm shear for given momentum bin
• —–> ncdm l>3 for given momentum bin
• —–> ncdm lmax for given momentum bin (truncation as in Ma and Bertschinger) but with curvature taken
into account a la arXiv:1305.3261
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• —–> jump to next momentum bin or species
• —> metric
• —> eta of synchronous gauge
• vector mode
• –> baryon velocity
• tensor modes:
• –> non-cold dark matter (ncdm): massive neutrinos, WDM, etc.
• —> loop over species
• -—> loop over momentum
• -—> define intermediate quantities
• -—> ncdm density for given momentum bin
• -—> ncdm l>0 for given momentum bin
• -—> ncdm lmax for given momentum bin (truncation as in Ma and Bertschinger) but with curvature taken into
account a la arXiv:1305.3261
• -—> jump to next momentum bin or species
• –> tensor metric perturbation h (gravitational waves)
• –> its time-derivative

5.13.2.23

perturb_tca_slip_and_shear()

int perturb_tca_slip_and_shear (
double ∗ y,
void ∗ parameters_and_workspace,
ErrorMsg error_message )

Summary:

• define local variables
• rename the fields of the input structure (just to avoid heavy notations)
• compute related background quantities
• –> (a) define short-cut notations for the scalar perturbations
• –> (b) define short-cut notations used only in tight-coupling approximation
• –> (c) compute metric-related quantities (depending on gauge; additional gauges can be coded below)
– Each continuity equation contains a term in (theta+metric_continuity) with metric_continuity = (h_←prime/2) in synchronous gauge, (-3 phi_prime) in newtonian gauge
– Each Euler equation contains a source term metric_euler with metric_euler = 0 in synchronous gauge,
(k2 psi) in newtonian gauge
– Each shear derivative equation contains a source term metric_shear equal to metric_shear = (h_←prime+6eta_prime)/2 in synchronous gauge, 0 in newtonian gauge
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– metric_shear_prime is the derivative of metric_shear
– In the ufa_class approximation, the leading-order source term is (h_prime/2) in synchronous gauge, (-3
(phi_prime+psi_prime)) in newtonian gauge: we approximate the later by (-6 phi_prime)
• –> (d) if some approximation schemes are turned on, enforce a few y[ ] values computed in perturb_einstein
• —> like Ma & Bertschinger
• —> relax assumption dkappa∼a −2 (like in CAMB)
• —> also relax assumption cb2∼a −1
• —> intermediate quantities for 2nd order tca: shear_g at first order in tight-coupling
• —> intermediate quantities for 2nd order tca: zero order for theta_b' = theta_g'
• -—> perturbed recombination has an impact
• —> intermediate quantities for 2nd order tca: shear_g_prime at first order in tight-coupling
• —> 2nd order as in CRS
• —> 2nd order like in CLASS paper
• —> add only the most important 2nd order terms
• —> store tight-coupling values of photon shear and its derivative

5.14

perturbations.h File Reference

#include "thermodynamics.h"
#include "evolver_ndf15.h"
#include "evolver_rkck.h"
Include dependency graph for perturbations.h:

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

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119

This graph shows which files directly or indirectly include this file:

perturbations.h

primordial.h

nonlinear_exp.h

nonlinear_test.h

primordial.c

perturbations.c

nonlinear.h

transfer.h

spectra.h

spectra.c

output.h

input.h

nonlinear.c

transfer.c

lensing.h

lensing.c

output.c

input.c

class.h

class.c

Data Structures
•
•
•
•

struct perturbs
struct perturb_vector
struct perturb_workspace
struct perturb_parameters_and_workspace

Macros

• #define _MAX_NUMBER_OF_K_FILES_ 30

Enumerations

• enum tca_flags

• enum tca_method

• enum possible_gauges { newtonian, synchronous }
• #define _SELECTION_NUM_MAX_ 100
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120

5.14.1

File Documentation

Detailed Description

Documented includes for perturbation module

5.14.2

Data Structure Documentation

5.14.2.1

struct perturbs

Structure containing everything about perturbations that other modules need to know, in particular tabled values of
the source functions S(k, τ ) for all requested modes (scalar/vector/tensor), initial conditions, types (temperature,
E-polarization, B-polarization, lensing potential, etc), multipole l and wavenumber k.
Data Fields
short

has_perturbations

short

has_cls

do we need to compute perturbations at
all ?
do we need any harmonic space
spectrum Cl (and hence Bessel
functions, transfer functions, ...)?

short

has_scalars

do we need scalars?

short

has_vectors

do we need vectors?

short

has_tensors

do we need tensors?

short

has_ad

do we need adiabatic mode?

short

has_bi

do we need isocurvature bi mode?

short

has_cdi

do we need isocurvature cdi mode?

short

has_nid

do we need isocurvature nid mode?

short

has_niv

do we need isocurvature niv mode?

short

has_perturbed_recombination

Do we want to consider perturbed
temperature and ionization fraction?

tensor_method

Neutrino contribution to tensors way to
treat neutrinos in tensor
perturbations(neglect, approximate as
massless, take exact equations)

short

evolve_tensor_ur

will we evolve ur tensor perturbations
(either because we have ur species, or
we have ncdm species with massless
approximation) ?

short

evolve_tensor_ncdm

will we evolve ncdm tensor perturbations
(if we have ncdm species and we use
the exact method) ?

short

has_cl_cmb_temperature

do we need Cl 's for CMB temperature?

short

has_cl_cmb_polarization

do we need Cl 's for CMB polarization?

short

has_cl_cmb_lensing_potential

do we need Cl 's for CMB lensing
potential?

short

has_cl_lensing_potential

do we need Cl 's for galaxy lensing
potential?

short

has_cl_number_count

short

has_pk_matter

do we need Cl 's for density number
count?
do we need matter Fourier spectrum?

short

has_density_transfers

enum tensor_methods

do we need to output individual matter
density transfer functions?

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Data Fields
short

has_velocity_transfers

do we need to output individual matter
velocity transfer functions?

short

has_metricpotential_transfers

do we need to output individual transfer
functions for scalar metric
perturbations?

short

has_nl_corrections_based_on_delta_m

do we want to compute non-linear
corrections with an algorithm relying on
delta_m (like halofit)?

short

has_nc_density

in dCl, do we want density terms ?

short

has_nc_rsd

short

has_nc_lens

in dCl, do we want redshift space
distortion terms ?
in dCl, do we want lensing terms ?

short

has_nc_gr

in dCl, do we want gravity terms ?

int

l_scalar_max

maximum l value for CMB scalars Cl 's

int

l_vector_max

maximum l value for CMB vectors Cl 's

int

l_tensor_max

maximum l value for CMB tensors Cl 's

int

l_lss_max

maximum l value for LSS Cl 's (density
and lensing potential in bins)

double

k_max_for_pk

maximum value of k in 1/Mpc in P(k) (if
Cl 's also requested, overseeded by
value kmax inferred from l_scalar_max if
it is bigger)

int

selection_num

number of selection functions (i.e. bins)
for matter density Cl 's

selection

type of selection functions

enum selection_type
double

selection_mean[_SELECTION_NUM_MAX_]
centers of selection functions

double

selection_width[_SELECTION_NUM_MAX_]widths of selection functions

int

switch_sw

int

switch_eisw

int

switch_lisw

int

switch_dop

int

switch_pol

in temperature calculation, do we want
to include the polarization-related term?

eisw_lisw_split_z

at which redshift do we define the cut
between eisw and lisw ?
Do we want to store perturbations?

double
int

store_perturbations

int

k_output_values_num

double

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in temperature calculation, do we want
to include the intrinsic temperature +
Sachs Wolfe term?
in temperature calculation, do we want
to include the early integrated Sachs
Wolfe term?
in temperature calculation, do we want
to include the late integrated Sachs
Wolfe term?
in temperature calculation, do we want
to include the Doppler term?

Number of perturbation outputs
(default=0)

k_output_values[_MAX_NUMBER_OF_K_FILES_]
List of k values where perturbation
output is requested.

int ∗

index_k_output_values

char

scalar_titles[_MAXTITLESTRINGLENGTH_]DELIMITER separated string of titles for
scalar perturbation output files.

List of indices corresponding to k-values
close to k_output_values for each mode.
[index_md∗k_output_values_num+ik]

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File Documentation

Data Fields
char

vector_titles[_MAXTITLESTRINGLENGTH_]DELIMITER separated string of titles for
vector perturbation output files.

char

tensor_titles[_MAXTITLESTRINGLENGTH_]DELIMITER separated string of titles for
tensor perturbation output files.

int

number_of_scalar_titles

number of titles/columns in scalar
perturbation output files

int

number_of_vector_titles

number of titles/columns in vector
perturbation output files

int

number_of_tensor_titles

number of titles/columns in tensor
perturbation output files

double ∗

scalar_perturbations_data[_MAX_NUMBER_OF_K_FILES_]
Array of double pointers to perturbation
output for scalars

double ∗

vector_perturbations_data[_MAX_NUMBER_OF_K_FILES_]
Array of double pointers to perturbation
output for vectors

double ∗

tensor_perturbations_data[_MAX_NUMBER_OF_K_FILES_]
Array of double pointers to perturbation
output for tensors

int

size_scalar_perturbation_data[_MAX_NUMBER_OF_K_FILES_]
Array of sizes of scalar double pointers

int

size_vector_perturbation_data[_MAX_NUMBER_OF_K_FILES_]
Array of sizes of vector double pointers

int

size_tensor_perturbation_data[_MAX_NUMBER_OF_K_FILES_]
Array of sizes of tensor double pointers

double

three_ceff2_ur

3 x effective squared sound speed for
the ultrarelativistic perturbations

double

three_cvis2_ur

3 x effective viscosity parameter for the
ultrarelativistic perturbations

double

z_max_pk

when we compute only the matter
spectrum / transfer functions, but not the
CMB, we are sometimes interested to
sample source functions at very high
redshift, way before recombination. This
z_max_pk will then fix the initial
sampling time of the sources.

short

has_cmb

do we need CMB-related sources
(temperature, polarization) ?

short

has_lss

do we need LSS-related sources
(lensing potential, ...) ?

gauge

enum possible_gauges
int

index_md_scalars

gauge in which to perform this
calculation
index value for scalars

int

index_md_tensors

index value for tensors

int

index_md_vectors

index value for vectors

int

md_size

number of modes included in
computation

int

index_ic_ad

index value for adiabatic

int

index_ic_cdi

index value for CDM isocurvature

int

index_ic_bi

index value for baryon isocurvature

int

index_ic_nid

int

index_ic_niv

int

index_ic_ten

index value for neutrino density
isocurvature
index value for neutrino velocity
isocurvature
index value for unique possibility for
tensors

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Data Fields
int ∗

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ic_size

for a given mode, ic_size[index_md] =
number of initial conditions included in
computation

short

has_source_t

do we need source for CMB
temperature?

short

has_source_p

do we need source for CMB
polarization?

short

has_source_delta_m

short

has_source_delta_g

do we need source for delta of total
matter?
do we need source for delta of gammas?

short

has_source_delta_b

do we need source for delta of baryons?

short

has_source_delta_cdm

short

has_source_delta_dcdm

do we need source for delta of cold dark
matter?
do we need source for delta of DCDM?

short

has_source_delta_fld

do we need source for delta of dark
energy?

short

has_source_delta_scf

short

has_source_delta_dr

short

has_source_delta_ur

short

has_source_delta_ncdm

do we need source for delta from scalar
field?
do we need source for delta of decay
radiation?
do we need source for delta of
ultra-relativistic neutrinos/relics?
do we need source for delta of all
non-cold dark matter species (e.g.
massive neutrinos)?

short

has_source_theta_m

short

has_source_theta_g

short

has_source_theta_b

do we need source for theta of baryons?

short

has_source_theta_cdm

short

has_source_theta_dcdm

do we need source for theta of cold dark
matter?
do we need source for theta of DCDM?

short

has_source_theta_fld

do we need source for theta of dark
energy?

short

has_source_theta_scf

short

has_source_theta_dr

short

has_source_theta_ur

short

has_source_theta_ncdm

do we need source for theta of scalar
field?
do we need source for theta of
ultra-relativistic neutrinos/relics?
do we need source for theta of
ultra-relativistic neutrinos/relics?
do we need source for theta of all
non-cold dark matter species (e.g.
massive neutrinos)?

short

has_source_phi

do we need source for metric fluctuation
phi?

short

has_source_phi_prime

do we need source for metric fluctuation
phi'?

short

has_source_phi_plus_psi

do we need source for metric fluctuation
(phi+psi)?

short

has_source_psi

do we need source for metric fluctuation
psi?

do we need source for theta of total
matter?
do we need source for theta of
gammas?

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File Documentation

Data Fields
short

has_source_h

int

index_tp_t0

do we need source for metric fluctuation
h?
do we need source for metric fluctuation
h'?
do we need source for metric fluctuation
eta?
do we need source for metric fluctuation
eta'?
index value for temperature (j=0 term)

short

has_source_h_prime

short

has_source_eta

short

has_source_eta_prime

int

index_tp_t1

index value for temperature (j=1 term)

int

index_tp_t2

index value for temperature (j=2 term)

int

index_tp_p

index value for polarization

int

index_tp_delta_m

index value for delta tot

int

index_tp_delta_g

index value for delta of gammas

int

index_tp_delta_b

index value for delta of baryons

int

index_tp_delta_cdm

index value for delta of cold dark matter

int

index_tp_delta_dcdm

index value for delta of DCDM

int

index_tp_delta_fld

index value for delta of dark energy

int

index_tp_delta_scf

index value for delta of scalar field

int

index_tp_delta_dr

index value for delta of decay radiation

int

index_tp_delta_ur

int

index_tp_delta_ncdm1

index value for delta of ultra-relativistic
neutrinos/relics
index value for delta of first non-cold
dark matter species (e.g. massive
neutrinos)

int

index_tp_perturbed_recombination_delta_temp
Gas temperature perturbation

int

index_tp_perturbed_recombination_delta_chi
Inionization fraction perturbation

int

index_tp_theta_m

index value for theta tot

int

index_tp_theta_g

index value for theta of gammas

int

index_tp_theta_b

index value for theta of baryons

int

index_tp_theta_cdm

index value for theta of cold dark matter

int

index_tp_theta_dcdm

index value for theta of DCDM

int

index_tp_theta_fld

index value for theta of dark energy

int

index_tp_theta_scf

index value for theta of scalar field

int

index_tp_theta_ur

int

index_tp_theta_dr

index value for theta of ultra-relativistic
neutrinos/relics
index value for F1 of decay radiation

int

index_tp_theta_ncdm1

index value for theta of first non-cold
dark matter species (e.g. massive
neutrinos)

int

index_tp_phi

index value for metric fluctuation phi

int

index_tp_phi_prime

index value for metric fluctuation phi'

int

index_tp_phi_plus_psi

index value for metric fluctuation phi+psi

int

index_tp_psi

index value for metric fluctuation psi

int

index_tp_h

index value for metric fluctuation h

int

index_tp_h_prime

index value for metric fluctuation h'

int

index_tp_eta

index value for metric fluctuation eta

int

index_tp_eta_prime

index value for metric fluctuation eta'

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Data Fields
int ∗

tp_size

number of types tp_size[index_md]
included in computation for each mode

int ∗

k_size_cmb

k_size_cmb[index_md] number of k
values used for CMB calculations,
requiring a fine sampling in k-space

int ∗

k_size_cl

k_size_cl[index_md] number of k values
used for non-CMB Cl calculations,
requiring a coarse sampling in k-space.

int ∗

k_size

k_size[index_md] = total number of k
values, including those needed for P(k)
but not for Cl 's

double ∗∗

k

k[index_md][index_k] = list of values

double

k_min

minimum value (over all modes)

double

k_max

maximum value (over all modes)

tau_size

tau_size = number of values

tau_sampling

tau_sampling[index_tau] = list of tau
values
used in presence of selection functions
(for matter density, cosmic shear...)

int
double ∗
double

selection_min_of_tau_min

double

selection_max_of_tau_max

used in presence of selection functions
(for matter density, cosmic shear...)

double

selection_delta_tau

used in presence of selection functions
(for matter density, cosmic shear...)

double ∗

selection_tau_min

double ∗

selection_tau_max

double ∗

selection_tau

double ∗

selection_function

value of conformal time below which
W(tau) is considered to vanish for each
bin
value of conformal time above which
W(tau) is considered to vanish for each
bin
value of conformal time at the center of
each bin
selection
function W(tau), normalized to
R
W (tau)dtau = 1, stored in
selection_function[bin∗ppt->tau_←size+index_tau]

double ∗∗∗

short

ErrorMsg

5.14.2.2

sources

Pointer towards the source interpolation
table sources[index_md] [index_ic ∗
ppt->tp_size[index_md] + index_type]
[index_tau ∗ ppt->k_size + index_k]

perturbations_verbose

flag regulating the amount of information
sent to standard output (none if set to
zero)

error_message

zone for writing error messages

struct perturb_vector

Structure containing the indices and the values of the perturbation variables which are integrated over time (as
well as their time-derivatives). For a given wavenumber, the size of these vectors changes when the approximation
scheme changes.

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Data Fields
int

index_pt_delta_g

photon density

int

index_pt_theta_g

photon velocity

int

index_pt_shear_g

photon shear

int

index_pt_l3_g

photon l=3

int

l_max_g

max momentum in Boltzmann hierarchy (at
least 3)

int

index_pt_pol0_g

photon polarization, l=0

int

index_pt_pol1_g

photon polarization, l=1

int

index_pt_pol2_g

photon polarization, l=2

int

index_pt_pol3_g

photon polarization, l=3

int

l_max_pol_g

max momentum in Boltzmann hierarchy (at
least 3)

int

index_pt_delta_b

baryon density

int

index_pt_theta_b

baryon velocity

int

index_pt_delta_cdm

cdm density

int

index_pt_theta_cdm

cdm velocity

int

index_pt_delta_dcdm

dcdm density

int

index_pt_theta_dcdm

dcdm velocity

int

index_pt_delta_fld

dark energy density in true fluid case

int

index_pt_theta_fld

dark energy velocity in true fluid case

int

index_pt_Gamma_fld

int

index_pt_phi_scf

unique dark energy dynamical variable in PPF
case
scalar field density

int

index_pt_phi_prime_scf

scalar field velocity

int

index_pt_delta_ur

density of ultra-relativistic neutrinos/relics

int

index_pt_theta_ur

velocity of ultra-relativistic neutrinos/relics

int

index_pt_shear_ur

shear of ultra-relativistic neutrinos/relics

int

index_pt_l3_ur

l=3 of ultra-relativistic neutrinos/relics

int

l_max_ur

max momentum in Boltzmann hierarchy (at
least 3)

int

index_pt_perturbed_recombination_delta_temp

Gas temperature perturbation

int

index_pt_perturbed_recombination_delta_chi

Inionization fraction perturbation

int

index_pt_F0_dr

The index to the first Legendre multipole of the
DR expansion. Not that this is not exactly the
usual delta, see Kaplinghat et al.,
astro-ph/9907388.

int

l_max_dr

max momentum in Boltzmann hierarchy for dr)

int

index_pt_psi0_ncdm1

first multipole of perturbation of first ncdm
species, Psi_0

int

N_ncdm

number of distinct non-cold-dark-matter (ncdm)
species

int ∗

l_max_ncdm

mutipole l at which Boltzmann hierarchy is
truncated (for each ncdm species)

int ∗

q_size_ncdm

number of discrete momenta (for each ncdm
species)

int

index_pt_eta

synchronous gauge metric perturbation eta

int

index_pt_phi

newtonian gauge metric perturbation phi

int

index_pt_hv_prime

vector metric perturbation h_v' in synchronous
gauge

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Data Fields
int

index_pt_V

vector metric perturbation V in Newtonian
gauge

int

index_pt_gw

tensor metric perturbation h (gravitational
waves)

int

index_pt_gwdot

its time-derivative

int

pt_size

size of perturbation vector

double ∗

y

vector of perturbations to be integrated

double ∗

dy

time-derivative of the same vector

used_in_sources

boolean array specifying which perturbations
enter in the calculation of source functions

int ∗

5.14.2.3

struct perturb_workspace

Workspace containing, among other things, the value at a given time of all background/perturbed quantities, as well
as their indices. There will be one such structure created for each mode (scalar/.../tensor) and each thread (in case
of parallel computing)
Data Fields
int

index_mt_psi

psi in longitudinal gauge

int

index_mt_phi_prime

(d phi/d conf.time) in longitudinal gauge

int

index_mt_h_prime

h' (wrt conf. time) in synchronous gauge

int

index_mt_h_prime_prime

h'' (wrt conf. time) in synchronous gauge

int

index_mt_eta_prime

eta' (wrt conf. time) in synchronous gauge

int

index_mt_alpha

α = (h0 + 6η 0 )/(2k 2 ) in synchronous gauge

int

index_mt_alpha_prime

α0 wrt conf. time) in synchronous gauge

int

index_mt_gw_prime_prime

second derivative wrt conformal time of gravitational
wave field, often called h

int

index_mt_V_prime

derivative of Newtonian gauge vector metric
perturbation V

int

index_mt_hv_prime_prime

Second derivative of Synchronous gauge vector
metric perturbation hv

int

mt_size

size of metric perturbation vector

double ∗

pvecback

background quantities

double ∗

pvecthermo

thermodynamics quantities

double ∗

pvecmetric

metric quantities

pv

struct perturb_vector ∗

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double

delta_rho

pointer to vector of integrated perturbations and their
time-derivatives
total density perturbation (gives delta Too)

double

rho_plus_p_theta

total (rho+p)∗theta perturbation (gives delta Toi)

double

rho_plus_p_shear

total (rho+p)∗shear (gives delta Tij)

double

delta_p

total pressure perturbation (gives Tii)

double

gw_source

stress-energy source term in Einstein's tensor
equations (gives Tij[tensor])

double

vector_source_pi

first stress-energy source term in Einstein's vector
equations

double

vector_source_v

second stress-energy source term in Einstein's vector
equations

128

File Documentation

Data Fields
double

tca_shear_g

photon shear in tight-coupling approximation

double

tca_slip

photon-baryon slip in tight-coupling approximation

double

rsa_delta_g

photon density in radiation streaming approximation

double

rsa_theta_g

photon velocity in radiation streaming approximation

double

rsa_delta_ur

photon density in radiation streaming approximation

double

rsa_theta_ur

photon velocity in radiation streaming approximation

double ∗

delta_ncdm

relative density perturbation of each ncdm species

double ∗

theta_ncdm

velocity divergence theta of each ncdm species

double ∗

shear_ncdm

shear for each ncdm species

double

delta_m

relative density perturbation of all non-relativistic
species

double

theta_m

velocity divergence theta of all non-relativistic species

double

delta_rho_fld

double

rho_plus_p_theta_fld

double

S_fld

density perturbation of fluid, not so trivial in PPF
scheme
velocity divergence of fluid, not so trivial in PPF
scheme
S quantity sourcing Gamma_prime evolution in PPF
scheme (equivalent to eq. 15 in 0808.3125)

double

Gamma_prime_fld

Gamma_prime in PPF scheme (equivalent to eq. 14 in
0808.3125)

FILE ∗

perturb_output_file

filepointer to output file

int

index_ikout

index for output k value (when k_output_values is set)

short

inter_mode

flag defining the method used for interpolation
background/thermo quantities tables

int

last_index_back

the background interpolation function
background_at_tau() keeps memory of the last point
called through this index

int

last_index_thermo

the thermodynamics interpolation function
thermodynamics_at_z() keeps memory of the last
point called through this index

int

index_ap_tca

index for tight-coupling approximation

int

index_ap_rsa

index for radiation streaming approximation

int

index_ap_ufa

index for ur fluid approximation

int

index_ap_ncdmfa

index for ncdm fluid approximation

int

ap_size

number of relevant approximations for a given mode

approx

array of approximation flags holding at a given time:
approx[index_ap]

max_l_max

maximum l_max for any multipole

s_l

array p
of freestreaming coefficients

int ∗
int
double ∗

sl =

5.14.2.4

1 − K ∗ (l2 − 1)/k 2

struct perturb_parameters_and_workspace

Structure pointing towards all what the function that perturb_derivs needs to know: fixed input parameters and
indices contained in the various structures, workspace, etc.

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129

Data Fields
struct precision ∗

ppr

pointer to the precision structure

struct background ∗

pba

pointer to the background structure

struct thermo ∗

pth

pointer to the thermodynamics structure

struct perturbs ∗

ppt

pointer to the precision structure

int

index_md

index of mode (scalar/.../vector/tensor)

int

index_ic

index of initial condition (adiabatic/isocurvature(s)/...)

int

index_k

index of wavenumber

k

current value of wavenumber in 1/Mpc

ppw

workspace defined above

double
struct perturb_workspace ∗

5.14.3

Macro Definition Documentation

5.14.3.1

_SELECTION_NUM_MAX_

#define _SELECTION_NUM_MAX_ 100

maximum number and types of selection function (for bins of matter density or cosmic shear)

5.14.3.2

_MAX_NUMBER_OF_K_FILES_

#define _MAX_NUMBER_OF_K_FILES_ 30

maximum number of k-values for perturbation output

5.14.4

Enumeration Type Documentation

5.14.4.1

tca_flags

enum tca_flags

flags for various approximation schemes (tca = tight-coupling approximation, rsa = radiation streaming approximation, ufa = massless neutrinos / ultra-relativistic relics fluid approximation)
CAUTION: must be listed below in chronological order, and cannot be reversible. When integrating equations for a
given mode, it is only possible to switch from left to right in the lists below.

5.14.4.2

tca_method

enum tca_method

labels for the way in which each approximation scheme is implemented

5.14.4.3

possible_gauges

enum possible_gauges

List of coded gauges. More gauges can in principle be defined.

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Enumerator
newtonian
synchronous

5.15

newtonian (or longitudinal) gauge
synchronous gauge with θcdm = 0 by convention

primordial.c File Reference

#include "primordial.h"
Include dependency graph for primordial.c:

primordial.c

primordial.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int primordial_spectrum_at_k (struct primordial ∗ppm, int index_md, enum linear_or_logarithmic mode, double input, double ∗output)
• int primordial_init (struct precision ∗ppr, struct perturbs ∗ppt, struct primordial ∗ppm)
• int primordial_free (struct primordial ∗ppm)
• int primordial_indices (struct perturbs ∗ppt, struct primordial ∗ppm)
• int primordial_get_lnk_list (struct primordial ∗ppm, double kmin, double kmax, double k_per_decade)
• int primordial_analytic_spectrum_init (struct perturbs ∗ppt, struct primordial ∗ppm)
• int primordial_analytic_spectrum (struct primordial ∗ppm, int index_md, int index_ic1_ic2, double k, double
∗pk)
• int primordial_inflation_potential (struct primordial ∗ppm, double phi, double ∗V, double ∗dV, double ∗ddV)
• int primordial_inflation_hubble (struct primordial ∗ppm, double phi, double ∗H, double ∗dH, double ∗ddH,
double ∗dddH)
• int primordial_inflation_indices (struct primordial ∗ppm)
• int primordial_inflation_solve_inflation (struct perturbs ∗ppt, struct primordial ∗ppm, struct precision ∗ppr)
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• int primordial_inflation_analytic_spectra (struct perturbs ∗ppt, struct primordial ∗ppm, struct precision ∗ppr,
double ∗y_ini)
• int primordial_inflation_spectra (struct perturbs ∗ppt, struct primordial ∗ppm, struct precision ∗ppr, double
∗y_ini)
• int primordial_inflation_one_wavenumber (struct perturbs ∗ppt, struct primordial ∗ppm, struct precision ∗ppr,
double ∗y_ini, int index_k)
• int primordial_inflation_one_k (struct primordial ∗ppm, struct precision ∗ppr, double k, double ∗y, double ∗dy,
double ∗curvature, double ∗tensor)
• int primordial_inflation_find_attractor (struct primordial ∗ppm, struct precision ∗ppr, double phi_0, double precision, double ∗y, double ∗dy, double ∗H_0, double ∗dphidt_0)
• int primordial_inflation_evolve_background (struct primordial ∗ppm, struct precision ∗ppr, double ∗y, double
∗dy, enum target_quantity target, double stop, short check_epsilon, enum integration_direction direction,
enum time_definition time)
• int primordial_inflation_check_potential (struct primordial ∗ppm, double phi, double ∗V, double ∗dV, double
∗ddV)
• int primordial_inflation_check_hubble (struct primordial ∗ppm, double phi, double ∗H, double ∗dH, double
∗ddH, double ∗dddH)
• int primordial_inflation_get_epsilon (struct primordial ∗ppm, double phi, double ∗epsilon)
• int primordial_inflation_find_phi_pivot (struct primordial ∗ppm, struct precision ∗ppr, double ∗y, double ∗dy)
• int primordial_inflation_derivs (double tau, double ∗y, double ∗dy, void ∗parameters_and_workspace, Error←Msg error_message)
• int primordial_external_spectrum_init (struct perturbs ∗ppt, struct primordial ∗ppm)

5.15.1

Detailed Description

Documented primordial module.

Julien Lesgourgues, 24.08.2010

This module computes the primordial spectra. It can be used in different modes: simple parametric form, evolving
inflaton perturbations, etc. So far only the mode corresponding to a simple analytic form in terms of amplitudes, tilts
and runnings has been developed.

The following functions can be called from other modules:

1. primordial_init() at the beginning (anytime after perturb_init() and before spectra_init())
2. primordial_spectrum_at_k() at any time for computing P(k) at any k
3. primordial_free() at the end

5.15.2

Function Documentation

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132

File Documentation

5.15.2.1

primordial_spectrum_at_k()

int primordial_spectrum_at_k (
struct primordial ∗ ppm,
int index_md,
enum linear_or_logarithmic mode,
double input,
double ∗ output )

Primordial spectra for arbitrary argument and for all initial conditions.
This routine evaluates the primordial spectrum at a given value of k by interpolating in the pre-computed table.
When k is not in the pre-computed range but the spectrum can be found analytically, it finds it. Otherwise returns
an error.
Can be called in two modes; linear or logarithmic:

• linear: takes k, returns P(k)
• logarithmic: takes ln(k), return ln(P(k))

One little subtlety: in case of several correlated initial conditions, the cross-correlation spectrum√can be negative.
Then, in logarithmic mode, the non-diagonal elements contain the cross-correlation angle P12 / P11 P22 (from -1
to 1) instead of ln P12
This function can be called from whatever module at whatever time, provided that primordial_init() has been called
before, and primordial_free() has not been called yet.
Parameters
ppm

Input: pointer to primordial structure containing tabulated primordial spectrum

index_md

Input: index of mode (scalar, tensor, ...)

mode

Input: linear or logarithmic

input

Input: wavenumber in 1/Mpc (linear mode) or its logarithm (logarithmic mode)

output

Output: for each pair of initial conditions, primordial spectra P(k) in M pc3 (linear mode), or their
logarithms and cross-correlation angles (logarithmic mode)

Returns
the error status
Summary:
• define local variables
• infer ln(k) from input. In linear mode, reject negative value of input k value.
• if ln(k) is not in the interpolation range, return an error, unless we are in the case of a analytic spectrum, for
which a direct computation is possible
• otherwise, interpolate in the pre-computed table

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5.15.2.2

133

primordial_init()

int primordial_init (
struct precision ∗ ppr,
struct perturbs ∗ ppt,
struct primordial ∗ ppm )

This routine initializes the primordial structure (in particular, it computes table of primordial spectrum values)
Parameters
ppr

Input: pointer to precision structure (defines method and precision for all computations)

ppt

Input: pointer to perturbation structure (useful for knowing k_min, k_max, etc.)

ppm

Output: pointer to initialized primordial structure

Returns
the error status
Summary:
• define local variables
• check that we really need to compute the primordial spectra
• get kmin and kmax from perturbation structure. Test that they make sense.
• allocate and fill values of ln k 's
• define indices and allocate tables in primordial structure
• deal with case of analytic primordial spectra (with amplitudes, tilts, runnings, etc.)
• deal with case of inflation with given V (φ) or H(φ)
• deal with the case of external calculation of Pk
• compute second derivative of each ln Pk versus lnk with spline, in view of interpolation
• derive spectral parameters from numerically computed spectra (not used by the rest of the code, but useful
to keep in memory for several types of investigation)
• expression for alpha_s comes from:

ns_2 = (lnpk_plus-lnpk_pivot)/(dlnk)+1
ns_1 = (lnpk_pivot-lnpk_minus)/(dlnk)+1
alpha_s = dns/dlnk = (ns_2-ns_1)/dlnk = (lnpk_plus-lnpk_pivot-lnpk_←pivot+lnpk_minus)/(dlnk)/(dlnk)
• expression for beta_s:

ppm->beta_s = (alpha_plus-alpha_minus)/dlnk = (lnpk_plusplus-2.∗lnpk_←plus+lnpk_pivot - (lnpk_pivot-2.∗lnpk_minus+lnpk_minusminus)/pow(dlnk,3)

5.15.2.3

primordial_free()

int primordial_free (
struct primordial ∗ ppm )

This routine frees all the memory space allocated by primordial_init().
To be called at the end of each run.

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Parameters
ppm

Input: pointer to primordial structure (which fields must be freed)

Returns
the error status

5.15.2.4

primordial_indices()

int primordial_indices (
struct perturbs ∗ ppt,
struct primordial ∗ ppm )

This routine defines indices and allocates tables in the primordial structure
Parameters
ppt

Input: pointer to perturbation structure

ppm

Input/output: pointer to primordial structure

Returns
the error status

5.15.2.5

primordial_get_lnk_list()

int primordial_get_lnk_list (
struct
double
double
double

primordial ∗ ppm,
kmin,
kmax,
k_per_decade )

This routine allocates and fills the list of wavenumbers k
Parameters
ppm

Input/output: pointer to primordial structure

kmin

Input: first value

kmax

Input: last value that we should encompass

k_per_decade

Input: number of k per decade

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Returns
the error status

5.15.2.6

primordial_analytic_spectrum_init()

int primordial_analytic_spectrum_init (
struct perturbs ∗ ppt,
struct primordial ∗ ppm )

This routine interprets and stores in a condensed form the input parameters in the case of a simple analytic spectra
with amplitudes, tilts, runnings, in such way that later on, the spectrum can be obtained by a quick call to the routine
primordial_analytic_spectrum(()
Parameters
ppt

Input: pointer to perturbation structure

ppm

Input/output: pointer to primordial structure

Returns
the error status

5.15.2.7

primordial_analytic_spectrum()

int primordial_analytic_spectrum (
struct primordial ∗ ppm,
int index_md,
int index_ic1_ic2,
double k,
double ∗ pk )

This routine returns the primordial spectrum in the simple analytic case with amplitudes, tilts, runnings, for each
mode (scalar/tensor...), pair of initial conditions, and wavenumber.
Parameters
ppm

Input/output: pointer to primordial structure

index_md

Input: index of mode (scalar, tensor, ...)

index_ic1_ic2

Input: pair of initial conditions (ic1, ic2)

k

Input: wavenumber in same units as pivot scale, i.e. in 1/Mpc

pk

Output: primordial power spectrum A (k/k_pivot)∧ (n+...)

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File Documentation

Returns
the error status

5.15.2.8

primordial_inflation_potential()

int primordial_inflation_potential (
struct primordial ∗ ppm,
double phi,
double ∗ V,
double ∗ dV,
double ∗ ddV )

This routine encodes the inflaton scalar potential
Parameters
ppm

Input: pointer to primordial structure

phi

Input: background inflaton field value in units of Mp

V

Output: inflaton potential in units of M p4

dV

Output: first derivative of inflaton potential wrt the field

ddV

Output: second derivative of inflaton potential wrt the field

Returns
the error status

5.15.2.9

primordial_inflation_hubble()

int primordial_inflation_hubble (
struct primordial ∗ ppm,
double phi,
double ∗ H,
double ∗ dH,
double ∗ ddH,
double ∗ dddH )

This routine encodes the function H(φ)
Parameters
ppm

Input: pointer to primordial structure

phi

Input: background inflaton field value in units of Mp

H

Output: Hubble parameters in units of Mp

dH

Output: dH/dφ

ddH

Output: d2 H/dφ2

dddH

Output: d3 H/dφ3
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Returns
the error status

5.15.2.10

primordial_inflation_indices()

int primordial_inflation_indices (
struct primordial ∗ ppm )

This routine defines indices used by the inflation simulator
Parameters
ppm

Input/output: pointer to primordial structure

Returns
the error status

5.15.2.11

primordial_inflation_solve_inflation()

int primordial_inflation_solve_inflation (
struct perturbs ∗ ppt,
struct primordial ∗ ppm,
struct precision ∗ ppr )

Main routine of inflation simulator. Its goal is to check the background evolution before and after the pivot value
phi=phi_pivot, and then, if this evolution is suitable, to call the routine primordial_inflation_spectra().
Parameters
ppt

Input: pointer to perturbation structure

ppm

Input/output: pointer to primordial structure

ppr

Input: pointer to precision structure

Returns
the error status

Summary:

• define local variables
• allocate vectors for background/perturbed quantities
• eventually, needs first to find phi_pivot
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• compute H_pivot at phi_pivot
• check positivity and negative slope of potential in field pivot value, and find value of phi_dot and H for field's
pivot value, assuming slow-roll attractor solution has been reached. If no solution, code will stop there.
• check positivity and negative slope of H(φ) in field pivot value, and get H_pivot
• find a_pivot, value of scale factor when k_pivot crosses horizon while phi=phi_pivot
• integrate background solution starting from phi_pivot and until k_max>>aH. This ensures that the inflationary
model considered here is valid and that the primordial spectrum can be computed. Otherwise, if slow-roll
brakes too early, model is not suitable and run stops.
• starting from this time, i.e. from y_ini[ ], we run the routine which takes care of computing the primordial
spectrum.
• before ending, we want to compute and store the values of φ corresponding to k=aH for k_min and k_max
• finally, we can de-allocate

5.15.2.12

primordial_inflation_analytic_spectra()

int primordial_inflation_analytic_spectra (
struct perturbs ∗ ppt,
struct primordial ∗ ppm,
struct precision ∗ ppr,
double ∗ y_ini )

Routine for the computation of an analytic apporoximation to the the primordial spectrum. In general, should be
used only for comparing with exact numerical computation performed by primordial_inflation_spectra().
Parameters
ppt

Input: pointer to perturbation structure

ppm

Input/output: pointer to primordial structure

ppr

Input: pointer to precision structure

y_ini

Input: initial conditions for the vector of background/perturbations, already allocated and filled

Returns
the error status
Summary
• allocate vectors for background/perturbed quantities
• initialize the background part of the running vector
• loop over Fourier wavenumbers
• read value of phi at time when k=aH
• get potential (and its derivatives) at this value
• calculate the analytic slow-roll formula for the spectra
• store the obtained result for curvature and tensor perturbations

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5.15.2.13

139

primordial_inflation_spectra()

int primordial_inflation_spectra (
struct
struct
struct
double

perturbs ∗ ppt,
primordial ∗ ppm,
precision ∗ ppr,
∗ y_ini )

Routine with a loop over wavenumbers for the computation of the primordial spectrum. For each wavenumber it
calls primordial_inflation_one_wavenumber()
Parameters
ppt

Input: pointer to perturbation structure

ppm

Input/output: pointer to primordial structure

ppr

Input: pointer to precision structure

y_ini

Input: initial conditions for the vector of background/perturbations, already allocated and filled

Returns
the error status

5.15.2.14

primordial_inflation_one_wavenumber()

int primordial_inflation_one_wavenumber (
struct perturbs ∗ ppt,
struct primordial ∗ ppm,
struct precision ∗ ppr,
double ∗ y_ini,
int index_k )

Routine coordinating the computation of the primordial spectrum for one wavenumber. It calls primordial_inflation←_one_k() to integrate the perturbation equations, and then it stores the result for the scalar/tensor spectra.
Parameters
ppt

Input: pointer to perturbation structure

ppm

Input/output: pointer to primordial structure

ppr

Input: pointer to precision structure

y_ini

Input: initial conditions for the vector of background/perturbations, already allocated and filled

index←_k

Input: index of wavenumber to be considered

Returns
the error status

Summary
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File Documentation

• allocate vectors for background/perturbed quantities
• initialize the background part of the running vector
– evolve the background until the relevant initial time for integrating perturbations
– evolve the background/perturbation equations from this time and until some time after Horizon crossing
• store the obtained result for curvature and tensor perturbations

5.15.2.15

primordial_inflation_one_k()

int primordial_inflation_one_k (
struct
struct
double
double
double
double

primordial ∗ ppm,
precision ∗ ppr,
k,
∗ y,
∗ dy,
∗ curvature,

double ∗ tensor )

Routine integrating the background plus perturbation equations for each wavenumber, and returning the scalar and
tensor spectrum.
Parameters
ppm

Input: pointer to primordial structure

ppr

Input: pointer to precision structure

k

Input: Fourier wavenumber

y

Input: running vector of background/perturbations, already allocated and initialized

dy

Input: running vector of background/perturbation derivatives, already allocated

curvature

Output: curvature perturbation

tensor

Output: tensor perturbation

Returns
the error status

Summary:

• define local variables
• initialize the generic integrator (same integrator already used in background, thermodynamics and perturbation modules)
• initialize variable used for deciding when to stop the calculation (= when the curvature remains stable)
• initialize conformal time to arbitrary value (here, only variations of tau matter: the equations that we integrate
do not depend explicitly on time)
• compute derivative of initial vector and infer first value of adaptive time-step
• loop over time
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• clean the generic integrator
• store final value of curvature for this wavenumber
• store final value of tensor perturbation for this wavenumber

5.15.2.16

primordial_inflation_find_attractor()

int primordial_inflation_find_attractor (
struct primordial ∗ ppm,
struct precision ∗ ppr,
double phi_0,
double
double
double
double
double

precision,
∗ y,
∗ dy,
∗ H_0,
∗ dphidt_0 )

Routine searching for the inflationary attractor solution at a given phi_0, by iterations, with a given tolerance. If no
solution found within tolerance, returns error message. The principle is the following. The code starts integrating
the background equations from various values of phi, corresponding to earlier and earlier value before phi_0, and
separated by a small arbitrary step size, corresponding roughly to 1 e-fold of inflation. Each time, the integration
starts with the initial condition φ = −V 0 /3H (slow-roll prediction). If the found value of φ0 in phi_0 is stable (up to
the parameter "precision"), the code considers that there is an attractor, and stops iterating. If this process does not
converge, it returns an error message.
Parameters
ppm

Input: pointer to primordial structure

ppr

Input: pointer to precision structure

phi_0

Input: field value at which we wish to find the solution

precision

Input: tolerance on output values (if too large, an attractor will always considered to be found)

y

Input: running vector of background variables, already allocated and initialized

dy

Input: running vector of background derivatives, already allocated

H_0

Output: Hubble value at phi_0 for attractor solution

dphidt←_0

Output: field derivative value at phi_0 for attractor solution

Returns
the error status

5.15.2.17

primordial_inflation_evolve_background()

int primordial_inflation_evolve_background (
struct primordial ∗ ppm,
struct precision ∗ ppr,
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double ∗ y,
double ∗ dy,
enum target_quantity target,
double stop,
short check_epsilon,
enum integration_direction direction,
enum time_definition time )

Routine integrating background equations only, from initial values stored in y, to a final value (if target = aH, until aH
= aH_stop; if target = phi, till phi = phi_stop; if target = end_inflation, until d2 a/dt2 = 0 (here t = proper time)). In
output, y contains the final background values. In addition, if check_epsilon is true, the routine controls at each step
that the expansion is accelerated and that inflation holds (wepsilon>1), otherwise it returns an error. Thanks to the
last argument, it is also possible to specify whether the integration should be carried forward or backward in time.
For the inflation_H case, only a 1st order differential equation is involved, so the forward and backward case can
be done exactly without problems. For the inflation_V case, the equation of motion is 2nd order. What the module
will do in the backward case is to search for an approximate solution, corresponding to the (first-order) attractor
inflationary solution. This approximate backward solution is used in order to estimate some initial times, but the
approximation made here will never impact the final result: the module is written in such a way that after using this
approximation, the code always computes (and relies on) the exact forward solution.
Parameters
ppm

Input: pointer to primordial structure

ppr

Input: pointer to precision structure

y

Input/output: running vector of background variables, already allocated and initialized

dy

Input: running vector of background derivatives, already allocated

target

Input: whether the goal is to reach a given aH or φ

stop

Input: the target value of either aH or φ

check_epsilon

Input: whether we should impose inflation (epsilon>1) at each step

direction

Input: whether we should integrate forward or backward in time

time

Input: definition of time (proper or conformal)

Returns
the error status

5.15.2.18

primordial_inflation_check_potential()

int primordial_inflation_check_potential (
struct
double
double
double
double

primordial ∗ ppm,
phi,
∗ V,
∗ dV,
∗ ddV )

Routine checking positivity and negative slope of potential. The negative slope is an arbitrary choice. Currently the
code can only deal with monotonic variations of the inflaton during inflation. So the slope had to be always negative
or always positive... we took the first option.

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Parameters
ppm

Input: pointer to primordial structure

phi

Input: field value where to perform the check

V

Output: inflaton potential in units of M p4

dV

Output: first derivative of inflaton potential wrt the field

ddV

Output: second derivative of inflaton potential wrt the field

Returns
the error status

5.15.2.19

primordial_inflation_check_hubble()

int primordial_inflation_check_hubble (
struct primordial ∗ ppm,
double
double
double
double
double

phi,
∗ H,
∗ dH,
∗ ddH,
∗ dddH )

Routine checking positivity and negative slope of H(φ). The negative slope is an arbitrary choice. Currently the
code can only deal with monotonic variations of the inflaton during inflation. And H can only decrease with time.
So the slope dH/dφ has to be always negative or always positive... we took the first option: phi increases, H
decreases.
Parameters
ppm

Input: pointer to primordial structure

phi

Input: field value where to perform the check

H

Output: Hubble parameters in units of Mp

dH

Output: dH/dφ

ddH

Output: d2 H/dφ2

dddH

Output: d3 H/dφ3

Returns
the error status

5.15.2.20

primordial_inflation_get_epsilon()

int primordial_inflation_get_epsilon (
struct primordial ∗ ppm,
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144

File Documentation

double phi,
double ∗ epsilon )

Routine computing the first slow-roll parameter epsilon

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145

Parameters
ppm

Input: pointer to primordial structure

phi

Input: field value where to compute epsilon

epsilon

Output: result

Returns
the error status

5.15.2.21

primordial_inflation_find_phi_pivot()

int primordial_inflation_find_phi_pivot (
struct primordial ∗ ppm,
struct precision ∗ ppr,
double ∗ y,
double ∗ dy )

Routine searching phi_pivot when a given amount of inflation is requested.
Parameters
ppm

Input/output: pointer to primordial structure

ppr

Input: pointer to precision structure

y

Input: running vector of background variables, already allocated and initialized

dy

Input: running vector of background derivatives, already allocated

Returns
the error status

Summary:

• define local variables
• check whether in vicinity of phi_end, inflation is still ongoing
• case in which epsilon>1: hence we must find the value phi_stop < phi_end where inflation ends up naturally
• –> find latest value of the field such that epsilon = primordial_inflation_small_epsilon (default: 0.1)
• –> bracketing right-hand value is phi_end (but the potential will not be evaluated exactly there, only closeby
• –> bracketing left-hand value is found by iterating with logarithmic step until epsilon < primordial_inflation←_small_epsilon
• –> find value such that epsilon = primordial_inflation_small_epsilon by bisection
• –> value found and stored as phi_small_epsilon
• –> find inflationary attractor in phi_small_epsilon (should exist since epsilon<<1 there)
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• --> compute amount of inflation between this phi_small_epsilon and the end of inflation
• –> by starting from phi_small_epsilon and integrating an approximate solution backward in time, try to estimate roughly a value close to phi_pivot but a bit smaller. This is done by trying to reach an amount of inflation
equal to the requested one, minus the amount after phi_small_epsilon, and plus primordial_inflation_extra←_efolds efolds (default: two). Note that it is not aggressive to require two extra e-folds of inflation before the
pivot, since the calculation of the spectrum in the observable range will require even more.
• –> find attractor in phi_try
• –> check the total amount of inflation between phi_try and the end of inflation
• –> go back to phi_try, and now find phi_pivot such that the amount of inflation between phi_pivot and the end
of inflation is exactly the one requested.
• case in which epsilon<1:
• –> find inflationary attractor in phi_small_epsilon (should exist since epsilon<1 there)
• --> by starting from phi_end and integrating an approximate solution backward in time, try to estimate roughly
a value close to phi_pivot but a bit smaller. This is done by trying to reach an amount of inflation equal to the
requested one, minus the amount after phi_small_epsilon, and plus primordial_inflation_extra_efolds efolds
(default: two). Note that it is not aggressive to require two extra e-folds of inflation before the pivot, since the
calculation of the spectrum in the observable range will require even more.
• –> we now have a value phi_try believed to be close to and slightly smaller than phi_pivot
• –> find attractor in phi_try
• –> check the total amount of inflation between phi_try and the end of inflation
• –> go back to phi_try, and now find phi_pivot such that the amount of inflation between phi_pivot and the end
of inflation is exactly the one requested.
• –> In verbose mode, check that phi_pivot is correct. Done by restarting from phi_pivot and going again till
the end of inflation.

5.15.2.22

primordial_inflation_derivs()

int primordial_inflation_derivs (
double tau,
double ∗ y,
double ∗ dy,
void ∗ parameters_and_workspace,
ErrorMsg error_message )

Routine returning derivative of system of background/perturbation variables. Like other routines used by the generic
integrator (background_derivs, thermodynamics_derivs, perturb_derivs), this routine has a generic list of arguments, and a slightly different error management, with the error message returned directly in an ErrMsg field.
Parameters
tau

Input: time (not used explicitly inside the routine, but requested by the generic
integrator)

y
dy

Input/output: running vector of background variables, already allocated and
initialized
Input: running vector of background derivatives, already allocated

parameters_and_workspace

Input: all necessary input variables apart from y

error_message

Output: error message
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Returns
the error status

5.15.2.23

primordial_external_spectrum_init()

int primordial_external_spectrum_init (
struct perturbs ∗ ppt,
struct primordial ∗ ppm )

This routine reads the primordial spectrum from an external command, and stores the tabulated values. The sampling of the k's given by the external command is preserved.
Author: Jesus Torrado (torradocacho@lorentz.leidenuniv.nl) Date: 2013-12-20
Parameters
ppt

Input/output: pointer to perturbation structure

ppm

Input/output: pointer to primordial structure

Returns
the error status

Summary:

• Initialization
• Launch the command and retrieve the output
• Store the read results into CLASS structures
• Make room
• Store values
• Release the memory used locally
• Tell CLASS that there are scalar (and tensor) modes

5.16

primordial.h File Reference

#include "perturbations.h"
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148

File Documentation

Include dependency graph for primordial.h:

primordial.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

This graph shows which files directly or indirectly include this file:

primordial.h

nonlinear.h

transfer.h

transfer.c

spectra.c

nonlinear_test.h

primordial.c

nonlinear.c

spectra.h

lensing.h

output.h

input.h

nonlinear_exp.h

lensing.c

output.c

input.c

class.h

class.c

Data Structures
• struct primordial

Enumerations
• enum primordial_spectrum_type
• enum linear_or_logarithmic
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5.16 primordial.h File Reference

•
•
•
•
•
•

149

enum potential_shape
enum target_quantity
enum integration_direction
enum time_definition
enum phi_pivot_methods
enum inflation_module_behavior

5.16.1

Detailed Description

Documented includes for primordial module.

5.16.2

Data Structure Documentation

5.16.2.1

struct primordial

Structure containing everything about primordial spectra that other modules need to know.
Once initialized by primordial_init(), contains a table of all primordial spectra as a function of wavenumber, mode,
and pair of initial conditions.
Data Fields
k_pivot

pivot scale in M pc−1

primordial_spec_type

type of primordial spectrum (simple analytic from,
integration of inflationary perturbations, etc.)

double

A_s

usual scalar amplitude = curvature power
spectrum at pivot scale

double

n_s

usual scalar tilt = [curvature power spectrum tilt at
pivot scale -1]

double

alpha_s

usual scalar running

double

beta_s

running of running

double

r

usual tensor to scalar ratio of power spectra,

double
enum primordial_spectrum_type

r = AT /AS = Ph /PR
double

n_t

usual tensor tilt = [GW power spectrum tilt at pivot
scale]

double

alpha_t

usual tensor running

double

f_bi

baryon isocurvature (BI) entropy-to-curvature ratio

Sbi /R
double

n_bi

BI tilt

double

alpha_bi

BI running

double

f_cdi

CDM isocurvature (CDI) entropy-to-curvature ratio

Scdi /R

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double

n_cdi

CDI tilt

double

alpha_cdi

CDI running

double

f_nid

neutrino density isocurvature (NID)
entropy-to-curvature ratio Snid /R

double

n_nid

NID tilt

double

alpha_nid

NID running

double

f_niv

neutrino velocity isocurvature (NIV)
entropy-to-curvature ratio Sniv /R

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Data Fields
double

n_niv

NIV tilt

double

alpha_niv

NIV running

double

c_ad_bi

ADxBI cross-correlation at pivot scale, from -1 to 1

double

n_ad_bi

ADxBI cross-correlation tilt

double

alpha_ad_bi

ADxBI cross-correlation running

double

c_ad_cdi

double

n_ad_cdi

ADxCDI cross-correlation at pivot scale, from -1 to
1
ADxCDI cross-correlation tilt

double

alpha_ad_cdi

ADxCDI cross-correlation running

double

c_ad_nid

double

n_ad_nid

ADxNID cross-correlation at pivot scale, from -1 to
1
ADxNID cross-correlation tilt

double

alpha_ad_nid

ADxNID cross-correlation running

double

c_ad_niv

double

n_ad_niv

ADxNIV cross-correlation at pivot scale, from -1 to
1
ADxNIV cross-correlation tilt

double

alpha_ad_niv

ADxNIV cross-correlation running

double

c_bi_cdi

double

n_bi_cdi

BIxCDI cross-correlation at pivot scale, from -1 to
1
BIxCDI cross-correlation tilt

double

alpha_bi_cdi

BIxCDI cross-correlation running

double

c_bi_nid

double

n_bi_nid

BIxNIV cross-correlation at pivot scale, from -1 to
1
BIxNIV cross-correlation tilt

double

alpha_bi_nid

BIxNIV cross-correlation running

double

c_bi_niv

double

n_bi_niv

BIxNIV cross-correlation at pivot scale, from -1 to
1
BIxNIV cross-correlation tilt

double

alpha_bi_niv

BIxNIV cross-correlation running

double

c_cdi_nid

double

n_cdi_nid

CDIxNID cross-correlation at pivot scale, from -1
to 1
CDIxNID cross-correlation tilt

double

alpha_cdi_nid

CDIxNID cross-correlation running

double

c_cdi_niv

double

n_cdi_niv

CDIxNIV cross-correlation at pivot scale, from -1
to 1
CDIxNIV cross-correlation tilt

double

alpha_cdi_niv

CDIxNIV cross-correlation running

double

c_nid_niv

double

n_nid_niv

NIDxNIV cross-correlation at pivot scale, from -1
to 1
NIDxNIV cross-correlation tilt

double

alpha_nid_niv

NIDxNIV cross-correlation running

potential

parameters describing the case
primordial_spec_type = inflation_V

double

V0

one parameter of the function V(phi)

double

V1

one parameter of the function V(phi)

double

V2

one parameter of the function V(phi)

double

V3

one parameter of the function V(phi)

double

V4

one parameter of the function V(phi)

double

H0

one parameter of the function H(phi)

double

H1

one parameter of the function H(phi)

enum potential_shape

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Data Fields
double

H2

one parameter of the function H(phi)

double

H3

one parameter of the function H(phi)

double

H4

one parameter of the function H(phi)

double

phi_end

value of inflaton at the end of inflation

phi_pivot_method

flag for method used to define and find the pivot
scale
For each of the above methods, critical value to
be reached between pivot and end of inflation
(N_star, [aH]ratio, etc.)

enum phi_pivot_methods
double

enum inflation_module_behavior

behavior

Specifies if the inflation module computes the
primordial spectrum numerically (default) or
analytically

char ∗

command

'external_Pk' mode: command generating the
table of Pk and custom parameters to be passed
to it string with the command for calling
'external_Pk'

double

custom1

one parameter of the primordial computed in
'external_Pk'

double

custom2

one parameter of the primordial computed in
'external_Pk'

double

custom3

one parameter of the primordial computed in
'external_Pk'

double

custom4

one parameter of the primordial computed in
'external_Pk'

double

custom5

one parameter of the primordial computed in
'external_Pk'

double

custom6

one parameter of the primordial computed in
'external_Pk'

double

custom7

one parameter of the primordial computed in
'external_Pk'

double

custom8

one parameter of the primordial computed in
'external_Pk'

double

custom9

one parameter of the primordial computed in
'external_Pk'

double

custom10

one parameter of the primordial computed in
'external_Pk'

int

md_size

number of modes included in computation

int ∗

ic_size

for a given mode, ic_size[index_md] = number of
initial conditions included in computation

int ∗

ic_ic_size

number of ordered pairs of (index_ic1, index_ic2);
this number is just N(N+1)/2 where N =
ic_size[index_md]

lnk_size

number of ln(k) values

lnk

list of ln(k) values lnk[index_k]

int
double ∗

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File Documentation

Data Fields
double ∗∗

lnpk

depends on indices index_md, index_ic1,
index_ic2, index_k as: lnpk[index_md][index_←k∗ppm->ic_ic_size[index_md]+index_ic1_ic2]
where index_ic1_ic2 labels ordered pairs
(index_ic1, index_ic2) (since the primordial
spectrum is symmetric in (index_ic1, index_ic2)).
• for diagonal elements (index_ic1 =
index_ic2) this arrays contains ln[P(k)]
where P(k) is positive by construction.
• for non-diagonal elements this arrays
contains the k-dependent cosine of the
correlation angle, namely P(k )_(index_ic1,
index_ic2)/sqrt[P(k)_index_ic1
P(k)_index_ic2] This choice is convenient
since the sign of the non-diagonal
cross-correlation is arbitrary. For fully
correlated or anti-correlated initial
conditions, this non -diagonal element is
independent on k, and equal to +1 or -1.

double ∗∗

ddlnpk

second derivative of above array, for spline
interpolation. So:
• for index_ic1 = index_ic, we spline ln[P(k)]
vs. ln(k), which is good since this function is
usually smooth.
• for non-diagonal coefficients, we spline
P(k)_(index_ic1,
index_ic2)/sqrt[P(k)_index_ic1
P(k)_index_ic2] vs. ln(k), which is fine since
this quantity is often assumed to be
constant (e.g for fully
correlated/anticorrelated initial conditions)
or nearly constant, and with arbitrary sign.

short ∗∗

is_non_zero

is_non_zero[index_md][index_ic1_ic2] set to false
if pair (index_ic1, index_ic2) is uncorrelated
(ensures more precision and saves time with
respect to the option of simply setting
P(k)_(index_ic1, index_ic2) to zero)

double ∗∗

amplitude

all amplitudes in matrix form:
amplitude[index_md][index_ic1_ic2]

double ∗∗

tilt

all tilts in matrix form: tilt[index_md][index_ic1_ic2]

double ∗∗

running

all runnings in matrix form:
running[index_md][index_ic1_ic2]

int

index_in_a

scale factor

int

index_in_phi

inflaton vev

int

index_in_dphi

its time derivative

int

index_in_ksi_re

Mukhanov variable (real part)

int

index_in_ksi_im

Mukhanov variable (imaginary part)

int

index_in_dksi_re

Mukhanov variable (real part, time derivative)

int

index_in_dksi_im

Mukhanov variable (imaginary part, time
derivative)

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Data Fields
int

index_in_ah_re

tensor perturbation (real part)

int

index_in_ah_im

tensor perturbation (imaginary part)

int

index_in_dah_re

tensor perturbation (real part, time derivative)

int

index_in_dah_im

tensor perturbation (imaginary part, time
derivative)

int

in_bg_size

size of vector of background quantities only

int

in_size

full size of vector

double

phi_pivot

in inflationary module, value of phi_pivot (set to 0
for inflation_V, inflation_H; found by code for
inflation_V_end)

double

phi_min

in inflationary module, value of phi when

kmin = aH
double

phi_max

double

phi_stop

in inflationary module, value of phi when

kmax = aH

short
ErrorMsg

primordial_verbose
error_message

5.16.3

Enumeration Type Documentation

5.16.3.1

primordial_spectrum_type

in inflationary module, value of phi at the end of
inflation
flag regulating the amount of information sent to
standard output (none if set to zero)
zone for writing error messages

enum primordial_spectrum_type

enum defining how the primordial spectrum should be computed

5.16.3.2

linear_or_logarithmic

enum linear_or_logarithmic

enum defining whether the spectrum routine works with linear or logarithmic input/output

5.16.3.3

potential_shape

enum potential_shape

enum defining the type of inflation potential function V(phi)
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5.16.3.4

File Documentation

target_quantity

enum target_quantity

enum defining which quantity plays the role of a target for evolving inflationary equations

5.16.3.5

integration_direction

enum integration_direction

enum specifying if we want to integrate equations forward or backward in time

5.16.3.6

time_definition

enum time_definition

enum specifying if we want to evolve quantities with conformal or proper time

5.16.3.7

phi_pivot_methods

enum phi_pivot_methods

enum specifying how, in the inflation_V_end case, the value of phi_pivot should calculated

5.16.3.8

inflation_module_behavior

enum inflation_module_behavior

enum specifying how the inflation module computes the primordial spectrum (default: numerical)
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5.17 spectra.c File Reference

5.17

155

spectra.c File Reference

#include "spectra.h"
Include dependency graph for spectra.c:

spectra.c

spectra.h

transfer.h

nonlinear.h

sys/shm.h

primordial.h

sys/stat.h

errno.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int spectra_cl_at_l (struct spectra ∗psp, double l, double ∗cl_tot, double ∗∗cl_md, double ∗∗cl_md_ic)
• int spectra_pk_at_z (struct background ∗pba, struct spectra ∗psp, enum linear_or_logarithmic mode, double
z, double ∗output_tot, double ∗output_ic)
• int spectra_pk_at_k_and_z (struct background ∗pba, struct primordial ∗ppm, struct spectra ∗psp, double k,
double z, double ∗pk_tot, double ∗pk_ic)
• int spectra_pk_nl_at_z (struct background ∗pba, struct spectra ∗psp, enum linear_or_logarithmic mode, double z, double ∗output_tot)
• int spectra_pk_nl_at_k_and_z (struct background ∗pba, struct primordial ∗ppm, struct spectra ∗psp, double
k, double z, double ∗pk_tot)
• int spectra_tk_at_z (struct background ∗pba, struct spectra ∗psp, double z, double ∗output)
• int spectra_tk_at_k_and_z (struct background ∗pba, struct spectra ∗psp, double k, double z, double ∗output)
• int spectra_init (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct primordial ∗ppm,
struct nonlinear ∗pnl, struct transfers ∗ptr, struct spectra ∗psp)
• int spectra_free (struct spectra ∗psp)
• int spectra_indices (struct background ∗pba, struct perturbs ∗ppt, struct transfers ∗ptr, struct primordial ∗ppm,
struct spectra ∗psp)
• int spectra_cls (struct background ∗pba, struct perturbs ∗ppt, struct transfers ∗ptr, struct primordial ∗ppm,
struct spectra ∗psp)
• int spectra_compute_cl (struct background ∗pba, struct perturbs ∗ppt, struct transfers ∗ptr, struct primordial
∗ppm, struct spectra ∗psp, int index_md, int index_ic1, int index_ic2, int index_l, int cl_integrand_num_←columns, double ∗cl_integrand, double ∗primordial_pk, double ∗transfer_ic1, double ∗transfer_ic2)
• int spectra_k_and_tau (struct background ∗pba, struct perturbs ∗ppt, struct spectra ∗psp)
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• int spectra_pk (struct background ∗pba, struct perturbs ∗ppt, struct primordial ∗ppm, struct nonlinear ∗pnl,
struct spectra ∗psp)
• int spectra_sigma (struct background ∗pba, struct primordial ∗ppm, struct spectra ∗psp, double R, double z,
double ∗sigma)
• int spectra_matter_transfers (struct background ∗pba, struct perturbs ∗ppt, struct spectra ∗psp)
• int spectra_output_tk_data (struct background ∗pba, struct perturbs ∗ppt, struct spectra ∗psp, enum file_←format output_format, double z, int number_of_titles, double ∗data)

5.17.1

Detailed Description

Documented spectra module
Julien Lesgourgues, 25.08.2010
This module computes the anisotropy and Fourier power spectra ClX , P (k), ...'s given the transfer and Bessel
functions (for anisotropy spectra), the source functions (for Fourier spectra) and the primordial spectra.
The following functions can be called from other modules:

1. spectra_init() at the beginning (but after transfer_init())
2. spectra_cl_at_l() at any time for computing Cl at any l
3. spectra_spectrum_at_z() at any time for computing P(k) at any z
4. spectra_spectrum_at_k_and z() at any time for computing P at any k and z
5. spectra_free() at the end

5.17.2

Function Documentation

5.17.2.1

spectra_cl_at_l()

int spectra_cl_at_l (
struct
double
double
double
double

spectra ∗ psp,
l,
∗ cl_tot,
∗∗ cl_md,
∗∗ cl_md_ic )

Anisotropy power spectra Cl 's for all types, modes and initial conditions.
This routine evaluates all the Cl 's at a given value of l by interpolating in the pre-computed table. When relevant, it
also sums over all initial conditions for each mode, and over all modes.
This function can be called from whatever module at whatever time, provided that spectra_init() has been called
before, and spectra_free() has not been called yet.
Parameters
psp

Input: pointer to spectra structure (containing pre-computed table)

l

Input: multipole number

cl_tot

Output: total Cl 's for all types (TT, TE, EE, etc..)

cl_md

Output: Cl 's for all types (TT, TE, EE, etc..) decomposed mode by mode (scalar, tensor, ...) when
relevant

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Returns
the error status

Summary:

• define local variables
• (a) treat case in which there is only one mode and one initial condition. Then, only cl_tot needs to be filled.
• (b) treat case in which there is only one mode with several initial condition. Fill cl_md_ic[index_md=0] and
sum it to get cl_tot.
• (c) loop over modes
• –> (c.1.) treat case in which the mode under consideration has only one initial condition. Fill cl_md[index_←md].
• –> (c.2.) treat case in which the mode under consideration has several initial conditions. Fill cl_md_ic[index←_md] and sum it to get cl_md[index_md]
• –> (c.3.) add contribution of cl_md[index_md] to cl_tot

5.17.2.2

spectra_pk_at_z()

int spectra_pk_at_z (
struct background ∗ pba,
struct spectra ∗ psp,
enum linear_or_logarithmic mode,
double z,
double ∗ output_tot,
double ∗ output_ic )

Matter power spectrum for arbitrary redshift and for all initial conditions.
This routine evaluates the matter power spectrum at a given value of z by interpolating in the pre-computed table (if
several values of z have been stored) or by directly reading it (if it only contains values at z=0 and we want P(k,z=0))
Can be called in two modes: linear or logarithmic.

• linear: returns P(k) (units: M pc3 )
• logarithmic: returns ln P (k)

One little subtlety: in case of several correlated initial conditions, the cross-correlation spectrum√can be negative.
Then, in logarithmic mode, the non-diagonal elements contain the cross-correlation angle P12 / P11 P22 (from -1
to 1) instead of ln P12
This function can be called from whatever module at whatever time, provided that spectra_init() has been called
before, and spectra_free() has not been called yet.

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Parameters
pba

Input: pointer to background structure (used for converting z into tau)

psp

Input: pointer to spectra structure (containing pre-computed table)

mode

Input: linear or logarithmic

z

Input: redshift

output_tot

Output: total matter power spectrum P(k) in M pc3 (linear mode), or its logarithms (logarithmic
mode)

output_ic

Output: for each pair of initial conditions, matter power spectra P(k) in M pc3 (linear mode), or
their logarithms and cross-correlation angles (logarithmic mode)

Returns
the error status
Summary:
• define local variables
• first step: convert z into ln τ
• second step: for both modes (linear or logarithmic), store the spectrum in logarithmic format in the output
array(s)
• –> (a) if only values at tau=tau_today are stored and we want P (k, z = 0), no need to interpolate
• –> (b) if several values of tau have been stored, use interpolation routine to get spectra at correct redshift
• third step: if there are several initial conditions, compute the total P(k) and set back all uncorrelated coefficients to exactly zero. Check positivity of total P(k).
• fourth step: depending on requested mode (linear or logarithmic), apply necessary transformation to the
output arrays
• –> (a) linear mode: if only one initial condition, convert output_pk to linear format; if several initial conditions,
convert output_ic to linear format, output_tot is already in this format
• –> (b) logarithmic mode: if only one initial condition, nothing to be done; if several initial conditions, convert
output_tot to logarithmic format, output_ic is already in this format

5.17.2.3

spectra_pk_at_k_and_z()

int spectra_pk_at_k_and_z (
struct
struct
struct
double
double

background ∗ pba,
primordial ∗ ppm,
spectra ∗ psp,
k,
z,

double ∗ pk_tot,
double ∗ pk_ic )

Matter power spectrum for arbitrary wavenumber, redshift and initial condition.
This routine evaluates the matter power spectrum at a given value of k and z by interpolating in a table of all P(k)'s
computed at this z by spectra_pk_at_z() (when kmin <= k <= kmax), or eventually by using directly the primordial
spectrum (when 0 <= k < kmin): the latter case is an approximation, valid when kmin << comoving Hubble scale
today. Returns zero when k=0. Returns an error when k<0 or k > kmax.
This function can be called from whatever module at whatever time, provided that spectra_init() has been called
before, and spectra_free() has not been called yet.

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Parameters
pba

Input: pointer to background structure (used for converting z into tau)

ppm

Input: pointer to primordial structure (used only in the case 0 < k < kmin)

psp

Input: pointer to spectra structure (containing pre-computed table)

k

Input: wavenumber in 1/Mpc

z

Input: redshift

pk_tot

Output: total matter power spectrum P(k) in M pc3

pk_ic

Output: for each pair of initial conditions, matter power spectra P(k) in M pc3

Returns
the error status
Summary:
• define local variables
• first step: check that k is in valid range [0:kmax] (the test for z will be done when calling spectra_pk_at_z())
• deal with case 0 <= k < kmin
• –> (a) subcase k=0: then P(k)=0
• –> (b) subcase 0 (a) if only values at tau=tau_today are stored and we want P(k,z=0), no need to interpolate
• –> (b) if several values of tau have been stored, use interpolation routine to get spectra at correct redshift
• fourth step: eventually convert to linear format

5.17.2.5

spectra_pk_nl_at_k_and_z()

int spectra_pk_nl_at_k_and_z (
struct background ∗ pba,
struct primordial ∗ ppm,
struct spectra ∗ psp,
double k,
double z,
double ∗ pk_tot )

Non-linear total matter power spectrum for arbitrary wavenumber and redshift.
This routine evaluates the matter power spectrum at a given value of k and z by interpolating in a table of all P(k)'s
computed at this z by spectra_pk_nl_at_z() (when kmin <= k <= kmax), or eventually by using directly the primordial
spectrum (when 0 <= k < kmin): the latter case is an approximation, valid when kmin << comoving Hubble scale
today. Returns zero when k=0. Returns an error when k<0 or k > kmax.
This function can be called from whatever module at whatever time, provided that spectra_init() has been called
before, and spectra_free() has not been called yet.
Parameters
pba

Input: pointer to background structure (used for converting z into tau)

ppm

Input: pointer to primordial structure (used only in the case 0 < k < kmin)

psp

Input: pointer to spectra structure (containing pre-computed table)

k

Input: wavenumber in 1/Mpc

z

Input: redshift

pk_tot

Output: total matter power spectrum P(k) in M pc3

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Returns
the error status

Summary:

• define local variables
• check that k is in valid range [0:kmax] (the test for z will be done when calling spectra_pk_at_z())
• compute P(k,z) (in logarithmic format for more accurate interpolation)
• get its second derivatives with spline, then interpolate, then convert to linear format

5.17.2.6

spectra_tk_at_z()

int spectra_tk_at_z
struct
struct
double
double

(
background ∗ pba,
spectra ∗ psp,
z,
∗ output )

Matter transfer functions Ti (k) for arbitrary redshift and for all initial conditions.
This routine evaluates the matter transfer functions at a given value of z by interpolating in the pre-computed table
(if several values of z have been stored) or by directly reading it (if it only contains values at z=0 and we want
Ti (k, z = 0))
This function can be called from whatever module at whatever time, provided that spectra_init() has been called
before, and spectra_free() has not been called yet.
Parameters
pba

Input: pointer to background structure (used for converting z into tau)

psp

Input: pointer to spectra structure (containing pre-computed table)

z

Input: redshift

output

Output: matter transfer functions

Returns
the error status
Summary:
• define local variables
• first step: convert z into ln(tau)
• second step: store the matter transfer functions in the output array
• –> (a) if only values at tau=tau_today are stored and we want Ti (k, z = 0), no need to interpolate
• –> (b) if several values of tau have been stored, use interpolation routine to get spectra at correct redshift

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File Documentation

5.17.2.7

spectra_tk_at_k_and_z()

int spectra_tk_at_k_and_z (
struct
struct
double
double
double

background ∗ pba,
spectra ∗ psp,
k,
z,
∗ output )

Matter transfer functions Ti (k) for arbitrary wavenumber, redshift and initial condition.
This routine evaluates the matter transfer functions at a given value of k and z by interpolating in a table of all
Ti (k, z)'s computed at this z by spectra_tk_at_z() (when kmin <= k <= kmax). Returns an error when k kmax.
This function can be called from whatever module at whatever time, provided that spectra_init() has been called
before, and spectra_free() has not been called yet.
Parameters
pba

Input: pointer to background structure (used for converting z into tau)

psp

Input: pointer to spectra structure (containing pre-computed table)

k

Input: wavenumber in 1/Mpc

z

Input: redshift

output

Output: matter transfer functions

Returns
the error status
Summary:
• define local variables
• check that k is in valid range [0:kmax] (the test for z will be done when calling spectra_tk_at_z())
• compute T_i(k,z)
• get its second derivatives w.r.t. k with spline, then interpolate

5.17.2.8

spectra_init()

int spectra_init (
struct precision ∗ ppr,
struct background ∗ pba,
struct
struct
struct
struct
struct

perturbs ∗ ppt,
primordial ∗ ppm,
nonlinear ∗ pnl,
transfers ∗ ptr,
spectra ∗ psp )

This routine initializes the spectra structure (in particular, computes table of anisotropy and Fourier spectra
ClX , P (k), ...)

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Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure (will provide H, Omega_m at redshift of interest)

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfer structure

ppm

Input: pointer to primordial structure

pnl

Input: pointer to nonlinear structure

psp

Output: pointer to initialized spectra structure

Returns
the error status

Summary:

• check that we really want to compute at least one spectrum
• initialize indices and allocate some of the arrays in the spectra structure
• deal with Cl 's, if any
• deal with P (k, τ ) and Ti (k, τ )

5.17.2.9

spectra_free()

int spectra_free (
struct spectra ∗ psp )

This routine frees all the memory space allocated by spectra_init().
To be called at the end of each run, only when no further calls to spectra_cls_at_l(), spectra_pk_at_z(), spectra_←pk_at_k_and_z() are needed.
Parameters
psp

Input: pointer to spectra structure (which fields must be freed)

Returns
the error status

5.17.2.10

spectra_indices()

int spectra_indices (
struct background ∗ pba,
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struct perturbs ∗ ppt,
struct transfers ∗ ptr,
struct primordial ∗ ppm,
struct spectra ∗ psp )

This routine defines indices and allocates tables in the spectra structure
Parameters
pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

ppm

Input: pointer to primordial structure

psp

Input/output: pointer to spectra structure

Returns
the error status

5.17.2.11

spectra_cls()

int spectra_cls (
struct
struct
struct
struct
struct

background ∗ pba,
perturbs ∗ ppt,
transfers ∗ ptr,
primordial ∗ ppm,
spectra ∗ psp )

This routine computes a table of values for all harmonic spectra Cl 's, given the transfer functions and primordial
spectra.
Parameters
pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

ppm

Input: pointer to primordial structure

psp

Input/Output: pointer to spectra structure

Returns
the error status

Summary:

• define local variables
• allocate pointers to arrays where results will be stored
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• store values of l
• loop over modes (scalar, tensors, etc). For each mode:
• –> (a) store number of l values for this mode
• –> (b) allocate arrays where results will be stored
• –> (c) loop over initial conditions
• —> loop over l values defined in the transfer module. For each l, compute the Cl 's for all types (TT, TE, ...) by
convolving primordial spectra with transfer functions. This elementary task is assigned to spectra_compute←_cl()
• –> (d) now that for a given mode, all possible Cl 's have been computed, compute second derivative of the
array in which they are stored, in view of spline interpolation.

5.17.2.12

spectra_compute_cl()

int spectra_compute_cl (
struct background ∗ pba,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
struct primordial ∗ ppm,
struct spectra ∗ psp,
int index_md,
int index_ic1,
int index_ic2,
int index_l,
int cl_integrand_num_columns,
double ∗ cl_integrand,
double ∗ primordial_pk,
double ∗ transfer_ic1,
double ∗ transfer_ic2 )

This routine computes the Cl 's for a given mode, pair of initial conditions and multipole, but for all types (TT, TE...),
by convolving the transfer functions with the primordial spectra.
Parameters
pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

ppm

Input: pointer to primordial structure

psp

Input/Output: pointer to spectra structure (result stored here)

index_md

Input: index of mode under consideration

index_ic1

Input: index of first initial condition in the correlator

index_ic2

Input: index of second initial condition in the correlator

index_l

Input: index of multipole under consideration

cl_integrand_num_columns

Input: number of columns in cl_integrand

cl_integrand

Input: an allocated workspace

primordial_pk

Input: table of primordial spectrum values

transfer_ic1

Input: table of transfer function values for first initial condition

transfer_ic2

Input: table of transfer function values for second initial condition

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Returns
the error status

5.17.2.13

spectra_k_and_tau()

int spectra_k_and_tau (
struct background ∗ pba,
struct perturbs ∗ ppt,
struct spectra ∗ psp )

This routine computes the values of k and tau at which the matter power spectra P (k, τ ) and the matter transfer
functions Ti (k, τ ) will be stored.
Parameters
pba

Input: pointer to background structure (for z to tau conversion)

ppt

Input: pointer to perturbation structure (contain source functions)

psp

Input/Output: pointer to spectra structure

Returns
the error status
Summary:
• define local variables
• check the presence of scalar modes
• check the maximum redshift z_max_pk at which P (k, z) and Ti (k, z) should be computable by interpolation.
If it is equal to zero, only P (k, z = 0) needs to be computed. If it is higher, we will store in a table various
P(k,tau) at several values of tau generously encompassing the range 0= index_ic1; this number is just N(N+1)/2
where N = ic_size[index_md]

is_non_zero

for a given mode, is_non_zero[index_md][index_ic1_ic2] is set to true if
the pair of initial conditions (index_ic1, index_ic2) are statistically
correlated, or to false if they are uncorrelated

int

has_tt

do we want ClT T ? (T = temperature)

int

has_ee

do we want ClEE ? (E = E-polarization)

int

has_te

do we want ClT E ?

int

has_bb

do we want ClBB ? (B = B-polarization)

int

has_pp

do we want Cl ? ( φ = CMB lensing potential)

int

has_tp

do we want Cl

int

has_ep

do we want Cl

int

has_dd

do we want Cldd ? (d = density)

int

has_td

do we want ClT d ?

int

has_pd

do we want Cl ?

int

has_ll

do we want Clll ? (l = galaxy lensing potential)

int

has_tl

do we want ClT l ?

int

has_dl

do we want Cldl ?

int

index_ct_tt

index for type ClT T

int

index_ct_ee

index for type ClEE

int

index_ct_te

index for type ClT E

int

index_ct_bb

index for type ClBB

int

index_ct_pp

index for type Cl

int

index_ct_tp

index for type Cl

int

index_ct_ep

index for type Cl

int

index_ct_dd

first index for type
Cldd ((d_size∗d_size-(d_size-non_diag)∗(d_size-non_diag-1)/2) values)

int

index_ct_td

first index for type ClT d (d_size values)

int

index_ct_pd

first index for type Cl (d_size values)

int

index_ct_ll

first index for type
Clll ((d_size∗d_size-(d_size-non_diag)∗(d_size-non_diag-1)/2) values)

int

index_ct_tl

first index for type ClT l (d_size values)

int

index_ct_dl

first index for type Cldl (d_size values)

short ∗∗

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φφ

Tφ

?

Eφ

?

φd

φφ
Tφ
Eφ

pd

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Data Fields
int

d_size

number of bins for which density Cl's are computed

int

ct_size

number of Cl types requested

l_size

number of multipole values for each requested mode, l_size[index_md]

l_size_max

greatest of all l_size[index_md]

l

list of multipole values l[index_l]

l_max_ct

last multipole (given as an input) at which we want to output Cl 's for a
given mode and type; l[index_md][l_size[index_md]-1] can be larger
than l_max[index_md], in order to ensure a better interpolation with no
boundary effects

l_max

last multipole (given as an input) at which we want to output Cl 's for a
given mode (maximized over types); l[index_md][l_size[index_md]-1]
can be larger than l_max[index_md], in order to ensure a better
interpolation with no boundary effects

l_max_tot

last multipole (given as an input) at which we want to output Cl 's
(maximized over modes and types); l[index_md][l_size[index_md]-1] can
be larger than l_max[index_md], in order to ensure a better interpolation
with no boundary effects

double ∗∗

cl

table of anisotropy spectra for each mode, multipole, pair of initial
conditions and types, cl[index_md][(index_l ∗
psp->ic_ic_size[index_md] + index_ic1_ic2) ∗ psp->ct_size + index_ct]

double ∗∗

ddcl

second derivatives of previous table with respect to l, in view of spline
interpolation

double

alpha_II_2_20

parameter describing adiabatic versus isocurvature contribution in
mutipole range [2,20] (see Planck parameter papers)

double

alpha_RI_2_20

parameter describing adiabatic versus isocurvature contribution in
mutipole range [2,20] (see Planck parameter papers)

double

alpha_RR_2_20

parameter describing adiabatic versus isocurvature contribution in
mutipole range [2,20] (see Planck parameter papers)

double

alpha_II_21_200

parameter describing adiabatic versus isocurvature contribution in
mutipole range [21,200] (see Planck parameter papers)

double

alpha_RI_21_200

parameter describing adiabatic versus isocurvature contribution in
mutipole range [21,200] (see Planck parameter papers)

double

alpha_RR_21_200

parameter describing adiabatic versus isocurvature contribution in
mutipole range [21,200] (see Planck parameter papers)

double

alpha_II_201_2500

parameter describing adiabatic versus isocurvature contribution in
mutipole range [201,2500] (see Planck parameter papers)

double

alpha_RI_201_2500

parameter describing adiabatic versus isocurvature contribution in
mutipole range [201,2500] (see Planck parameter papers)

double

alpha_RR_201_2500

parameter describing adiabatic versus isocurvature contribution in
mutipole range [201,2500] (see Planck parameter papers)

double

alpha_II_2_2500

parameter describing adiabatic versus isocurvature contribution in
mutipole range [2,2500] (see Planck parameter papers)

double

alpha_RI_2_2500

parameter describing adiabatic versus isocurvature contribution in
mutipole range [2,2500] (see Planck parameter papers)

double

alpha_RR_2_2500

parameter describing adiabatic versus isocurvature contribution in
mutipole range [2,2500] (see Planck parameter papers)

double

alpha_kp

parameter describing adiabatic versus isocurvature contribution at pivot
scale (see Planck parameter papers)

double

alpha_k1

parameter describing adiabatic versus isocurvature contribution at scale
k1 (see Planck parameter papers)

int ∗
int
double ∗
int ∗∗

int ∗

int

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Data Fields
double

alpha_k2

parameter describing adiabatic versus isocurvature contribution at scale
k2 (see Planck parameter papers)

int

ln_k_size

number ln(k) values

ln_k

list of ln(k) values ln_k[index_k]

ln_tau_size

number ln(tau) values (only one if z_max_pk = 0)

double ∗

ln_tau

list of ln(tau) values ln_tau[index_tau]

double ∗

ln_pk

Matter power spectrum. depends on indices index_md, index_ic1,
index_ic2, index_k, index_tau as: ln_pk[(index_tau ∗ psp->k_size +
index_k)∗ psp->ic_ic_size[index_md] + index_ic1_ic2] where
index_ic1_ic2 labels ordered pairs (index_ic1, index_ic2) (since the
primordial spectrum is symmetric in (index_ic1, index_ic2)).

double ∗
int

• for diagonal elements (index_ic1 = index_ic2) this arrays contains
ln[P(k)] where P(k) is positive by construction.
• for non-diagonal elements this arrays contains the k-dependent
cosine of the correlation angle, namely P(k)_(index_ic1,
index_ic2)/sqrt[P(k)_index_ic1 P(k)_index_ic2] This choice is
convenient since the sign of the non-diagonal cross-correlation is
arbitrary. For fully correlated or anti-correlated initial conditions,
this non-diagonal element is independent on k, and equal to +1 or
-1.
double ∗

ddln_pk

second derivative of above array with respect to log(tau), for spline
interpolation. So:
• for index_ic1 = index_ic, we spline ln[P(k)] vs. ln(k), which is good
since this function is usually smooth.
• for non-diagonal coefficients, we spline P(k)_(index_ic1,
index_ic2)/sqrt[P(k)_index_ic1 P(k)_index_ic2] vs. ln(k), which is
fine since this quantity is often assumed to be constant (e.g for
fully correlated/anticorrelated initial conditions) or nearly constant,
and with arbitrary sign.

sigma8

sigma8 parameter

double ∗

double

ln_pk_nl

Non-linear matter power spectrum. depends on indices index_k,
index_tau as: ln_pk_nl[index_tau ∗ psp->k_size + index_k]

double ∗

ddln_pk_nl

second derivative of above array with respect to log(tau), for spline
interpolation.

int

index_tr_delta_g

index of gamma density transfer function

int

index_tr_delta_b

index of baryon density transfer function

int

index_tr_delta_cdm

index of cold dark matter density transfer function

int

index_tr_delta_dcdm

index of decaying cold dark matter density transfer function

int

index_tr_delta_scf

index of scalar field phi transfer function

int

index_tr_delta_fld

index of dark energy fluid density transfer function

int

index_tr_delta_ur

index of ultra-relativistic neutrinos/relics density transfer function

int

index_tr_delta_dr

index of decay radiation density transfer function

int

index_tr_delta_ncdm1

index of first species of non-cold dark matter (massive neutrinos, ...)
density transfer function

int

index_tr_delta_tot

index of total matter density transfer function

int

index_tr_theta_g

index of gamma velocity transfer function

int

index_tr_theta_b

index of baryon velocity transfer function

int

index_tr_theta_cdm

index of cold dark matter velocity transfer function

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Data Fields
int

index_tr_theta_dcdm

index of decaying cold dark matter velocity transfer function

int

index_tr_theta_scf

index of derivative of scalar field phi transfer function

int

index_tr_theta_fld

index of dark energy fluid velocity transfer function

int

index_tr_theta_ur

index of ultra-relativistic neutrinos/relics velocity transfer function

int

index_tr_theta_dr

index of decay radiation velocity transfer function

int

index_tr_theta_ncdm1

index of first species of non-cold dark matter (massive neutrinos, ...)
velocity transfer function

int

index_tr_theta_tot

index of total matter velocity transfer function

int

index_tr_phi

index of Bardeen potential phi

int

index_tr_psi

index of Bardeen potential psi

int

index_tr_phi_prime

index of derivative of Bardeen potential phi

int

index_tr_h

index of synchronous gauge metric perturbation h

int

index_tr_h_prime

index of synchronous gauge metric perturbation h'

int

index_tr_eta

index of synchronous gauge metric perturbation eta

int

index_tr_eta_prime

index of synchronous gauge metric perturbation eta'

int

tr_size

total number of species in transfer functions

double ∗

matter_transfer

Matter transfer functions. Depends on indices
index_md,index_tau,index_ic,index_k, index_tr as:
matter_transfer[((index_tau∗psp->ln_k_size + index_k) ∗
psp->ic_size[index_md] + index_ic) ∗ psp->tr_size + index_tr]

double ∗

ddmatter_transfer

second derivative of above array with respect to log(tau), for spline
interpolation.

spectra_verbose

flag regulating the amount of information sent to standard output (none if
set to zero)

error_message

zone for writing error messages

short
ErrorMsg

5.19

thermodynamics.c File Reference

#include "thermodynamics.h"
#include "hyrec.h"
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175

Include dependency graph for thermodynamics.c:

thermodynamics.c

thermodynamics.h

hyrec.h

background.h

quadrature.h

growTable.h

arrays.h

dei_rkck.h

parser.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int thermodynamics_at_z (struct background ∗pba, struct thermo ∗pth, double z, short inter_mode, int ∗last←_index, double ∗pvecback, double ∗pvecthermo)
• int thermodynamics_init (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth)
• int thermodynamics_free (struct thermo ∗pth)
• int thermodynamics_indices (struct thermo ∗pth, struct recombination ∗preco, struct reionization ∗preio)
• int thermodynamics_helium_from_bbn (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth)
• int thermodynamics_onthespot_energy_injection (struct precision ∗ppr, struct background ∗pba, struct recombination ∗preco, double z, double ∗energy_rate, ErrorMsg error_message)
• int thermodynamics_energy_injection (struct precision ∗ppr, struct background ∗pba, struct recombination
∗preco, double z, double ∗energy_rate, ErrorMsg error_message)
• int thermodynamics_reionization_function (double z, struct thermo ∗pth, struct reionization ∗preio, double
∗xe)
• int thermodynamics_get_xe_before_reionization (struct precision ∗ppr, struct thermo ∗pth, struct recombination ∗preco, double z, double ∗xe)
• int thermodynamics_reionization (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct
recombination ∗preco, struct reionization ∗preio, double ∗pvecback)
• int thermodynamics_reionization_sample (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth,
struct recombination ∗preco, struct reionization ∗preio, double ∗pvecback)
• int thermodynamics_recombination (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct
recombination ∗preco, double ∗pvecback)
• int thermodynamics_recombination_with_hyrec (struct precision ∗ppr, struct background ∗pba, struct thermo
∗pth, struct recombination ∗preco, double ∗pvecback)
• int thermodynamics_recombination_with_recfast (struct precision ∗ppr, struct background ∗pba, struct
thermo ∗pth, struct recombination ∗preco, double ∗pvecback)
• int thermodynamics_derivs_with_recfast (double z, double ∗y, double ∗dy, void ∗parameters_and_workspace,
ErrorMsg error_message)
• int thermodynamics_merge_reco_and_reio (struct precision ∗ppr, struct thermo ∗pth, struct recombination
∗preco, struct reionization ∗preio)
• int thermodynamics_output_titles (struct background ∗pba, struct thermo ∗pth, char titles[_MAXTITLESTR←INGLENGTH_])
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File Documentation

5.19.1

Detailed Description

Documented thermodynamics module
Julien Lesgourgues, 6.09.2010
Deals with the thermodynamical evolution. This module has two purposes:

• at the beginning, to initialize the thermodynamics, i.e. to integrate the thermodynamical equations, and store
all thermodynamical quantities as a function of redshift inside an interpolation table. The current version of
recombination is based on RECFAST v1.5. The current version of reionization is based on exactly the same
reionization function as in CAMB, in order to make allow for comparison. It should be easy to generalize the
module to more complicated reionization histories.
• to provide a routine which allow other modules to evaluate any thermodynamical quantities at a given redshift
value (by interpolating within the interpolation table).

The logic is the following:

• in a first step, the code assumes that there is no reionization, and computes the ionization fraction, Thomson scattering rate, baryon temperature, etc., using RECFAST. The result is stored in a temporary table
'recombination_table' (within a temporary structure of type 'recombination') for each redshift in a range 0 < z
< z_initial. The sampling in z space is done with a simple linear step size.
• in a second step, the code adds the reionization history, starting from a redshift z_reio_start. The ionization
fraction at this redshift is read in the previous recombination table in order to ensure a perfect matching.
The code computes the ionization fraction, Thomson scattering rate, baryon temperature, etc., using a given
parametrization of the reionization history. The result is stored in a temporary table 'reionization_table' (within
a temporary structure of type 'reionization') for each redshift in the range 0 < z < z_reio_start. The sampling
in z space is found automatically, given the precision parameter 'reionization_sampling'.
• in a third step, the code merges the two tables 'recombination_table' and 'reionization_table' inside the table 'thermodynamics_table', and the temporary structures 'recombination' and 'reionization' are freed. In
'thermodynamics_table', the sampling in z space is the one defined in the recombination algorithm for z_←reio_start < z < z_initial, and the one defined in the reionization algorithm for 0 < z < z_reio_start.
• at this stage, only a few columns in the table 'thermodynamics_table' have been filled. In a fourth step,
the remaining columns are filled, using some numerical integration/derivation routines from the 'array.c' tools
module.
• small detail: one of the columns contains the maximum variation rate of a few relevant thermodynamical
quantities. This rate will be used for defining automatically the sampling step size in the perturbation module.
Hence, the exact value of this rate is unimportant, but its order of magnitude at a given z defines the sampling
precision of the perturbation module. Hence, it is harmless to use a smoothing routine in order to make this
rate look nicer, although this will not affect the final result significantly. The last step in the thermodynamics←_init module is to perform this smoothing.

In summary, the following functions can be called from other modules:

1. thermodynamics_init() at the beginning (but after background_init())
2. thermodynamics_at_z() at any later time
3. thermodynamics_free() at the end, when no more calls to thermodynamics_at_z() are needed
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5.19.2

Function Documentation

5.19.2.1

thermodynamics_at_z()

177

int thermodynamics_at_z (
struct background ∗ pba,
struct thermo ∗ pth,
double z,
short inter_mode,
int ∗ last_index,
double ∗ pvecback,
double ∗ pvecthermo )

Thermodynamics quantities at given redshift z.
Evaluates all thermodynamics quantities at a given value of the redshift by reading the pre-computed table and
interpolating.
Parameters
pba

Input: pointer to background structure

pth

Input: pointer to the thermodynamics structure (containing pre-computed table)

z

Input: redshift

inter_mode

Input: interpolation mode (normal or growing_closeby)

last_index

Input/Output: index of the previous/current point in the interpolation array (input only for closeby
mode, output for both)

pvecback

Input: vector of background quantities (used only in case z>z_initial for getting ddkappa and
dddkappa; in that case, should be already allocated and filled, with format short_info or larger; in
other cases, will be ignored)

pvecthermo

Output: vector of thermodynamics quantities (assumed to be already allocated)

Returns
the error status
Summary:
• define local variables
• interpolate in table with array_interpolate_spline() (normal mode) or array_interpolate_spline_growing_←closeby() (closeby mode)

5.19.2.2

thermodynamics_init()

int thermodynamics_init (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth )

Initialize the thermo structure, and in particular the thermodynamics interpolation table.

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Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input/Output: pointer to initialized thermo structure

Returns
the error status

Summary:

• define local variables
• initialize pointers, allocate background vector
• compute and check primordial Helium fraction
• check energy injection parameters
• assign values to all indices in the structures with thermodynamics_indices()
• solve recombination and store values of z, xe , dκ/dτ, Tb , c2b with thermodynamics_recombination()
• if there is reionization, solve reionization and store values of z, xe , dκ/dτ, Tb , c2b with thermodynamics_←reionization()
• merge tables in recombination and reionization structures into a single table in thermo structure
• compute table of corresponding conformal times
• store initial value of conformal time in the structure
• fill missing columns (quantities not computed previously but related)
• –> baryon drag interaction rate time minus one, -[R ∗ kappa'], stored temporarily in column ddkappa
• –> second derivative of this rate, -[R ∗ kappa']'', stored temporarily in column dddkappa
• –> compute tau_d = [int_{tau_today}∧ {tau} dtau -dkappa_d/dtau]
• –> compute r_d = [int_{tau_ini}∧ {tau} dtau [1/kappa']
• –> second derivative with respect to tau of dkappa (in view of spline interpolation)
• –> first derivative with respect to tau of dkappa (using spline interpolation)
• –> compute -kappa = [int_{tau_today}∧ {tau} dtau dkappa/dtau], store temporarily in column "g"
• –> derivatives of baryon sound speed (only computed if some non-minimal tight-coupling schemes is requested)
• —> second derivative with respect to tau of cb2
• —> first derivative with respect to tau of cb2 (using spline interpolation)
• –> compute visibility: g = (dκ/dτ )e−κ
• —> compute g
• —> compute exp(-kappa)
• —> compute g' (the plus sign of the second term is correct, see def of -kappa in thermodynamics module!)
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• —> compute g''
• —> store g
• —> compute variation rate
• smooth the rate (details of smoothing unimportant: only the order of magnitude of the rate matters)
• fill tables of second derivatives with respect to z (in view of spline interpolation)
• find maximum of g
• find conformal recombination time using background_tau_of_z()
• find damping scale at recombination (using linear interpolation)
• find time (always after recombination) at which tau_c/tau falls below some threshold, defining tau_free_←streaming
• find baryon drag time (when tau_d crosses one, using linear interpolation) and sound horizon at that time
• find time above which visibility falls below a given fraction of its maximum
• if verbose flag set to next-to-minimum value, print the main results

5.19.2.3

thermodynamics_free()

int thermodynamics_free (
struct thermo ∗ pth )

Free all memory space allocated by thermodynamics_init().
Parameters
pth

Input/Output: pointer to thermo structure (to be freed)

Returns
the error status

5.19.2.4

thermodynamics_indices()

int thermodynamics_indices (
struct thermo ∗ pth,
struct recombination ∗ preco,
struct reionization ∗ preio )

Assign value to each relevant index in vectors of thermodynamical quantities, as well as in vector containing reionization parameters.

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Parameters
pth

Input/Output: pointer to thermo structure

preco

Input/Output: pointer to recombination structure

preio

Input/Output: pointer to reionization structure

Returns
the error status

Summary:

• define local variables
• initialization of all indices and flags in thermo structure
• initialization of all indices and flags in recombination structure
• initialization of all indices and flags in reionization structure
• same with parameters of the function Xe (z)

5.19.2.5

thermodynamics_helium_from_bbn()

int thermodynamics_helium_from_bbn (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth )

Infer the primordial helium fraction from standard BBN, as a function of the baryon density and expansion rate during
BBN.
This module is simpler then the one used in arXiv:0712.2826 because it neglects the impact of a possible significant
chemical potentials for electron neutrinos. The full code with xi_nu_e could be introduced here later.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input/Output: pointer to initialized thermo structure

Returns
the error status

Summary:

• Infer effective number of neutrinos at the time of BBN
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• 8.6173e-11 converts from Kelvin to MeV. We randomly choose 0.1 MeV to be the temperature of BBN
• compute Delta N_eff as defined in bbn file, i.e. ∆Nef f = 0 means Nef f = 3.046
• spline in one dimension (along deltaN)
• interpolate in one dimension (along deltaN)
• spline in remaining dimension (along omegab)
• interpolate in remaining dimension (along omegab)
• deallocate arrays

5.19.2.6

thermodynamics_onthespot_energy_injection()

int thermodynamics_onthespot_energy_injection (
struct precision ∗ ppr,
struct background ∗ pba,
struct recombination ∗ preco,
double z,
double ∗ energy_rate,
ErrorMsg error_message )

In case of non-minimal cosmology, this function determines the energy rate injected in the IGM at a given redshift z
(= on-the-spot annihilation). This energy injection may come e.g. from dark matter annihilation or decay.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

preco

Input: pointer to recombination structure

z

Input: redshift

energy_rate

Output: energy density injection rate

error_message

Output: error message

Returns
the error status

5.19.2.7

thermodynamics_energy_injection()

int thermodynamics_energy_injection (
struct precision ∗ ppr,
struct background ∗ pba,
struct recombination ∗ preco,
double z,
double ∗ energy_rate,
ErrorMsg error_message )
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In case of non-minimal cosmology, this function determines the effective energy rate absorbed by the IGM at a given
redshift (beyond the on-the-spot annihilation). This energy injection may come e.g. from dark matter annihilation or
decay.

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Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

preco

Input: pointer to recombination structure

z

Input: redshift

energy_rate

Output: energy density injection rate

error_message

Output: error message

Returns
the error status

5.19.2.8

thermodynamics_reionization_function()

int thermodynamics_reionization_function (
double z,
struct thermo ∗ pth,
struct reionization ∗ preio,
double ∗ xe )

This subroutine contains the reionization function Xe (z) (one for each scheme; so far, only the function corresponding to the reio_camb scheme is coded)
Parameters
z

Input: redshift

pth

Input: pointer to thermo structure, to know which scheme is used

preio

Input: pointer to reionization structure, containing the parameters of the function Xe (z)

xe

Output: Xe (z)

Summary:

• define local variables
• implementation of ionization function similar to the one in CAMB
• –> case z > z_reio_start
• –> case z < z_reio_start: hydrogen contribution (tanh of complicated argument)
• –> case z < z_reio_start: helium contribution (tanh of simpler argument)
• implementation of binned ionization function similar to astro-ph/0606552
• –> case z > z_reio_start
• implementation of many tanh jumps
• –> case z > z_reio_start
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• implementation of reio_inter
• –> case z > z_reio_start

5.19.2.9

thermodynamics_get_xe_before_reionization()

int thermodynamics_get_xe_before_reionization (
struct precision ∗ ppr,
struct thermo ∗ pth,
struct recombination ∗ preco,
double z,
double ∗ xe )

This subroutine reads Xe (z) in the recombination table at the time at which reionization starts. Hence it provides
correct initial conditions for the reionization function.
Parameters
ppr

Input: pointer to precision structure

pth

Input: pointer to thermo structure

preco

Input: pointer to recombination structure

z

Input: redshift z_reio_start

xe

Output: Xe (z) at z

5.19.2.10

thermodynamics_reionization()

int thermodynamics_reionization (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct recombination ∗ preco,
struct reionization ∗ preio,
double ∗ pvecback )

This routine computes the reionization history. In the reio_camb scheme, this is straightforward if the input parameter
is the reionization redshift. If the input is the optical depth, need to find z_reio by dichotomy (trying several z_reio
until the correct tau_reio is approached).
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermo structure

preco

Input: pointer to filled recombination structure

preio

Input/Output: pointer to reionization structure (to be filled)

pvecback

Input: vector of background quantities (used as workspace: must be already allocated, with format
short_info or larger, but does not need to be filled)
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Returns
the error status

Summary:

• define local variables
• allocate the vector of parameters defining the function Xe (z)
• (a) if reionization implemented like in CAMB
• –> set values of these parameters, excepted those depending on the reionization redshift
• –> if reionization redshift given as an input, initialize the remaining values and fill reionization table
• –> if reionization optical depth given as an input, find reionization redshift by dichotomy and initialize the
remaining values
• (b) if reionization implemented with reio_bins_tanh scheme
• (c) if reionization implemented with reio_many_tanh scheme
• (d) if reionization implemented with reio_inter scheme

5.19.2.11

thermodynamics_reionization_sample()

int thermodynamics_reionization_sample (
struct precision ∗ ppr,
struct
struct
struct
struct
double

background ∗ pba,
thermo ∗ pth,
recombination ∗ preco,
reionization ∗ preio,
∗ pvecback )

For fixed input reionization parameters, this routine computes the reionization history and fills the reionization table.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermo structure

preco

Input: pointer to filled recombination structure

preio

Input/Output: pointer to reionization structure (to be filled)

pvecback

Input: vector of background quantities (used as workspace: must be already allocated, with format
short_info or larger, but does not need to be filled)

Returns
the error status

Summary:
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• define local variables
• (a) allocate vector of values related to reionization
• (b) create a growTable with gt_init()
• (c) first line is taken from thermodynamics table, just before reionization starts
• –> look where to start in current thermodynamics table
• –> get redshift
• –> get Xe
• –> get dκ/dz = (dκ/dτ ) ∗ (dτ /dz) = −(dκ/dτ )/H
• –> get baryon temperature
• –> after recombination, Tb scales like (1+z)∗∗2. Compute constant factor Tb/(1+z)∗∗2.
• –> get baryon sound speed
• –> store these values in growing table
• (d) set the maximum step value (equal to the step in thermodynamics table)
• (e) loop over redshift values in order to find values of z, x_e, kappa' (Tb and cb2 found later by integration).
The sampling in z space is found here.
• (f) allocate reionization_table with correct size
• (g) retrieve data stored in the growTable with gt_getPtr()
• (h) copy growTable to reionization_temporary_table (invert order of lines, so that redshift is growing, like in
recombination table)
• (i) free the growTable with gt_free() , free vector of reionization variables
• (j) another loop on z, to integrate equation for Tb and to compute cb2
• –> derivative of baryon temperature
• –> increment baryon temperature
• –> get baryon sound speed
• –> spline dτ /dz with respect to z in view of integrating for optical depth
• –> integrate for optical depth

5.19.2.12

thermodynamics_recombination()

int thermodynamics_recombination (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
struct recombination ∗ preco,
double ∗ pvecback )

Integrate thermodynamics with your favorite recombination code.
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5.19.2.13

187

thermodynamics_recombination_with_hyrec()

int thermodynamics_recombination_with_hyrec (
struct
struct
struct
struct
double

precision ∗ ppr,
background ∗ pba,
thermo ∗ pth,
recombination ∗ preco,
∗ pvecback )

Integrate thermodynamics with HyRec.
Integrate thermodynamics with HyRec, allocate and fill the part of the thermodynamics interpolation table (the rest
is filled in thermodynamics_init()). Called once by thermodynamics_recombination(), from thermodynamics_init().

HYREC: Hydrogen and Helium Recombination Code
Written by Yacine Ali-Haimoud and Chris Hirata (Caltech)

Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

preco

Output: pointer to recombination structure

pvecback

Input: pointer to an allocated (but empty) vector of background variables

Summary:

• Fill hyrec parameter structure
• Build effective rate tables
• distribute addresses for each table
• Normalize 2s–1s differential decay rate to L2s1s (can be set by user in hydrogen.h)
• Compute the recombination history by calling a function in hyrec (no CLASS-like error management here)
• fill a few parameters in preco and pth
• allocate memory for thermodynamics interpolation tables (size known in advance) and fill it
• –> get redshift, corresponding results from hyrec, and background quantities
• –> store the results in the table

5.19.2.14

thermodynamics_recombination_with_recfast()

int thermodynamics_recombination_with_recfast (
struct precision ∗ ppr,
struct background ∗ pba,
struct thermo ∗ pth,
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struct recombination ∗ preco,
double ∗ pvecback )

Integrate thermodynamics with RECFAST.
Integrate thermodynamics with RECFAST, allocate and fill the part of the thermodynamics interpolation table (the
rest is filled in thermodynamics_init()). Called once by thermodynamics_recombination, from thermodynamics_←init().
RECFAST is an integrator for Cosmic Recombination of Hydrogen and Helium, developed by Douglas Scott
(dscott@astro.ubc.ca) based on calculations in the paper Seager, Sasselov & Scott (ApJ, 523, L1, 1999).
and "fudge" updates in Wong, Moss & Scott (2008).
Permission to use, copy, modify and distribute without fee or royalty at any tier, this software and its documentation,
for any purpose and without fee or royalty is hereby granted, provided that you agree to comply with the following
copyright notice and statements, including the disclaimer, and that the same appear on ALL copies of the software
and documentation, including modifications that you make for internal use or for distribution:
Copyright 1999-2010 by University of British Columbia. All rights reserved.
THIS SOFTWARE IS PROVIDED "AS IS", AND U.B.C. MAKES NO REPRESENTATIONS OR WARRANTIES,
EXPRESS OR IMPLIED. BY WAY OF EXAMPLE, BUT NOT LIMITATION, U.B.C. MAKES NO REPRESENTAT←IONS OR WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PARTICULAR PURPOSE OR THAT
THE USE OF THE LICENSED SOFTWARE OR DOCUMENTATION WILL NOT INFRINGE ANY THIRD PARTY
PATENTS, COPYRIGHTS, TRADEMARKS OR OTHER RIGHTS.
Version 1.5: includes extra fitting function from Rubino-Martin et al. arXiv:0910.4383v1 [astro-ph.CO]
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

preco

Output: pointer to recombination structure

pvecback

Input: pointer to an allocated (but empty) vector of background variables

Returns
the error status

Summary:

• define local variables
• allocate memory for thermodynamics interpolation tables (size known in advance)
• initialize generic integrator with initialize_generic_integrator()
• read a few precision/cosmological parameters
• define the fields of the 'thermodynamics parameter and workspace' structure
• impose initial conditions at early times
• loop over redshift steps Nz; integrate over each step with generic_integrator(), store the results in the table
using thermodynamics_derivs_with_recfast()
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• –> first approximation: H and Helium fully ionized
• –> second approximation: first Helium recombination (analytic approximation)
• –> third approximation: first Helium recombination completed
• –> fourth approximation: second Helium recombination starts (analytic approximation)
• –> fifth approximation: second Helium recombination (full evolution for Helium), H recombination starts (analytic approximation)
• –> last case: full evolution for H and Helium
• –> store the results in the table
• cleanup generic integrator with cleanup_generic_integrator()

5.19.2.15

thermodynamics_derivs_with_recfast()

int thermodynamics_derivs_with_recfast (
double z,
double ∗ y,
double ∗ dy,
void ∗ parameters_and_workspace,
ErrorMsg error_message )

Subroutine evaluating the derivative with respect to redshift of thermodynamical quantities (from RECFAST version
1.4).
Computes derivatives of the three variables to integrate: dxH /dz, dxHe /dz, dTmat /dz .
This is one of the few functions in the code which are passed to the generic_integrator() routine. Since generic_←integrator() should work with functions passed from various modules, the format of the arguments is a bit special:

• fixed parameters and workspaces are passed through a generic pointer. Here, this pointer contains the precision, background and recombination structures, plus a background vector, but generic_integrator() doesn't
know its fine structure.
• the error management is a bit special: errors are not written as usual to pth->error_message, but to a generic
error_message passed in the list of arguments.

Parameters
z

Input: redshift

y

Input: vector of variable to integrate

dy

Output: its derivative (already allocated)

parameters_and_workspace

Input: pointer to fixed parameters (e.g. indices) and workspace (already allocated)

error_message

Output: error message

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5.19.2.16

thermodynamics_merge_reco_and_reio()

int thermodynamics_merge_reco_and_reio (
struct
struct
struct
struct

precision ∗ ppr,
thermo ∗ pth,
recombination ∗ preco,
reionization ∗ preio )

This routine merges the two tables 'recombination_table' and 'reionization_table' inside the table 'thermodynamics←_table', and frees the temporary structures 'recombination' and 'reionization'.
Parameters
ppr

Input: pointer to precision structure

pth

Input/Output: pointer to thermo structure

preco

Input: pointer to filled recombination structure

preio

Input: pointer to reionization structure

Returns
the error status

Summary:

• define local variables
• first, a little check that the two tables match each other and can be merged
• find number of redshift in full table = number in reco + number in reio - overlap
• allocate arrays in thermo structure
• fill these arrays
• free the temporary structures

5.19.2.17

thermodynamics_output_titles()

int thermodynamics_output_titles (
struct background ∗ pba,
struct thermo ∗ pth,
char titles[_MAXTITLESTRINGLENGTH_] )

Subroutine for formatting thermodynamics output
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5.20 thermodynamics.h File Reference

5.20

191

thermodynamics.h File Reference

#include "background.h"
Include dependency graph for thermodynamics.h:

thermodynamics.h

background.h

quadrature.h

growTable.h

arrays.h

dei_rkck.h

parser.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

This graph shows which files directly or indirectly include this file:

thermodynamics.h

perturbations.h

perturbations.c

nonlinear.h

transfer.h

transfer.c

spectra.h

lensing.h

spectra.c

lensing.c

input.h

primordial.h

nonlinear_exp.h

nonlinear.c

output.h

output.c

input.c

class.h

class.c

Data Structures
•
•
•
•

struct thermo
struct recombination
struct reionization
struct thermodynamics_parameters_and_workspace

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thermodynamics.c

nonlinear_test.h

primordial.c

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File Documentation

Macros
• #define f1(x) (-0.75∗x∗(x∗x/3.-1.)+0.5)
• #define f2(x) (x∗x∗(0.5-x/3.)∗6.)

• #define _YHE_BIG_ 0.5
• #define _YHE_SMALL_ 0.01

Enumerations
• enum recombination_algorithm
• enum reionization_parametrization {
reio_none, reio_camb, reio_bins_tanh, reio_half_tanh,
reio_many_tanh, reio_inter }
• enum reionization_z_or_tau { reio_z, reio_tau }

5.20.1

Detailed Description

Documented includes for thermodynamics module

5.20.2

Data Structure Documentation

5.20.2.1

struct thermo

All thermodynamics parameters and evolution that other modules need to know.
Once initialized by thermodynamics_init(), contains all the necessary information on the thermodynamics, and in
particular, a table of thermodynamical quantities as a function of the redshift, used for interpolation in other modules.
Data Fields
YHe

YHe : primordial helium fraction

recombination

recombination code

reio_parametrization

reionization scheme

reio_z_or_tau

is the input parameter the
reionization redshift or optical depth?

double

tau_reio

if above set to tau, input value of
reionization optical depth

double

z_reio

if above set to z, input value of
reionization redshift
do we want to include in computation
derivatives of baryon sound speed?

double
enum recombination_algorithm
enum reionization_parametrization
enum reionization_z_or_tau

short

compute_cb2_derivatives

short

compute_damping_scale

do we want to compute the simplest
analytic approximation to the photon
damping (or diffusion) scale?

reionization_width

parameters for reio_camb width of H
reionization

double

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Data Fields
double

reionization_exponent

shape of H reionization

double

helium_fullreio_redshift

redshift for of helium reionization

double

helium_fullreio_width

width of helium reionization

binned_reio_num

parameters for reio_bins_tanh with
how many bins do we want to
describe reionization?
central z value for each bin

int

double ∗

binned_reio_z

double ∗

binned_reio_xe

double
int

many_tanh_num

double ∗

many_tanh_z

double ∗

many_tanh_xe

imposed Xe (z) value at the end of
each jump (ie at later times)

many_tanh_width

sharpness of tanh() steps

reio_inter_num

double
int

double ∗

reio_inter_z

parameters for reio_inter with how
many jumps do we want to describe
reionization?
discrete z values

double ∗

reio_inter_xe

discrete Xe (z) values

annihilation

parameters for energy injection

has_on_the_spot

parameter describing CDM
annihilation (f  / m_cdm,
see e.g. 0905.0003)

double

decay

flag to specify if we want to use the
on-the-spot approximation

double

annihilation_variation

parameter describing CDM decay
(f/tau, see e.g. 1109.6322)

double

annihilation_z

if this parameter is non-zero, the
function F(z)=(f  /
m_cdm)(z) will be a parabola in
log-log scale between zmin and
zmax, with a curvature given by
annihlation_variation (must be
negative), and with a maximum in
zmax; it will be constant outside this
range

double

annihilation_zmax

if annihilation_variation is non-zero,
this is the value of z at which the
parameter annihilation is defined, i.e.
F(annihilation_z)=annihilation

double

annihilation_zmin

double

annihilation_f_halo

double

annihilation_z_halo

if annihilation_variation is non-zero,
redshift above which annihilation rate
is maximal
if annihilation_variation is non-zero,
redshift below which annihilation rate
is constant
takes the contribution of DM
annihilation in halos into account
ionization fraction xe

double
short

int

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binned_reio_step_sharpness

imposed Xe (z) value at center of
each bin
sharpness of tanh() step interpolating
between binned values
parameters for reio_many_tanh with
how many jumps do we want to
describe reionization?
central z value for each tanh jump

index_th_xe

194

File Documentation

Data Fields
int

index_th_dkappa

Thomson scattering rate dκ/dτ
(units 1/Mpc)

int

index_th_tau_d

Baryon drag optical depth

int

index_th_ddkappa

scattering rate derivative d2 κ/dτ 2

int

index_th_dddkappa

scattering rate second derivative

d3 κ/dτ 3
int

index_th_exp_m_kappa

exp−κ

int

index_th_g

visibility function

g = (dκ/dτ ) ∗ exp−κ
int

index_th_dg

visibility function derivative (dg/dτ )

int

index_th_ddg

visibility function second derivative

(d2 g/dτ 2 )
int

index_th_Tb

baryon temperature Tb

int

index_th_cb2

squared baryon sound speed c2b

int

index_th_dcb2

derivative wrt conformal time of
squared baryon sound speed
d[c2b ]/dτ (only computed if some
non-minimal tight-coupling schemes
is requested)

int

index_th_ddcb2

second derivative wrt conformal time
of squared baryon sound speed
d2 [c2b ]/dτ 2 (only computed if some
non0-minimal tight-coupling schemes
is requested)

int

index_th_rate

int

index_th_r_d

maximum variation rate of exp−κ , g
and (dg/dτ ), used for computing
integration step in perturbation
module
simple analytic approximation to the
photon comoving damping scale

int

th_size

size of thermodynamics vector

int

tt_size

number of lines (redshift steps) in the
tables
vector z_table[index_z] with values of
redshift (vector of size tt_size)

double ∗

z_table

double ∗

thermodynamics_table

table thermodynamics_table[index←_z∗pth->tt_size+pba->index_th]
with all other quantities (array of size
th_size∗tt_size)

double ∗

d2thermodynamics_dz2_table

table d2thermodynamics_dz2_←table[index_z∗pth->tt_size+pba>index_th] with values of d2 ti /dz 2
(array of size th_size∗tt_size)

double

z_rec

z at which the visibility reaches its
maximum (= recombination redshift)

double

tau_rec

conformal time at which the visibility
reaches its maximum (=
recombination time)

double

rs_rec

comoving sound horizon at
recombination

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Data Fields
double

ds_rec

double

ra_rec

double

da_rec

double

rd_rec

double

z_d

physical sound horizon at
recombination
conformal angular diameter distance
to recombination
physical angular diameter distance to
recombination
comoving photon damping scale at
recombination
baryon drag redshift

double

tau_d

baryon drag time

double

ds_d

physical sound horizon at baryon
drag

double

rs_d

comoving sound horizon at baryon
drag

double

tau_cut

at at which the visibility goes below a
fixed fraction of the maximum
visibility, used for an approximation in
perturbation module

double

angular_rescaling

[ratio ra_rec / (tau0-tau_rec)]: gives
CMB rescaling in angular space
relative to flat model (=1 for curvature
K=0)

double

tau_free_streaming

double

tau_ini

minimum value of tau at which
sfree-streaming approximation can
be switched on
initial conformal time at which
thermodynamical variables have
been be integrated

double

n_e

total number density of electrons
today (free or not)

short

inter_normal

flag for calling thermodynamics_at_z
and find position in interpolation table
normally

short

inter_closeby

flag for calling thermodynamics_at_z
and find position in interpolation table
starting from previous position in
previous call

short

thermodynamics_verbose

flag regulating the amount of
information sent to standard output
(none if set to zero)

error_message

zone for writing error messages

ErrorMsg

5.20.2.2

struct recombination

Temporary structure where all the recombination history is defined and stored.
This structure is used internally by the thermodynamics module, but never passed to other modules.
Data Fields
int

index_re_z

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redshift z

196

File Documentation

Data Fields
int

index_re_xe

ionization fraction xe

int

index_re_Tb

baryon temperature Tb

int

index_re_cb2

squared baryon sound speed c2b

int

index_re_dkappadtau

Thomson scattering rate dκ/dτ (units 1/Mpc)

int

re_size

size of this vector

int

rt_size

number of lines (redshift steps) in the table

recombination_table

table recombination_table[index_z∗preco->re_size+index_re] with all
other quantities (array of size preco->rt_size∗preco->re_size)

double
double
double
double
double
double
double

CDB
CR
CK
CL
CT
fHe
CDB_He

defined as in RECFAST
defined as in RECFAST
defined as in RECFAST
defined as in RECFAST
defined as in RECFAST
defined as in RECFAST
defined as in RECFAST

double

CK_He

defined as in RECFAST

double

CL_He

defined as in RECFAST

double
double

fu
H_frac

defined as in RECFAST
defined as in RECFAST

double
double
double
double
double

Tnow
Nnow
Bfact
CB1
CB1_He1

defined as in RECFAST
defined as in RECFAST
defined as in RECFAST
defined as in RECFAST
defined as in RECFAST

double

CB1_He2

defined as in RECFAST

double
double
double

H0
YHe
annihilation

defined as in RECFAST
defined as in RECFAST
parameter describing CDM annihilation (f  / m_cdm, see e.g.
0905.0003)

has_on_the_spot

flag to specify if we want to use the on-the-spot approximation

double

decay

parameter describing CDM decay (f/tau, see e.g. 1109.6322)

double

annihilation_variation

if this parameter is non-zero, the function F(z)=(f  / m_cdm)(z)
will be a parabola in log-log scale between zmin and zmax, with a
curvature given by annihlation_variation (must be negative), and with a
maximum in zmax; it will be constant outside this range

double

annihilation_z

if annihilation_variation is non-zero, this is the value of z at which the
parameter annihilation is defined, i.e. F(annihilation_z)=annihilation

double

annihilation_zmax

double

annihilation_zmin

double

annihilation_f_halo

if annihilation_variation is non-zero, redshift above which annihilation rate
is maximal
if annihilation_variation is non-zero, redshift below which annihilation rate
is constant
takes the contribution of DM annihilation in halos into account

double

annihilation_z_halo

characteristic redshift for DM annihilation in halos

double ∗

short

5.20.2.3

struct reionization

Temporary structure where all the reionization history is defined and stored.
This structure is used internally by the thermodynamics module, but never passed to other modules.

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Data Fields
int

index_re_z

redshift z

int

index_re_xe

ionization fraction xe

int

index_re_Tb

baryon temperature Tb

int

index_re_cb2

squared baryon sound speed c2b

int

index_re_dkappadtau

Thomson scattering rate dκ/dτ (units 1/Mpc)

int

index_re_dkappadz

Thomson scattering rate with respect to redshift dκ/dz (units
1/Mpc)

int

index_re_d3kappadz3

second derivative of previous quantity with respect to redshift

int

re_size

size of this vector

int

rt_size

number of lines (redshift steps) in the table

reionization_table

table reionization_table[index_z∗preio->re_size+index_re] with all
other quantities (array of size preio->rt_size∗preio->re_size)

reionization_optical_depth

reionization optical depth inferred from reionization history

int

index_reio_redshift

hydrogen reionization redshift

int

index_reio_exponent

an exponent used in the function x_e(z) in the reio_camb scheme

int

index_reio_width

a width defining the duration of hydrogen reionization in the
reio_camb scheme

int

index_reio_xe_before

ionization fraction at redshift 'reio_start'

int

index_reio_xe_after

ionization fraction after full reionization

int

index_helium_fullreio_fraction

int

index_helium_fullreio_redshift

helium full reionization fraction inferred from primordial helium
fraction
helium full reionization redshift

int

index_helium_fullreio_width

a width defining the duration of helium full reionization in the
reio_camb scheme

int

reio_num_z

number of reionization jumps

int

index_reio_first_z

redshift at which we start to impose reionization function

int

index_reio_first_xe

ionization fraction at redshift first_z (inferred from recombination
code)

int

index_reio_step_sharpness

sharpness of tanh jump

int

index_reio_start

redshift above which hydrogen reionization neglected

reionization_parameters

vector containing all reionization parameters necessary to
compute xe(z)

int

reio_num_params

length of vector reionization_parameters

int

index_reco_when_reio_start

index of line in recombination table corresponding to first line of
reionization table

double ∗
double

double ∗

5.20.2.4

struct thermodynamics_parameters_and_workspace

temporary parameters and workspace passed to the thermodynamics_derivs function

5.20.3

Macro Definition Documentation

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File Documentation

5.20.3.1

f1

#define f1(
x ) (-0.75∗x∗(x∗x/3.-1.)+0.5)

Two useful smooth step functions, for smoothing transitions in recfast.goes from 0 to 1 when x goes from -1 to 1

5.20.3.2

f2

#define f2(
x ) (x∗x∗(0.5-x/3.)∗6.)

goes from 0 to 1 when x goes from 0 to 1

5.20.3.3

_YHE_BIG_

#define _YHE_BIG_ 0.5

maximal YHe

5.20.3.4

_YHE_SMALL_

#define _YHE_SMALL_ 0.01

minimal YHe

5.20.4

Enumeration Type Documentation

5.20.4.1

recombination_algorithm

enum recombination_algorithm

List of possible recombination algorithms.

5.20.4.2

reionization_parametrization

enum reionization_parametrization

List of possible reionization schemes.
Enumerator
reio_none

no reionization

reio_camb

reionization parameterized like in CAMB

reio_bins_tanh

binned reionization history with tanh inteprolation between bins

reio_half_tanh

half a tanh, instead of the full tanh

reio_many_tanh
reio_inter

similar to reio_camb but with more than one tanh
linear interpolation between specified points

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199

reionization_z_or_tau

enum reionization_z_or_tau

Is the input parameter the reionization redshift or optical depth?
Enumerator
reio_z
reio_tau

5.21

input = redshift
input = tau

transfer.c File Reference

#include "transfer.h"
Include dependency graph for transfer.c:

transfer.c

transfer.h

nonlinear.h

sys/shm.h

primordial.h

sys/stat.h

errno.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

Functions
• int transfer_functions_at_q (struct transfers ∗ptr, int index_md, int index_ic, int index_tt, int index_l, double q,
double ∗transfer_function)
• int transfer_init (struct precision ∗ppr, struct background ∗pba, struct thermo ∗pth, struct perturbs ∗ppt, struct
nonlinear ∗pnl, struct transfers ∗ptr)
• int transfer_free (struct transfers ∗ptr)
• int transfer_indices_of_transfers (struct precision ∗ppr, struct perturbs ∗ppt, struct transfers ∗ptr, double q_←period, double K, int sgnK)
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File Documentation
• int transfer_get_l_list (struct precision ∗ppr, struct perturbs ∗ppt, struct transfers ∗ptr)
• int transfer_get_q_list (struct precision ∗ppr, struct perturbs ∗ppt, struct transfers ∗ptr, double q_period, double K, int sgnK)
• int transfer_get_k_list (struct perturbs ∗ppt, struct transfers ∗ptr, double K)
• int transfer_get_source_correspondence (struct perturbs ∗ppt, struct transfers ∗ptr, int ∗∗tp_of_tt)
• int transfer_source_tau_size (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct transfers ∗ptr, double tau_rec, double tau0, int index_md, int index_tt, int ∗tau_size)
• int transfer_compute_for_each_q (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct
transfers ∗ptr, int ∗∗tp_of_tt, int index_q, int tau_size_max, double tau_rec, double ∗∗∗pert_sources, double
∗∗∗pert_sources_spline, struct transfer_workspace ∗ptw)
• int transfer_interpolate_sources (struct perturbs ∗ppt, struct transfers ∗ptr, int index_q, int index_md, int
index_ic, int index_type, double ∗pert_source, double ∗pert_source_spline, double ∗interpolated_sources)
• int transfer_sources (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct transfers ∗ptr,
double ∗interpolated_sources, double tau_rec, int index_q, int index_md, int index_tt, double ∗sources, double ∗tau0_minus_tau, double ∗w_trapz, int ∗tau_size_out)
• int transfer_selection_function (struct precision ∗ppr, struct perturbs ∗ppt, struct transfers ∗ptr, int bin, double
z, double ∗selection)
• int transfer_dNdz_analytic (struct transfers ∗ptr, double z, double ∗dNdz, double ∗dln_dNdz_dz)
• int transfer_selection_sampling (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct
transfers ∗ptr, int bin, double ∗tau0_minus_tau, int tau_size)
• int transfer_lensing_sampling (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct
transfers ∗ptr, int bin, double tau0, double ∗tau0_minus_tau, int tau_size)
• int transfer_source_resample (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct transfers ∗ptr, int bin, double ∗tau0_minus_tau, int tau_size, int index_md, double tau0, double ∗interpolated_←sources, double ∗sources)
• int transfer_selection_times (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct transfers ∗ptr, int bin, double ∗tau_min, double ∗tau_mean, double ∗tau_max)
• int transfer_selection_compute (struct precision ∗ppr, struct background ∗pba, struct perturbs ∗ppt, struct
transfers ∗ptr, double ∗selection, double ∗tau0_minus_tau, double ∗w_trapz, int tau_size, double ∗pvecback,
double tau0, int bin)
• int transfer_compute_for_each_l (struct transfer_workspace ∗ptw, struct precision ∗ppr, struct perturbs ∗ppt,
struct transfers ∗ptr, int index_q, int index_md, int index_ic, int index_tt, int index_l, double l, double q_max←_bessel, radial_function_type radial_type)
• int transfer_integrate (struct perturbs ∗ppt, struct transfers ∗ptr, struct transfer_workspace ∗ptw, int index_q,
int index_md, int index_tt, double l, int index_l, double k, radial_function_type radial_type, double ∗trsf)
• int transfer_limber (struct transfers ∗ptr, struct transfer_workspace ∗ptw, int index_md, int index_q, double l,
double q, radial_function_type radial_type, double ∗trsf)
• int transfer_limber_interpolate (struct transfers ∗ptr, double ∗tau0_minus_tau, double ∗sources, int tau_size,
double tau0_minus_tau_limber, double ∗S)
• int transfer_limber2 (int tau_size, struct transfers ∗ptr, int index_md, int index_k, double l, double k, double
∗tau0_minus_tau, double ∗sources, radial_function_type radial_type, double ∗trsf)

5.21.1

Detailed Description

Documented transfer module.
Julien Lesgourgues, 28.07.2013
This module has two purposes:

• at the beginning, to compute the transfer functions ∆X
l (q), and store them in tables used for interpolation in
other modules.
• at any time in the code, to evaluate the transfer functions (for a given mode, initial condition, type and multipole
l) at any wavenumber q (by interpolating within the interpolation table).
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Hence the following functions can be called from other modules:

1. transfer_init() at the beginning (but after perturb_init() and bessel_init())
2. transfer_functions_at_q() at any later time
3. transfer_free() at the end, when no more calls to transfer_functions_at_q() are needed

Note that in the standard implementation of CLASS, only the pre-computed values of the transfer functions are
used, no interpolation is necessary; hence the routine transfer_functions_at_q() is actually never called.

5.21.2

Function Documentation

5.21.2.1

transfer_functions_at_q()

int transfer_functions_at_q (
struct transfers ∗ ptr,
int index_md,
int index_ic,
int index_tt,
int index_l,
double q,
double ∗ transfer_function )

Transfer function ∆X
l (q) at a given wavenumber q.
For a given mode (scalar, vector, tensor), initial condition, type (temperature, polarization, lensing, etc) and multipole,
computes the transfer function for an arbitrary value of q by interpolating between pre-computed values of q. This
function can be called from whatever module at whatever time, provided that transfer_init() has been called before,
and transfer_free() has not been called yet.
Wavenumbers are called q in this module and k in the perturbation module. In flat universes k=q. In non-flat
universes q and k differ through q2 = k2 + K(1 + m), where m=0,1,2 for scalar, vector, tensor. q should be
used throughout the transfer module, excepted when interpolating or manipulating the source functions S(k,tau)
calculated in the perturbation module: for a given value of q, this should be done at the corresponding k(q).
Parameters
ptr

Input: pointer to transfer structure

index_md

Input: index of requested mode

index_ic

Input: index of requested initial condition

index_tt

Input: index of requested type

index_l

Input: index of requested multipole

q

Input: any wavenumber

transfer_function

Output: transfer function

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File Documentation

Returns
the error status

Summary:

• interpolate in pre-computed table using array_interpolate_two()

5.21.2.2

transfer_init()

int transfer_init (
struct
struct
struct
struct

precision ∗ ppr,
background ∗ pba,
thermo ∗ pth,
perturbs ∗ ppt,

struct nonlinear ∗ pnl,
struct transfers ∗ ptr )

This routine initializes the transfers structure, (in particular, computes table of transfer functions ∆X
l (q))
Main steps:

• initialize all indices in the transfers structure and allocate all its arrays using transfer_indices_of_transfers().
• for each thread (in case of parallel run), initialize the fields of a memory zone called the transfer_workspace
with transfer_workspace_init()
• loop over q values. For each q, compute the Bessel functions if needed with transfer_update_HIS(), and defer
the calculation of all transfer functions to transfer_compute_for_each_q()
• for each thread, free the the workspace with transfer_workspace_free()

Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

pth

Input: pointer to thermodynamics structure

ppt

Input: pointer to perturbation structure

pnl

Input: pointer to nonlinear structure

ptr

Output: pointer to initialized transfers structure

Returns
the error status

Summary:

• define local variables
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• array with the correspondence between the index of sources in the perturbation module and in the transfer
module, tp_of_tt[index_md][index_tt]
• check whether any spectrum in harmonic space (i.e., any Cl 's) is actually requested
• get number of modes (scalars, tensors...)
• get conformal age / recombination time from background / thermodynamics structures (only place where
these structures are used in this module)
• correspondence between k and l depend on angular diameter distance, i.e. on curvature.
• order of magnitude of the oscillation period of transfer functions
• initialize all indices in the transfers structure and allocate all its arrays using transfer_indices_of_transfers()
• copy sources to a local array sources (in fact, only the pointers are copied, not the data), and eventually apply
non-linear corrections to the sources
• spline all the sources passed by the perturbation module with respect to k (in order to interpolate later at a
given value of k)
• allocate and fill array describing the correspondence between perturbation types and transfer types
• evaluate maximum number of sampled times in the transfer sources: needs to be known here, in order to
allocate a large enough workspace
• compute flat spherical bessel functions
• eventually read the selection and evolution functions
• loop over all wavenumbers (parallelized).
• finally, free arrays allocated outside parallel zone

5.21.2.3

transfer_free()

int transfer_free (
struct transfers ∗ ptr )

This routine frees all the memory space allocated by transfer_init().
To be called at the end of each run, only when no further calls to transfer_functions_at_k() are needed.
Parameters
ptr

Input: pointer to transfers structure (which fields must be freed)

Returns
the error status

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File Documentation

5.21.2.4

transfer_indices_of_transfers()

int transfer_indices_of_transfers (
struct
struct
struct
double
double

precision ∗ ppr,
perturbs ∗ ppt,
transfers ∗ ptr,
q_period,
K,

int sgnK )

This routine defines all indices and allocates all tables in the transfers structure
Compute list of (k, l) values, allocate and fill corresponding arrays in the transfers structure. Allocate the array of
transfer function tables.
Parameters
ppr

Input: pointer to precision structure

ppt

Input: pointer to perturbation structure

ptr

Input/Output: pointer to transfer structure

q_period

Input: order of magnitude of the oscillation period of transfer functions

K

Input: spatial curvature (in absolute value)

sgnK

Input: spatial curvature sign (open/closed/flat)

Returns
the error status

Summary:

• define local variables
• define indices for transfer types
• type indices common to scalars and tensors
• type indices for scalars
• type indices for vectors
• type indices for tensors
• allocate arrays of (k, l) values and transfer functions
• get q values using transfer_get_q_list()
• get k values using transfer_get_k_list()
• get l values using transfer_get_l_list()
• loop over modes (scalar, etc). For each mode:
• allocate arrays of transfer functions, (ptr->transfer[index_md])[index_ic][index_tt][index_l][index_k]

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205

transfer_get_l_list()

int transfer_get_l_list (
struct precision ∗ ppr,
struct perturbs ∗ ppt,
struct transfers ∗ ptr )

This routine defines the number and values of multipoles l for all modes.
Parameters
ppr

Input: pointer to precision structure

ppt

Input: pointer to perturbation structure

ptr

Input/Output: pointer to transfers structure containing l's

Returns
the error status

Summary:

• allocate and fill l array
• start from l = 2 and increase with logarithmic step
• when the logarithmic step becomes larger than some linear step, stick to this linear step till l_max
• last value set to exactly l_max
• so far we just counted the number of values. Now repeat the whole thing but fill array with values.

5.21.2.6

transfer_get_q_list()

int transfer_get_q_list (
struct precision ∗ ppr,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
double q_period,
double K,
int sgnK )

This routine defines the number and values of wavenumbers q for each mode (goes smoothly from logarithmic step
for small q's to linear step for large q's).
Parameters
ppr

Input: pointer to precision structure

ppt

Input: pointer to perturbation structure

ptr

Input/Output: pointer to transfers structure containing q's

q_period

Input: order of magnitude of the oscillation period of transfer functions

K

Input: spatial curvature (in absolute value)

sgnK

Input: spatial curvature sign (open/closed/flat)

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File Documentation

Returns
the error status

5.21.2.7

transfer_get_k_list()

int transfer_get_k_list (
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
double K )

This routine infers from the q values a list of corresponding k values for each mode.
Parameters
ppt

Input: pointer to perturbation structure

ptr

Input/Output: pointer to transfers structure containing q's

K

Input: spatial curvature

Returns
the error status

5.21.2.8

transfer_get_source_correspondence()

int transfer_get_source_correspondence (
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
int ∗∗ tp_of_tt )

This routine defines the correspondence between the sources in the perturbation and transfer module.
Parameters
ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure containing l's

tp_of←_tt

Input/Output: array with the correspondence (allocated before, filled here)

Returns
the error status

Summary:
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207

• running index on modes
• running index on transfer types
• which source are we considering? Define correspondence between transfer types and source types

5.21.2.9

transfer_source_tau_size()

int transfer_source_tau_size (
struct precision ∗ ppr,
struct background ∗ pba,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
double tau_rec,
double tau0,
int index_md,
int index_tt,
int ∗ tau_size )

the code makes a distinction between "perturbation sources" (e.g. gravitational potential) and "transfer sources"
(e.g. total density fluctuations, obtained through the Poisson equation, and observed with a given selection function).
This routine computes the number of sampled time values for each type of transfer sources.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

tau_rec

Input: recombination time

tau0

Input: time today

index_md

Input: index of the mode (scalar, tensor)

index_tt

Input: index of transfer type

tau_size

Output: pointer to number of sampled times

Returns
the error status

5.21.2.10

transfer_compute_for_each_q()

int transfer_compute_for_each_q (
struct precision ∗ ppr,
struct background ∗ pba,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
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int ∗∗ tp_of_tt,
int index_q,
int tau_size_max,
double
double
double
struct

tau_rec,
∗∗∗ pert_sources,
∗∗∗ pert_sources_spline,
transfer_workspace ∗ ptw )

Summary:

• define local variables
– we deal with workspaces, i.e. with contiguous memory zones (one per thread) containing various fields
used by the integration routine
• for a given l, maximum value of k such that we can convolve the source with Bessel functions j_l(x) without
reaching x_max
• store the sources in the workspace and define all fields in this workspace
• loop over all modes. For each mode
• loop over initial conditions.
• check if we must now deal with a new source with a new index ppt->index_type. If yes, interpolate it at the
right values of k.
• Select radial function type

5.21.2.11

transfer_interpolate_sources()

int transfer_interpolate_sources (
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
int index_q,
int index_md,
int index_ic,
int index_type,
double ∗ pert_source,
double ∗ pert_source_spline,
double ∗ interpolated_sources )

This routine interpolates sources S(k, τ ) for each mode, initial condition and type (of perturbation module), to get
them at the right values of k, using the spline interpolation method.
Parameters
ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

index_q

Input: index of wavenumber

index_md

Input: index of mode

index_ic

Input: index of initial condition

index_type

Input: index of type of source (in perturbation module)

pert_source

Input: array of sources

pert_source_spline

Input: array of second derivative of sources

interpolated_sources

Output: array of interpolated sources (filled here but allocated in transfer_init()
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Returns
the error status

Summary:

• define local variables
• interpolate at each k value using the usual spline interpolation algorithm.

5.21.2.12

transfer_sources()

int transfer_sources (
struct precision ∗ ppr,
struct background ∗ pba,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
double ∗ interpolated_sources,
double tau_rec,
int index_q,
int index_md,
int index_tt,
double ∗ sources,
double ∗ tau0_minus_tau,
double ∗ w_trapz,
int ∗ tau_size_out )

The code makes a distinction between "perturbation sources" (e.g. gravitational potential) and "transfer sources"
(e.g. total density fluctuations, obtained through the Poisson equation, and observed with a given selection function).
This routine computes the transfer source given the interpolated perturbation source, and copies it in the workspace.
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

interpolated_sources

Input: interpolated perturbation source

tau_rec

Input: recombination time

index_q

Input: index of wavenumber

index_md

Input: index of mode

index_tt

Input: index of type of (transfer) source

sources

Output: transfer source

tau0_minus_tau

Output: values of (tau0-tau) at which source are sample

w_trapz

Output: trapezoidal weights for integration over tau

tau_size_out

Output: pointer to size of previous two arrays, converted to double

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Returns
the error status

Summary:

• define local variables
– in which cases are perturbation and transfer sources are different? I.e., in which case do we need
to multiply the sources by some background and/or window function, and eventually to resample it, or
redefine its time limits?
– case where we need to redefine by a window function (or any function of the background and of k)
• case where we do not need to redefine
– return tau_size value that will be stored in the workspace (the workspace wants a double)

5.21.2.13

transfer_selection_function()

int transfer_selection_function (
struct precision ∗ ppr,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
int bin,
double z,
double ∗ selection )

Arbitrarily normalized selection function dN/dz(z,bin)
Parameters
ppr

Input: pointer to precision structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

bin

Input: redshift bin number

z

Input: one value of redshift

selection

Output: pointer to selection function

Returns
the error status

5.21.2.14

transfer_dNdz_analytic()

int transfer_dNdz_analytic (
struct transfers ∗ ptr,
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5.21 transfer.c File Reference

double z,
double ∗ dNdz,
double ∗ dln_dNdz_dz )

Analytic form for dNdz distribution, from arXiv:1004.4640
Parameters
ptr

Input: pointer to transfer structure

z

Input: redshift

dNdz

Output: density per redshift, dN/dZ

dln_dNdz_dz

Output: dln(dN/dz)/dz, used optionally for the source evolution

Returns
the error status

5.21.2.15

transfer_selection_sampling()

int transfer_selection_sampling (
struct precision ∗ ppr,
struct background ∗ pba,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
int bin,
double ∗ tau0_minus_tau,
int tau_size )

For sources that need to be multiplied by a selection function, redefine a finer time sampling in a small range
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

bin

Input: redshift bin number

tau0_minus_tau

Output: values of (tau0-tau) at which source are sample

tau_size

Output: pointer to size of previous array

Returns
the error status

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File Documentation

5.21.2.16

transfer_lensing_sampling()

int transfer_lensing_sampling (
struct precision ∗ ppr,
struct background ∗ pba,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
int bin,
double tau0,
double ∗ tau0_minus_tau,
int tau_size )

For lensing sources that need to be convolved with a selection function, redefine the sampling within the range
extending from the tau_min of the selection function up to tau0
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

bin

Input: redshift bin number

tau0

Input: time today

tau0_minus_tau

Output: values of (tau0-tau) at which source are sample

tau_size

Output: pointer to size of previous array

Returns
the error status

5.21.2.17

transfer_source_resample()

int transfer_source_resample (
struct precision ∗ ppr,
struct background ∗ pba,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
int bin,
double ∗ tau0_minus_tau,
int tau_size,
int index_md,
double tau0,
double ∗ interpolated_sources,
double ∗ sources )

For sources that need to be multiplied by a selection function, redefine a finer time sampling in a small range, and
resample the perturbation sources at the new value by linear interpolation
Parameters
ppr

Input: pointer to precision structure
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Parameters
pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

bin

Input: redshift bin number

tau0_minus_tau

Output: values of (tau0-tau) at which source are sample

tau_size

Output: pointer to size of previous array

index_md

Input: index of mode

tau0

Input: time today

interpolated_sources

Input: interpolated perturbation source

sources

Output: resampled transfer source

Returns
the error status

5.21.2.18

transfer_selection_times()

int transfer_selection_times (
struct precision ∗ ppr,
struct background ∗ pba,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
int bin,
double ∗ tau_min,
double ∗ tau_mean,
double ∗ tau_max )

For each selection function, compute the min, mean and max values of conformal time (associated to the min, mean
and max values of redshift specified by the user)
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

bin

Input: redshift bin number

tau_min

Output: smallest time in the selection interval

tau_mean

Output: time corresponding to z_mean

tau_max

Output: largest time in the selection interval

Returns
the error status

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5.21.2.19

transfer_selection_compute()

int transfer_selection_compute (
struct
struct
struct
struct
double

precision ∗ ppr,
background ∗ pba,
perturbs ∗ ppt,
transfers ∗ ptr,
∗ selection,

double ∗ tau0_minus_tau,
double ∗ w_trapz,
int tau_size,
double ∗ pvecback,
double tau0,
int bin )

Compute and normalize selection function for a set of time values
Parameters
ppr

Input: pointer to precision structure

pba

Input: pointer to background structure

ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

selection

Output: normalized selection function

tau0_minus_tau

Input: values of (tau0-tau) at which source are sample

w_trapz

Input: trapezoidal weights for integration over tau

tau_size

Input: size of previous two arrays

pvecback

Input: allocated array of background values

tau0

Input: time today

bin

Input: redshift bin number

Returns
the error status

5.21.2.20

transfer_compute_for_each_l()

int transfer_compute_for_each_l (
struct transfer_workspace ∗ ptw,
struct precision ∗ ppr,
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
int index_q,
int index_md,
int index_ic,
int index_tt,
int index_l,
double l,
double q_max_bessel,
radial_function_type radial_type )
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This routine computes the transfer functions ∆X
l (k)) as a function of wavenumber k for a given mode, initial condition, type and multipole l passed in input.
For a given value of k, the transfer function is inferred from the source function (passed in input in the array
interpolated_sources) and from Bessel functions (passed in input in the bessels structure), either by convolving
them along tau, or by a Limber approximation. This elementary task is distributed either to transfer_integrate() or to
transfer_limber(). The task of this routine is mainly to loop over k values, and to decide at which k_max the calculation can be stopped, according to some approximation scheme designed to find a compromise between execution
time and precision. The approximation scheme is defined by parameters in the precision structure.
Parameters
ptw

Input: pointer to transfer_workspace structure (allocated in transfer_init() to avoid numerous
reallocation)

ppr

Input: pointer to precision structure

ppt

Input: pointer to perturbation structure

ptr

Input/output: pointer to transfers structure (result stored there)

index_q

Input: index of wavenumber

index_md

Input: index of mode

index_ic

Input: index of initial condition

index_tt

Input: index of type of transfer

index_l

Input: index of multipole

l

Input: multipole

q_max_bessel

Input: maximum value of argument q at which Bessel functions are computed

radial_type

Input: type of radial (Bessel) functions to convolve with

Returns
the error status

Summary:

• define local variables
• return zero transfer function if l is above l_max
• store transfer function in transfer structure

5.21.2.21

transfer_integrate()

int transfer_integrate (
struct perturbs ∗ ppt,
struct transfers ∗ ptr,
struct transfer_workspace ∗ ptw,
int index_q,
int index_md,
int index_tt,
double l,
int index_l,
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double k,
radial_function_type radial_type,
double ∗ trsf )

This routine computes the transfer functions ∆X
l (k)) for each mode, initial condition, type, multipole l and wavenumber k, by convolving the source function (passed in input in the array interpolated_sources) with Bessel functions
(passed in input in the bessels structure).

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Parameters
ppt

Input: pointer to perturbation structure

ptr

Input: pointer to transfers structure

ptw

Input: pointer to transfer_workspace structure (allocated in transfer_init() to avoid numerous
reallocation)

index_q

Input: index of wavenumber

index_md

Input: index of mode

index_tt

Input: index of type

l

Input: multipole

index_l

Input: index of multipole

k

Input: wavenumber

radial_type

Input: type of radial (Bessel) functions to convolve with

trsf

Output: transfer function ∆l (k)

Returns
the error status

Summary:

• define local variables
• find minimum value of (tau0-tau) at which jl (k[τ0 − τ ]) is known, given that jl (x) is sampled above some
finite value xmin (below which it can be approximated by zero)
• if there is no overlap between the region in which bessels and sources are non-zero, return zero
• if there is an overlap:
• –> trivial case: the source is a Dirac function and is sampled in only one point
• –> other cases
• —> (a) find index in the source's tau list corresponding to the last point in the overlapping region. After this
step, index_tau_max can be as small as zero, but not negative.
• —> (b) the source function can vanish at large τ . Check if further points can be eliminated. After this step
and if we did not return a null transfer function, index_tau_max can be as small as zero, but not negative.
• Compute the radial function:
• Now we do most of the convolution integral:
• This integral is correct for the case where no truncation has occurred. If it has been truncated at some
index_tau_max because f[index_tau_max+1]==0, it is still correct. The 'mistake' in using the wrong weight
w_trapz[index_tau_max] is exactly compensated by the triangle we miss. However, for the Bessel cut off, we
must subtract the wrong triangle and add the correct triangle.

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5.21.2.22

transfer_limber()

int transfer_limber (
struct transfers ∗ ptr,
struct transfer_workspace ∗ ptw,
int index_md,
int index_q,
double l,
double q,
radial_function_type radial_type,
double ∗ trsf )

This routine computes the transfer functions ∆X
l (k)) for each mode, initial condition, type, multipole l and wavenumber k, by using the Limber approximation, i.e by evaluating the source function (passed in input in the array
interpolated_sources) at a single value of tau (the Bessel function being approximated as a Dirac distribution).
Parameters
ptr

Input: pointer to transfers structure

ptw

Input: pointer to transfer workspace structure

index_md

Input: index of mode

index_q

Input: index of wavenumber

l

Input: multipole

q

Input: wavenumber

radial_type

Input: type of radial (Bessel) functions to convolve with

trsf

Output: transfer function ∆l (k)

Returns
the error status

Summary:

• define local variables
• get k, l and infer tau such that k(tau0-tau)=l+1/2; check that tau is in appropriate range
• get transfer = source ∗

5.21.2.23

p
p
π/(2l + 1)/q = source∗[tau0-tau] ∗ π/(2l + 1)/(l + 1/2)

transfer_limber_interpolate()

int transfer_limber_interpolate (
struct transfers ∗ ptr,
double ∗ tau0_minus_tau,
double ∗ sources,
int tau_size,
double tau0_minus_tau_limber,
double ∗ S )
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219

• find bracketing indices. index_tau must be at least 1 (so that index_tau-1 is at least 0) and at most tau_size-2
(so that index_tau+1 is at most tau_size-1).
• interpolate by fitting a polynomial of order two; get source and its first two derivatives. Note that we are
not interpolating S, but the product S∗(tau0-tau). Indeed this product is regular in tau=tau0, while S alone
diverges for lensing.

5.21.2.24

transfer_limber2()

int transfer_limber2 (
int tau_size,
struct transfers ∗ ptr,
int index_md,
int index_k,
double l,
double k,
double ∗ tau0_minus_tau,
double ∗ sources,
radial_function_type radial_type,
double ∗ trsf )

This routine computes the transfer functions ∆X
l (k)) for each mode, initial condition, type, multipole l and wavenumber k, by using the Limber approximation at order two, i.e as a function of the source function and its first two
derivatives at a single value of tau
Parameters
tau_size

Input: size of conformal time array

ptr

Input: pointer to transfers structure

index_md

Input: index of mode

index_k

Input: index of wavenumber

l

Input: multipole

k

Input: wavenumber

tau0_minus_tau

Input: array of values of (tau_today - tau)

sources

Input: source functions

radial_type

Input: type of radial (Bessel) functions to convolve with

trsf

Output: transfer function ∆l (k)

Returns
the error status
Summary:
• define local variables
• get k, l and infer tau such that k(tau0-tau)=l+1/2; check that tau is in appropriate range
• find bracketing indices
• interpolate by fitting a polynomial of order two; get source and its first two derivatives
• get transfer from 2nd order Limber approx (inferred from 0809.5112 [astro-ph])
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5.22

File Documentation

transfer.h File Reference

#include
#include
#include
#include
#include

"nonlinear.h"
"hyperspherical.h"


"errno.h"

Include dependency graph for transfer.h:

transfer.h

nonlinear.h

sys/shm.h

primordial.h

sys/stat.h

errno.h

hyperspherical.h

perturbations.h

thermodynamics.h

evolver_rkck.h

background.h

quadrature.h

growTable.h

arrays.h

evolver_ndf15.h

parser.h

dei_rkck.h

sparse.h

common.h

stdio.h

stdlib.h

math.h

string.h

float.h

svnversion.h

stdarg.h

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221

This graph shows which files directly or indirectly include this file:

transfer.h

spectra.h

lensing.h

lensing.c

output.c

class.h

transfer.c

spectra.c

output.h

input.h

input.c

class.c

Data Structures
• struct transfers
• struct transfer_workspace

Enumerations
• enum radial_function_type

5.22.1

Detailed Description

Documented includes for transfer module.

5.22.2

Data Structure Documentation

5.22.2.1

struct transfers

Structure containing everything about transfer functions in harmonic space ∆X
l (q) that other modules need to know.
Once initialized by transfer_init(), contains all tables of transfer functions used for interpolation in other modules, for
all requested modes (scalar/vector/tensor), initial conditions, types (temperature, polarization, etc), multipoles l, and
wavenumbers q.
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Wavenumbers are called q in this module and k in the perturbation module. In flat universes k=q. In non-flat
universes q and k differ through q2 = k2 + K(1+m), where m=0,1,2 for scalar, vector, tensor. q should be used
throughout the transfer module, except when interpolating or manipulating the source functions S(k,tau) calculated
in the perturbation module: for a given value of q, this should be done at the corresponding k(q).
The content of this structure is entirely computed in this module, given the content of the 'precision', 'bessels',
'background', 'thermodynamics' and 'perturbation' structures.

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Data Fields
double

lcmb_rescale

normally set to one, can be used exceptionally
to rescale by hand the CMB lensing potential

double

lcmb_tilt

normally set to zero, can be used exceptionally
to tilt by hand the CMB lensing potential

double

lcmb_pivot

if lcmb_tilt non-zero, corresponding pivot scale

double

selection_bias[_SELECTION_NUM_MAX_]

light-to-mass bias in the transfer function of
density number count

double

selection_magnification_bias[_SELECTION_NUM_MAX_]
magnification bias in the transfer function of
density number count

short

has_nz_file

Has dN/dz (selection function) input file?

short

has_nz_analytic

Use analytic form for dN/dz (selection function)
distribution?
dN/dz (selection function) input file name

FileName
int

nz_file_name
nz_size

double ∗

nz_z

double ∗

nz_nz

number of redshift values in input tabulated
selection function
redshift values in input tabulated selection
function
input tabulated values of selection function

double ∗

nz_ddnz

second derivatives in splined selection function

short

has_nz_evo_file

Has dN/dz (evolution function) input file?

short

has_nz_evo_analytic

Use analytic form for dN/dz (evolution function)
distribution?
dN/dz (evolution function) input file name

FileName
int

nz_evo_file_name
nz_evo_size

double ∗

nz_evo_z

double ∗

nz_evo_nz

number of redshift values in input tabulated
evolution function
redshift values in input tabulated evolution
function
input tabulated values of evolution function

double ∗

nz_evo_dlog_nz

log of tabulated values of evolution function

double ∗

nz_evo_dd_dlog_nz
has_cls

second derivatives in splined log of evolution
function
copy of same flag in perturbation structure

int

md_size

number of modes included in computation

int

index_tt_t0

index for transfer type = temperature (j=0 term)

int

index_tt_t1

index for transfer type = temperature (j=1 term)

int

index_tt_t2

index for transfer type = temperature (j=2 term)

int

index_tt_e

index for transfer type = E-polarization

int

index_tt_b

index for transfer type = B-polarization

int

index_tt_lcmb

index for transfer type = CMB lensing

int

index_tt_density

index for first bin of transfer type = matter
density

int

index_tt_lensing

index for first bin of transfer type = galaxy
lensing

int

index_tt_rsd

index for first bin of transfer type = redshift
space distortion of number count

int

index_tt_d0

index for first bin of transfer type = doppler
effect for of number count (j=0 term)

int

index_tt_d1

index for first bin of transfer type = doppler
effect for of number count (j=1 term)

short

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Data Fields
int

index_tt_nc_lens

int

index_tt_nc_g1

int

index_tt_nc_g2

int

index_tt_nc_g3

int

index_tt_nc_g4

int

index_tt_nc_g5

int ∗
int ∗∗

tt_size

index for first bin of transfer type = lensing for
of number count
index for first bin of transfer type = gravity term
G1 for of number count
index for first bin of transfer type = gravity term
G2 for of number count
index for first bin of transfer type = gravity term
G3 for of number count
index for first bin of transfer type = gravity term
G3 for of number count
index for first bin of transfer type = gravity term
G3 for of number count
number of requested transfer types
tt_size[index_md] for each mode

l_size_tt

number of multipole values for which we
effectively compute the transfer
function,l_size_tt[index_md][index_tt]

l_size

number of multipole values for each requested
mode, l_size[index_md]

l_size_max

greatest of all l_size[index_md]

l

list of multipole values l[index_l]

angular_rescaling

correction between l and k space due to
curvature (= comoving angular diameter
distance to recombination / comoving radius to
recombination)

q_size

number of wavenumber values

double ∗

q

list of wavenumber values, q[index_q]

double ∗∗

k

list of wavenumber values for each requested
mode, k[index_md][index_q]. In flat universes
k=q. In non-flat universes q and k differ through
q2 = k2 + K(1+m), where m=0,1,2 for scalar,
vector, tensor. q should be used throughout the
transfer module, excepted when interpolating
or manipulating the source functions S(k,tau):
for a given value of q this should be done in
k(q).

index_q_flat_approximation

index of the first q value using the flat rescaling
approximation

transfer

table of transfer functions for each mode, initial
condition, type, multipole and wavenumber,
with argument transfer[index_md][((index_ic ∗
ptr->tt_size[index_md] + index_tt) ∗
ptr->l_size[index_md] + index_l) ∗ ptr->q_size
+ index_q]

short

initialise_HIS_cache

short

transfer_verbose

only true if we are using CLASS for setting up a
cache of HIS structures
flag regulating the amount of information sent
to standard output (none if set to zero)

int ∗
int
int ∗
double

size_t

int
double ∗∗

ErrorMsg

error_message

zone for writing error messages

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5.22.2.2

225

struct transfer_workspace

Structure containing all the quantities that each thread needs to know for computing transfer functions (but that can
be forgotten once the transfer functions are known, otherwise they would be stored in the transfer module)
Data Fields
HyperInterpStruct

int

HIS

structure containing all hyperspherical bessel functions (flat
case) or all hyperspherical bessel functions for a given value of
beta=q/sqrt(|K|) (non-flat case). HIS = Hyperspherical
Interpolation Structure.

HIS_allocated

int

l_size

flag specifying whether the previous structure has been
allocated
pointer to structure containing all the spherical bessel functions
of the flat case (used even in the non-flat case, for
approximation schemes). pBIS = pointer to Bessel Interpolation
Structure.
number of l values

int

tau_size

number of discrete time values for a given type

int

tau_size_max

maximum number of discrete time values for all types

double ∗

interpolated_sources

interpolated_sources[index_tau]: sources interpolated from the
perturbation module at the right value of k

double ∗

sources

sources[index_tau]: sources used in transfer module, possibly
differing from those in the perturbation module by some
resampling or rescaling

double ∗

tau0_minus_tau

tau0_minus_tau[index_tau]: values of (tau0 - tau)

double ∗

w_trapz

w_trapz[index_tau]: values of weights in trapezoidal integration
(related to time steps)

double ∗

chi

chi[index_tau]: value of argument of bessel function:
k(tau0-tau) (flat case) or sqrt(|K|)(tau0-tau) (non-flat case)

double ∗

cscKgen

cscKgen[index_tau]: useful trigonometric function

double ∗

cotKgen

cotKgen[index_tau]: useful trigonometric function

K

curvature parameter (see background module for details)

sgnK

0 (flat), 1 (positive curvature, spherical, closed), -1 (negative
curvature, hyperbolic, open)

double

tau0_minus_tau_cut

short

neglect_late_source

critical value of (tau0-tau) in time cut approximation for the
wavenumber at hand
flag stating whether we use the time cut approximation for the
wavenumber at hand

HyperInterpStruct ∗

double
int

pBIS

5.22.3

Enumeration Type Documentation

5.22.3.1

radial_function_type

enum radial_function_type

enumeration of possible source types. This looks redundant with respect to the definition of indices index_tt_... This
definition is however convenient and time-saving: it allows to use a "case" statement in transfer_radial_function()
Generated by Doxygen

226

File Documentation

Generated by Doxygen

Chapter 6

The ‘external_Pk‘ mode

• Author: Jesus Torrado (torradocacho [@] lorentz.leidenuniv.nl)
• Date: 2013-12-20

Introduction

This mode allows for an arbitrary primordial spectrum P(k) to be calculated by an external command and passed
to CLASS. That external command may be anything that can be run in the shell: a python script, some compiled
C or Fortran code... This command is executed from within CLASS, and CLASS is able to pass it a number of
parameters defining the spectrum (an amplitude, a tilt...). Those parameters can be used in a Markov chain search
performed by MontePython.

This mode includes the simple case of a precomputed primordial spectrum stored in a text file. In that case, the
cat shell command will do the trick (see below).

Currently, scalar and tensor spectra of perturbations of adiabatic modes are supported.

Use case #1: reading the spectrum from a table

In this case, say the file with the table is called spectrum.txt, located under /path/to, simply include in the
.ini file

command = cat path/to/spectrum.txt

It is necessary that 1st 4 characters are exactly cat.

228

The ‘external_Pk‘ mode

Use case #2: getting the spectrum from an external command
Here an external command is called to generate the spectrum; it may be some compiled C or Fortran code, a python
script... This command may be passed up to 10 floating point arguments, named custom1 to custom10, which
are assigned values inside the .ini file of CLASS. The command parameter would look like

command = /path/to/example.py

if it starts with #/usr/bin/python, otherwise

command = python /path/to/example.py

As an example of the 1st use case, one may use the included script generate_Pk_example.py, which implements a single-field slow-roll spectrum without running, and takes 3 arguments:

• custom1 – the pivot scale (k_0 = 0.05 1/Mpc for Planck).
• custom2 – the amplitude of the scalar power spectrum.
• custom3 – the scalar spectral index.

In order to use it, the following lines must be present in the parameter file:

P_k_ini
command
custom1
custom2
custom3

type = external_Pk
= /path/to/CLASS/external_Pk/generate_Pk_example.py
= 0.05
= 2.2e-9
= 1.

Defined or not (in that case, 0-valued), parameters from custom4 to custom10 will be passed to the example
script, which should ignore them. In this case, CLASS will run in the shell the command

/path/to/CLASS/external_Pk/generate_Pk_example.py 0.05 2.2e-9 1. 0 0 0 0 0 0 0

If CLASS fails to run the command, try to do it directly yourself by hand, using exactly the same string that was
given in command.

Output of the command / format of the table
The command must generate an output separated into lines, each containing a tuple (k, P(k)). The following
requirements must be fulfilled:

• Each line must contain 2 (3, if tensors) floating point numbers: k (in 1/Mpc units) and P_s(k) (and P←_t(k), if tensors), separated by any number of spaces or tabs. The numbers can be in scientific notation,
e.g. 1.4e-3.
• The lines must be sorted in increasing values of k.
• There must be at least two points (k, P(k)) before and after the interval of k requested by CLASS, in
order not to introduce unnecessary interpolation error. Otherwise, an error will be raised. In most of the
cases, generating the spectrum between 1e-6 and 1 1/Mpc should be more than enough.
Generated by Doxygen

229

Precision
This implementation properly handles double-precision floating point numbers (i.e. about 17 significant figures),
both for the input parameters of the command and for the output of the command (or the table).
The sampling of k given by the command (or table) is preserved to be used internally by CLASS. It must be fine
enough a sampling to clearly show the features of the spectrum. The best way to test this is to plot the output/table
and check it with the naked eye.
Another thing to have in mind arises at the time of convolving with the transfer functions. Two precision parameters
are implied: the sampling of k in the integral, given by k_step_trans, and the sampling of the transfer functions
in l, given by l_logstep and l_linstep. In general, it will be enough to reduce the values of the first and
the third parameters. A good start is to give them rather small values, say k_step_trans=0.01 and l_←linstep=1, and to increase them slowly until the point at which the effect of increasing them gets noticeable.

Parameter fit with MontePython
(MontePython)[http://montepython.net/] is able to interact with the external_Pk mode transparently,
using the custom parameters in an MCMC fit. One must just add the appropriate lines to the input file of Monte←Python. For our example, if we wanted to fit the amplitude and spectral index of the primordial spectrum, it would
be:

data.cosmo_arguments[’P_k_ini type’] = ’external_Pk’
data.cosmo_arguments[’command’] = ’/path/to/CLASS/external_Pk/generate_Pk_example.py’
data.cosmo_arguments[’custom1’] = 0.05
# k_pivot
data.parameters[’custom2’]
= [ 2.2, 0, -1, 0.055, 1.e-9, ’cosmo’] # A_s
data.parameters[’custom3’]
= [ 1., 0, -1, 0.0074,
1, ’cosmo’] # n_s

Notice that since in our case custom1 represents the pivot scale, it is passed as a (non-varying) argument, instead
of as a (varying) parameter.
In this case, one would not include the corresponding lines for the primordial parameters of CLASS: k_pivot,

A_s, n_s, alpha_s, etc. They would simply be ignored.

Limitations
• So far, this mode cannot handle vector perturbations, nor isocurvature initial conditions.
• The external script knows nothing about the rest of the CLASS parameters, so if it needs, e.g., k_pivot, it
should be either hard coded, or its value passed as one of the custom parameters.

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230

The ‘external_Pk‘ mode

Generated by Doxygen

Chapter 7

Updating the manual
Author: D. C. Hooper (hooper@physik.rwth-aachen.de)
This pdf manual and accompanying web version have been generated using the doxygen software (http←://www.doxygen.org). This software directly reads the code and extracts the necessary comments to form
the manual, meaning it is very easy to generate newer versions of the manual as desired.

For CLASS developpers:
To maintain the usefulness of the manual, a new version should be generated after any major upgrade to CLASS.
To keep track of how up-to-date the manual is the title page also displays the last modification date. The manual is
generated automatically from the code, excepted a few chapters written manually in the files

README.md
doc/input/chap2.md
doc/input/chap3.md
doc/input/mod.md
external_Pk/README.md

You can update these files, or add new ones that should be declared in the INPUT= field of doc/input/doxyconf.
Generating a new version of this manual is straightforward. First, you need to install the doxygen software, which
can be done by following the instructions on the software's webpage. The location where you install this software
is irrelevant; it doesn't need to be in the same folder as CLASS. For Mac OSX, homebrew users can install the
software with brew install doxygen --with-graphviz.
Once installed, navigate to the class/doc/input directory and run the first script

.

make1.sh

This will generate a new version of the html manual and the necessary files to make the pdf version. Unfortunately,
doxygen does not yet offer the option to automatically order the output chapters in the pdf version of the manual.
Hence, before compiling the pdf, this must be done manually. To do this you need to find the refman.tex file in
class/doc/manual/latex. With this file you can modify the title page, headers, footers, and chapter ordering for the
final pdf. Usually we just make two things: add manually the line

\vspace*{1cm}
{\large Last updated \today}\\

after

232

Updating the manual

{\Large C\+L\+A\+SS M\+A\+N\+U\+AL }\\

and move manually the chapters "The external Pk mode" and "Updating the manual" to the end,
after the automatically generated part. Once you have this file with your desired configuration, navigate back to the
class/doc/input directory, and run the second script

.

make2.sh

You should now be able to find the finished pdf in class/doc/manual/CLASS_MANUAL.pdf. Finally you
can commit the changes to git, but not all the content of doc/ is necessary: only doc/README, doc/input/,
doc/manual/CLASS_MANUAL.pdf, doc/manual/html/. This means that before committing you will have
to do a: git add doc/manual/html/, but NOT a: git add doc/manual/latex/!
As a final comment, doxygen uses two main configuration files: doxyconf and doxygen.sty, both located in
class/doc/input. Changes to these files can dramatically impact the outcome, so any modifications to these files
should be done with great care.

Generated by Doxygen

Index
_MAX_NUMBER_OF_K_FILES_
perturbations.h, 129
_M_EV_TOO_BIG_FOR_HALOFIT_
nonlinear.h, 80
_SELECTION_NUM_MAX_
perturbations.h, 129
_YHE_BIG_
thermodynamics.h, 198
_YHE_SMALL_
thermodynamics.h, 198
_Z_PK_NUM_MAX_
output.h, 89
background, 37
background.c, 19
background_at_tau, 21
background_derivs, 33
background_free, 26
background_free_input, 26
background_functions, 22
background_indices, 27
background_init, 25
background_initial_conditions, 31
background_ncdm_M_from_Omega, 30
background_ncdm_distribution, 27
background_ncdm_init, 28
background_ncdm_momenta, 29
background_ncdm_test_function, 28
background_output_data, 33
background_output_titles, 32
background_solve, 30
background_tau_of_z, 22
background_w_fld, 23
V_e_scf, 34
V_p_scf, 35
V_scf, 35
background.h, 36
background_at_tau
background.c, 21
background_derivs
background.c, 33
background_free
background.c, 26
background_free_input
background.c, 26
background_functions
background.c, 22
background_indices
background.c, 27
background_init

background.c, 25
background_initial_conditions
background.c, 31
background_ncdm_M_from_Omega
background.c, 30
background_ncdm_distribution
background.c, 27
background_ncdm_init
background.c, 28
background_ncdm_momenta
background.c, 29
background_ncdm_test_function
background.c, 28
background_output_data
background.c, 33
background_output_titles
background.c, 32
background_parameters_and_workspace, 41
background_parameters_for_distributions, 41
background_solve
background.c, 30
background_tau_of_z
background.c, 22
background_w_fld
background.c, 23
class.c, 41
class_fzero_ridder
input.c, 59
common.h, 42
evolver_type, 51
file_format, 52
pk_def, 51
error_message
nonlinear, 17
evolver_type
common.h, 51
f1
thermodynamics.h, 197
f2
thermodynamics.h, 198
file_format
common.h, 52
get_machine_precision
input.c, 59
inflation_module_behavior
primordial.h, 154

234

INDEX

input.c, 52
class_fzero_ridder, 59
get_machine_precision, 59
input_default_params, 57
input_default_precision, 58
input_find_root, 60
input_get_guess, 60
input_init, 54
input_init_from_arguments, 53
input_read_parameters, 55
input_try_unknown_parameters, 59
input.h, 61
target_names, 62
input_default_params
input.c, 57
input_default_precision
input.c, 58
input_find_root
input.c, 60
input_get_guess
input.c, 60
input_init
input.c, 54
input_init_from_arguments
input.c, 53
input_read_parameters
input.c, 55
input_try_unknown_parameters
input.c, 59
integration_direction
primordial.h, 154
k
nonlinear, 16
k_nl
nonlinear, 16
k_size
nonlinear, 16
lensing, 76
lensing.c, 63
lensing_addback_cl_ee_bb, 69
lensing_addback_cl_te, 68
lensing_addback_cl_tt, 67
lensing_cl_at_l, 64
lensing_d00, 69
lensing_d11, 70
lensing_d1m1, 70
lensing_d20, 71
lensing_d22, 71
lensing_d2m2, 71
lensing_d31, 72
lensing_d3m1, 72
lensing_d3m3, 73
lensing_d40, 73
lensing_d4m2, 73
lensing_d4m4, 74
lensing_free, 66
lensing_indices, 66

lensing_init, 65
lensing_lensed_cl_ee_bb, 68
lensing_lensed_cl_te, 67
lensing_lensed_cl_tt, 67
lensing.h, 74
lensing_addback_cl_ee_bb
lensing.c, 69
lensing_addback_cl_te
lensing.c, 68
lensing_addback_cl_tt
lensing.c, 67
lensing_cl_at_l
lensing.c, 64
lensing_d00
lensing.c, 69
lensing_d11
lensing.c, 70
lensing_d1m1
lensing.c, 70
lensing_d20
lensing.c, 71
lensing_d22
lensing.c, 71
lensing_d2m2
lensing.c, 71
lensing_d31
lensing.c, 72
lensing_d3m1
lensing.c, 72
lensing_d3m3
lensing.c, 73
lensing_d40
lensing.c, 73
lensing_d4m2
lensing.c, 73
lensing_d4m4
lensing.c, 74
lensing_free
lensing.c, 66
lensing_indices
lensing.c, 66
lensing_init
lensing.c, 65
lensing_lensed_cl_ee_bb
lensing.c, 68
lensing_lensed_cl_te
lensing.c, 67
lensing_lensed_cl_tt
lensing.c, 67
linear_or_logarithmic
primordial.h, 153
method
nonlinear, 16
nl_corr_density
nonlinear, 16
nonlinear, 15
error_message, 17
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INDEX

k, 16
k_nl, 16
k_size, 16
method, 16
nl_corr_density, 16
nonlinear_verbose, 16
tau, 16
tau_size, 16
nonlinear.c, 77
nonlinear_halofit, 78
nonlinear_init, 78
nonlinear.h, 79
_M_EV_TOO_BIG_FOR_HALOFIT_, 80
nonlinear_halofit
nonlinear.c, 78
nonlinear_init
nonlinear.c, 78
nonlinear_verbose
nonlinear, 16
output, 89
output.c, 80
output_cl, 82
output_init, 81
output_one_line_of_cl, 85
output_one_line_of_pk, 87
output_open_cl_file, 85
output_open_pk_file, 86
output_pk, 83
output_pk_nl, 83
output_print_data, 84
output_tk, 84
output.h, 87
_Z_PK_NUM_MAX_, 89
output_cl
output.c, 82
output_init
output.c, 81
output_one_line_of_cl
output.c, 85
output_one_line_of_pk
output.c, 87
output_open_cl_file
output.c, 85
output_open_pk_file
output.c, 86
output_pk
output.c, 83
output_pk_nl
output.c, 83
output_print_data
output.c, 84
output_tk
output.c, 84
perturb_approximations
perturbations.c, 107
perturb_derivs
perturbations.c, 113
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235

perturb_einstein
perturbations.c, 110
perturb_find_approximation_number
perturbations.c, 99
perturb_find_approximation_switches
perturbations.c, 100
perturb_free
perturbations.c, 93
perturb_get_k_list
perturbations.c, 95
perturb_indices_of_perturbs
perturbations.c, 93
perturb_init
perturbations.c, 92
perturb_initial_conditions
perturbations.c, 104
perturb_parameters_and_workspace, 128
perturb_prepare_output
perturbations.c, 99
perturb_print_variables
perturbations.c, 112
perturb_solve
perturbations.c, 98
perturb_sources
perturbations.c, 111
perturb_sources_at_tau
perturbations.c, 91
perturb_tca_slip_and_shear
perturbations.c, 117
perturb_timesampling_for_sources
perturbations.c, 94
perturb_timescale
perturbations.c, 109
perturb_total_stress_energy
perturbations.c, 111
perturb_vector, 125
perturb_vector_free
perturbations.c, 103
perturb_vector_init
perturbations.c, 101
perturb_workspace, 127
perturb_workspace_free
perturbations.c, 97
perturb_workspace_init
perturbations.c, 96
perturbations.c, 90
perturb_approximations, 107
perturb_derivs, 113
perturb_einstein, 110
perturb_find_approximation_number, 99
perturb_find_approximation_switches, 100
perturb_free, 93
perturb_get_k_list, 95
perturb_indices_of_perturbs, 93
perturb_init, 92
perturb_initial_conditions, 104
perturb_prepare_output, 99
perturb_print_variables, 112

236

perturb_solve, 98
perturb_sources, 111
perturb_sources_at_tau, 91
perturb_tca_slip_and_shear, 117
perturb_timesampling_for_sources, 94
perturb_timescale, 109
perturb_total_stress_energy, 111
perturb_vector_free, 103
perturb_vector_init, 101
perturb_workspace_free, 97
perturb_workspace_init, 96
perturbations.h, 118
_MAX_NUMBER_OF_K_FILES_, 129
_SELECTION_NUM_MAX_, 129
possible_gauges, 129
tca_flags, 129
tca_method, 129
perturbs, 120
phi_pivot_methods
primordial.h, 154
pk_def
common.h, 51
possible_gauges
perturbations.h, 129
potential_shape
primordial.h, 153
precision, 43
primordial, 149
primordial.c, 130
primordial_analytic_spectrum, 135
primordial_analytic_spectrum_init, 135
primordial_external_spectrum_init, 147
primordial_free, 133
primordial_get_lnk_list, 134
primordial_indices, 134
primordial_inflation_analytic_spectra, 138
primordial_inflation_check_hubble, 143
primordial_inflation_check_potential, 142
primordial_inflation_derivs, 146
primordial_inflation_evolve_background, 141
primordial_inflation_find_attractor, 141
primordial_inflation_find_phi_pivot, 145
primordial_inflation_get_epsilon, 143
primordial_inflation_hubble, 136
primordial_inflation_indices, 137
primordial_inflation_one_k, 140
primordial_inflation_one_wavenumber, 139
primordial_inflation_potential, 136
primordial_inflation_solve_inflation, 137
primordial_inflation_spectra, 138
primordial_init, 132
primordial_spectrum_at_k, 131
primordial.h, 147
inflation_module_behavior, 154
integration_direction, 154
linear_or_logarithmic, 153
phi_pivot_methods, 154
potential_shape, 153

INDEX

primordial_spectrum_type, 153
target_quantity, 153
time_definition, 154
primordial_analytic_spectrum
primordial.c, 135
primordial_analytic_spectrum_init
primordial.c, 135
primordial_external_spectrum_init
primordial.c, 147
primordial_free
primordial.c, 133
primordial_get_lnk_list
primordial.c, 134
primordial_indices
primordial.c, 134
primordial_inflation_analytic_spectra
primordial.c, 138
primordial_inflation_check_hubble
primordial.c, 143
primordial_inflation_check_potential
primordial.c, 142
primordial_inflation_derivs
primordial.c, 146
primordial_inflation_evolve_background
primordial.c, 141
primordial_inflation_find_attractor
primordial.c, 141
primordial_inflation_find_phi_pivot
primordial.c, 145
primordial_inflation_get_epsilon
primordial.c, 143
primordial_inflation_hubble
primordial.c, 136
primordial_inflation_indices
primordial.c, 137
primordial_inflation_one_k
primordial.c, 140
primordial_inflation_one_wavenumber
primordial.c, 139
primordial_inflation_potential
primordial.c, 136
primordial_inflation_solve_inflation
primordial.c, 137
primordial_inflation_spectra
primordial.c, 138
primordial_init
primordial.c, 132
primordial_spectrum_at_k
primordial.c, 131
primordial_spectrum_type
primordial.h, 153
radial_function_type
transfer.h, 225
recombination, 195
recombination_algorithm
thermodynamics.h, 198
reionization, 196
reionization_parametrization
Generated by Doxygen

INDEX

thermodynamics.h, 198
reionization_z_or_tau
thermodynamics.h, 199
spectra, 170
spectra.c, 155
spectra_cl_at_l, 156
spectra_cls, 164
spectra_compute_cl, 165
spectra_free, 163
spectra_indices, 163
spectra_init, 162
spectra_k_and_tau, 166
spectra_matter_transfers, 168
spectra_output_tk_data, 168
spectra_pk, 166
spectra_pk_at_k_and_z, 158
spectra_pk_at_z, 157
spectra_pk_nl_at_k_and_z, 160
spectra_pk_nl_at_z, 159
spectra_sigma, 167
spectra_tk_at_k_and_z, 161
spectra_tk_at_z, 161
spectra.h, 169
spectra_cl_at_l
spectra.c, 156
spectra_cls
spectra.c, 164
spectra_compute_cl
spectra.c, 165
spectra_free
spectra.c, 163
spectra_indices
spectra.c, 163
spectra_init
spectra.c, 162
spectra_k_and_tau
spectra.c, 166
spectra_matter_transfers
spectra.c, 168
spectra_output_tk_data
spectra.c, 168
spectra_pk
spectra.c, 166
spectra_pk_at_k_and_z
spectra.c, 158
spectra_pk_at_z
spectra.c, 157
spectra_pk_nl_at_k_and_z
spectra.c, 160
spectra_pk_nl_at_z
spectra.c, 159
spectra_sigma
spectra.c, 167
spectra_tk_at_k_and_z
spectra.c, 161
spectra_tk_at_z
spectra.c, 161
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237

target_names
input.h, 62
target_quantity
primordial.h, 153
tau
nonlinear, 16
tau_size
nonlinear, 16
tca_flags
perturbations.h, 129
tca_method
perturbations.h, 129
thermo, 192
thermodynamics.c, 174
thermodynamics_at_z, 177
thermodynamics_derivs_with_recfast, 189
thermodynamics_energy_injection, 181
thermodynamics_free, 179
thermodynamics_get_xe_before_reionization, 184
thermodynamics_helium_from_bbn, 180
thermodynamics_indices, 179
thermodynamics_init, 177
thermodynamics_merge_reco_and_reio, 189
thermodynamics_onthespot_energy_injection, 181
thermodynamics_output_titles, 190
thermodynamics_recombination, 186
thermodynamics_recombination_with_hyrec, 186
thermodynamics_recombination_with_recfast, 187
thermodynamics_reionization, 184
thermodynamics_reionization_function, 183
thermodynamics_reionization_sample, 185
thermodynamics.h, 191
_YHE_BIG_, 198
_YHE_SMALL_, 198
f1, 197
f2, 198
recombination_algorithm, 198
reionization_parametrization, 198
reionization_z_or_tau, 199
thermodynamics_at_z
thermodynamics.c, 177
thermodynamics_derivs_with_recfast
thermodynamics.c, 189
thermodynamics_energy_injection
thermodynamics.c, 181
thermodynamics_free
thermodynamics.c, 179
thermodynamics_get_xe_before_reionization
thermodynamics.c, 184
thermodynamics_helium_from_bbn
thermodynamics.c, 180
thermodynamics_indices
thermodynamics.c, 179
thermodynamics_init
thermodynamics.c, 177
thermodynamics_merge_reco_and_reio
thermodynamics.c, 189
thermodynamics_onthespot_energy_injection

238

thermodynamics.c, 181
thermodynamics_output_titles
thermodynamics.c, 190
thermodynamics_parameters_and_workspace, 197
thermodynamics_recombination
thermodynamics.c, 186
thermodynamics_recombination_with_hyrec
thermodynamics.c, 186
thermodynamics_recombination_with_recfast
thermodynamics.c, 187
thermodynamics_reionization
thermodynamics.c, 184
thermodynamics_reionization_function
thermodynamics.c, 183
thermodynamics_reionization_sample
thermodynamics.c, 185
time_definition
primordial.h, 154
transfer.c, 199
transfer_compute_for_each_l, 214
transfer_compute_for_each_q, 207
transfer_dNdz_analytic, 210
transfer_free, 203
transfer_functions_at_q, 201
transfer_get_k_list, 206
transfer_get_l_list, 204
transfer_get_q_list, 205
transfer_get_source_correspondence, 206
transfer_indices_of_transfers, 203
transfer_init, 202
transfer_integrate, 215
transfer_interpolate_sources, 208
transfer_lensing_sampling, 211
transfer_limber, 217
transfer_limber2, 219
transfer_limber_interpolate, 218
transfer_selection_compute, 213
transfer_selection_function, 210
transfer_selection_sampling, 211
transfer_selection_times, 213
transfer_source_resample, 212
transfer_source_tau_size, 207
transfer_sources, 209
transfer.h, 220
radial_function_type, 225
transfer_compute_for_each_l
transfer.c, 214
transfer_compute_for_each_q
transfer.c, 207
transfer_dNdz_analytic
transfer.c, 210
transfer_free
transfer.c, 203
transfer_functions_at_q
transfer.c, 201
transfer_get_k_list
transfer.c, 206
transfer_get_l_list

INDEX

transfer.c, 204
transfer_get_q_list
transfer.c, 205
transfer_get_source_correspondence
transfer.c, 206
transfer_indices_of_transfers
transfer.c, 203
transfer_init
transfer.c, 202
transfer_integrate
transfer.c, 215
transfer_interpolate_sources
transfer.c, 208
transfer_lensing_sampling
transfer.c, 211
transfer_limber
transfer.c, 217
transfer_limber2
transfer.c, 219
transfer_limber_interpolate
transfer.c, 218
transfer_selection_compute
transfer.c, 213
transfer_selection_function
transfer.c, 210
transfer_selection_sampling
transfer.c, 211
transfer_selection_times
transfer.c, 213
transfer_source_resample
transfer.c, 212
transfer_source_tau_size
transfer.c, 207
transfer_sources
transfer.c, 209
transfer_workspace, 224
transfers, 221
V_e_scf
background.c, 34
V_p_scf
background.c, 35
V_scf
background.c, 35

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