pyccl.cosmology module

The core functionality of CCL, including the core data types lives in this module. Its focus is the Cosmology class, which plays a central role, carrying the information on cosmological parameters and derived quantities needed in most of the calculations carried out by CCL.

Note

All of the standalone functions in other modules, which take cosmo as their first argument, are methods of Cosmology.

Some important CCL parameters, governing for example the precision and speed of some calculations, are independent of the Cosmology objects, and instead can be accessed at a global level. You can do so as e.g. pyccl.gsl_params['ODE_GROWTH_EPSREL'], pyccl.spline_params['K_MIN'], pyccl.physical_constants['CLIGHT'], or pyccl.gsl_params.ODE_GROWTH_EPSREL, pyccl.spline_params.K_MIN, pyccl.physical_constants.CLIGHT.

class pyccl.cosmology.TransferFunctions(value, names=None, *, module=None, qualname=None, type=None, start=1, boundary=None)[source]

Bases: Enum

BBKS = 'bbks'

EISENSTEIN_HU = 'eisenstein_hu'

EISENSTEIN_HU_NOWIGGLES = 'eisenstein_hu_nowiggles'

BOLTZMANN_CLASS = 'boltzmann_class'

BOLTZMANN_CAMB = 'boltzmann_camb'

BOLTZMANN_ISITGR = 'boltzmann_isitgr'

CALCULATOR = 'calculator'

EMULATOR_LINPK = 'emulator'

class pyccl.cosmology.MatterPowerSpectra(value, names=None, *, module=None, qualname=None, type=None, start=1, boundary=None)[source]

Bases: Enum

LINEAR = 'linear'

HALOFIT = 'halofit'

CAMB = 'camb'

CALCULATOR = 'calculator'

EMULATOR_NLPK = 'emulator'

class pyccl.cosmology.Cosmology(self, *, Omega_c=None, Omega_b=None, h=None, n_s=None, sigma8=None, A_s=None, Omega_k=0.0, Omega_g=None, Neff=None, m_nu=0.0, mass_split='normal', w0=-1.0, wa=0.0, T_CMB=2.7255, T_ncdm=0.71611, transfer_function='boltzmann_camb', matter_power_spectrum='halofit', baryonic_effects=None, mg_parametrization=None, extra_parameters=None)[source]

Bases: CCLObject

Stores information about cosmological parameters and associated data (e.g. distances, power spectra).

The values of cosmological parameters may be looked up by name (e.g. cosmo["sigma8"]). Note that some of the parameters accessible this way are not contained in the signature of Cosmology, but are derived during initialization.

Note

Although some arguments default to None, they will raise a ValueError inside this function if not specified, so they are not optional.

Note

The parameter Omega_g can be used to set the radiation density (not including relativistic neutrinos) to zero. Doing this will give you a model that is physically inconsistent since the temperature of the CMB will still be non-zero.

Parameters:

Omega_c (float) – Cold dark matter density fraction.
Omega_b (float) – Baryonic matter density fraction.
h (float) – Hubble constant divided by 100 km/s/Mpc; unitless.
A_s (float) – Power spectrum normalization. Exactly one of A_s and sigma_8 is required.
sigma8 (float) – Variance of matter density perturbations at an 8 Mpc/h scale. Exactly one of A_s and sigma_8 is required. Note that, if a value of sigma8 is passed, CCL will enforce the linear matter power spectrum to be correctly normalised to this value of \(\sigma_8\), even in the presence of other parameters (e.g. modified gravity parameters) that might affect the overall power spectrum normalization.
n_s (float) – Primordial scalar perturbation spectral index.
Omega_k (float) – Curvature density fraction. Defaults to 0.
Omega_g (float) – Density in relativistic species except massless neutrinos. The default of None corresponds to setting this from the CMB temperature. Note that if a non-None value is given, this may result in a physically inconsistent model because the CMB temperature will still be non-zero in the parameters.
Neff (float) – Effective number of massless neutrinos present. Defaults to 3.044.
m_nu (float or array) – Mass in eV of the massive neutrinos present. Defaults to 0. If a sequence is passed, it is assumed that the elements of the sequence represent the individual neutrino masses.
mass_split (str) – Type of massive neutrinos. Should be one of ‘single’, ‘equal’, ‘normal’, ‘inverted’. ‘single’ treats the mass as being held by one massive neutrino. The other options split the mass into 3 massive neutrinos. Ignored if a sequence is passed in m_nu. Default is ‘normal’.
w0 (float) – First order term of dark energy equation of state. Defaults to -1.
wa (float) – Second order term of dark energy equation of state. Defaults to 0.
T_CMB (float) – The CMB temperature today. The default value is 2.7255.
T_ncdm (float) – Non-CDM temperature in units of photon temperature. The default is 0.71611.
transfer_function (str or EmulatorPk) – The transfer function to use. Defaults to ‘boltzmann_camb’.
matter_power_spectrum (str or EmulatorPk) – The matter power spectrum to use. Defaults to ‘halofit’.
baryonic_effects (Baryons or None) – The baryonic effects model to use. Options are None (no baryonic effects), or a Baryons object.
mg_parametrization (ModifiedGravity or None) – The modified gravity parametrization to use. Options are None (no MG), or a ModifiedGravity object. Currently, only MuSigmaMG is supported.
extra_parameters (dict) – Dictionary holding extra parameters. Currently supports extra parameters for CAMB. Details described below. Defaults to None.

Currently supported extra parameters for CAMB are:

halofit_version

HMCode_A_baryon

HMCode_eta_baryon

HMCode_logT_AGN

kmax

lmax

dark_energy_model

Consult the CAMB documentation for their usage. These parameters are passed in a dict to extra_parameters as:

extra_parameters = {"camb": {"halofit_version": "mead2020_feedback",
                             "HMCode_logT_AGN": 7.8}}

Note

If using camb to compute the non-linear power spectrum with HMCode to include baryonic effects, you should not include any extra baryonic effects (i.e. set baryonic_effects=None).

write_yaml(filename, *, sort_keys=False)[source]

Write a YAML representation of the parameters to file.

Parameters:: filename (str) – file name, file pointer, or stream to write parameters to.

classmethod read_yaml(filename, **kwargs)[source]

Read the parameters from a YAML file.

Parameters:

filename (str) – file name, file pointer, or stream to read parameters from.
**kwargs (dict) – additional keywords that supersede file contents

compute_growth()[source]: Compute the growth function.

compute_linear_power()[source]: Compute the linear power spectrum.

compute_nonlin_power()[source]: Compute the non-linear power spectrum.

compute_sigma()[source]: Compute the sigma(M) spline.

get_linear_power(name='delta_matter:delta_matter')[source]

Get the Pk2D object associated with the linear power spectrum with name name.

Parameters:: name (str or None) – name of the power spectrum to return.
Returns:: Pk2D object containing the linear power spectrum with name name.

get_nonlin_power(name='delta_matter:delta_matter')[source]

Get the Pk2D object associated with the non-linear power spectrum with name name.

Parameters:: name (str or None) – name of the power spectrum to return.
Returns:: Pk2D object containing the non-linear power spectrum with name name.

property has_distances: Checks if the distances have been precomputed.

property has_growth: Checks if the growth function has been precomputed.

property has_linear_power: Checks if the linear power spectra have been precomputed.

property has_nonlin_power: Checks if the non-linear power spectra have been precomputed.

property has_sigma: Checks if sigma(M) is precomputed.

CIBTracer(*, z_min=0.0, z_max=6.0, n_chi=1024)

Specific Tracer associated with the cosmic infrared background (CIB). The radial kernel for this tracer is simply

\[W(\chi) = \frac{1}{1+z}.\]

Any angular power spectra computed with this tracer, should use a three-dimensional power spectrum involving the CIB emissivity density in units of \({\rm Jy}\,{\rm Mpc}^{-1}\,{\rm srad}^{-1}\) (or multiples thereof – see e.g. HaloProfileCIBShang12).

Parameters:

cosmo (Cosmology) – Cosmology object.
z_min (float) – minimum redshift down to which we define the kernel.
z_max (float) – maximum redshift up to which we define the kernel.
n_chi (float) – number of intervals in the radial comoving distance on which we sample the kernel.

CMBLensingTracer(*, z_source, n_samples=100)

A Tracer for CMB lensing convergence \(\kappa\). The associated kernel and transfer function are described in Eq. 31 of the CCL paper.

Parameters:

cosmo (Cosmology) – Cosmology object.
z_source (float) – Redshift of source plane for CMB lensing.
n_samples (int) – number of samples over which the kernel is desired. These will be equi-spaced in radial distance. The kernel is quite smooth, so usually O(100) samples is enough.

ISWTracer(*, z_max=6.0, n_chi=1024)

Specific Tracer associated with the integrated Sachs-Wolfe effect (ISW). Useful when cross-correlating any low-redshift probe with the primary CMB anisotropies. The ISW contribution to the temperature fluctuations is:

\[\Delta T_{\rm CMB} = 2T_{\rm CMB} \int_0^{\chi_{LSS}}d\chi a\,\dot{\phi}\]

Any angular power spectra computed with this tracer, should use a three-dimensional power spectrum involving the matter power spectrum.

Warning

The current implementation of this tracer assumes a standard Poisson equation relating \(\phi\) and \(\delta\), and linear, scale-independent structure growth. Although this should be valid in \(\Lambda\) CDM and on the large scales the ISW is sensitive to, these approximations must be borne in mind.

Parameters:

cosmo (Cosmology) – Cosmology object.
z_max (float) – maximum redshift up to which we define the kernel.
n_chi (float) – number of intervals in the radial comoving distance on which we sample the kernel.

NumberCountsTracer(*, dndz, bias=None, mag_bias=None, has_rsd, n_samples=256)

Specific Tracer associated to galaxy clustering with linear scale-independent bias, including redshift-space distortions and magnification. The associated contributions are described in detail in Section 2.4.1 of the CCL paper.

Warning

When including redshift-space distortions, the current implementation assumes linear, scale-independent growth Although this should be valid in \(\Lambda\) CDM and on the large scales (especially when considering a broad \(N(z)\)), this approximation should be borne in mind.

Parameters:

cosmo (Cosmology) – Cosmology object.
dndz (tuple) – A tuple of arrays (z, N(z)) giving the redshift distribution of the objects. The units are arbitrary; N(z) will be normalized to unity.
bias (tuple) – A tuple of arrays (z, b(z)) giving the galaxy bias. If None, this tracer won’t include a term proportional to the matter density contrast.
mag_bias (tuple) – A tuple of arrays (z, s(z)) giving the magnification bias as a function of redshift. If None, the tracer is assumed to not have magnification bias terms.
has_rsd (bool) – If True, this tracer will include a redshift-space distortion term.
n_samples (int) – number of samples over which the magnification lensing kernel is desired. These will be equi-spaced in radial distance. The kernel is quite smooth, so usually O(100) samples is enough.

WeakLensingTracer(*, dndz, has_shear=True, ia_bias=None, use_A_ia=True, n_samples=256)

Specific Tracer associated to galaxy shape distortions including lensing shear and intrinsic alignments within the L-NLA model. The associated contributions are described in detail in Section 2.4.1 of the CCL paper.

Parameters:

cosmo (Cosmology) – Cosmology object.
dndz (tuple) – A tuple of arrays (z, N(z)) giving the redshift distribution of the objects. The units are arbitrary; N(z) will be normalized to unity.
has_shear (bool) – set to False if you want to omit the lensing shear contribution from this tracer.
ia_bias (tuple) – A tuple of arrays (z, A_IA(z)) giving the intrinsic alignment amplitude A_IA(z). If None, the tracer is assumed to not have intrinsic alignments.
use_A_ia (bool) – set to True to use the conventional IA normalization. Set to False to use the raw input amplitude, which will usually be 1 for use with perturbation theory IA modeling.
n_samples (int) – number of samples over which the lensing kernel is desired. These will be equi-spaced in radial distance. The kernel is quite smooth, so usually O(100) samples is enough.

age_of_universe(a)

Age of the Universe at some scale factor, in \(\rm Gyr\).

Parameters:

cosmo (Cosmology) – Cosmological parameters.
a (float or (na,) array_like) – Scale factor(s) normalized to 1 today.

Returns:

t_age – Age of the Universe at a. nan if a is out of bounds of the spline parametets stored in cosmo.

Return type:

float or (na,) ndarray

angular_cl(tracer1, tracer2, ell, *, p_of_k_a='delta_matter:delta_matter', l_limber=-1, limber_max_error=0.01, limber_integration_method='qag_quad', non_limber_integration_method='FKEM', p_of_k_a_lin='delta_matter:delta_matter', return_meta=False)

Calculate the angular (cross-)power spectrum for a pair of tracers.

Parameters:

cosmo (Cosmology) – A Cosmology object.
tracer1 (Tracer) – a Tracer object, of any kind.
tracer2 (Tracer) – a second Tracer object.
ell (float or array) – Angular multipole(s) at which to evaluate the angular power spectrum.
p_of_k_a (Pk2D, str or None) – 3D Power spectrum to project. If a string, it must correspond to one of the non-linear power spectra stored in cosmo (e.g. ‘delta_matter:delta_matter’).
l_limber (int, float or 'auto') – Angular wavenumber beyond which Limber’s approximation will be used. Defaults to -1. If ‘auto’, then the non-limber integrator will be used to compute the right transition point given the value of limber_max_error.
limber_max_error (float) – Maximum fractional error for Limber integration.
limber_integration_method (string) – integration method to be used for the Limber integrals. Possibilities: ‘qag_quad’ (GSL’s qag method backed up by quad when it fails) and ‘spline’ (the integrand is splined and then integrated numerically).
non_limber_integration_method (string) – integration method to be used for the non-Limber integrals. Currently the only method implemented is 'FKEM' (see the N5K paper for details).
p_of_k_a_lin (Pk2D, str or None) – 3D linear Power spectrum to project, for special use in PT calculations using the FKEM non-limber integration technique. If a string, it must correspond to one of the linear power spectra stored in cosmo (e.g. ‘delta_matter:delta_matter’).
return_meta (bool) – if True, also return a dictionary with various metadata about the calculation, such as l_limber as calculated by the non-limber integrator.

Returns:

Angular (cross-)power spectrum values, \(C_\ell\), for the pair of tracers, as a function of \(\ell\).

Return type:

float or array

angular_cl_cov_SSC(tracer1, tracer2, *, ell, t_of_kk_a, tracer3=None, tracer4=None, ell2=None, sigma2_B=None, fsky=1.0, integration_method='qag_quad')

Calculate the super-sample contribution to the connected non-Gaussian covariance for a pair of power spectra \(C_{\ell_1}^{ab}\) and \(C_{\ell_2}^{cd}\), between two pairs of tracers (\((a,b)\) and \((c,d)\)).

Specifically, it computes:

\[{\rm Cov}_{\rm SSC}(\ell_1,\ell_2)= \int \frac{d\chi}{\chi^4} \tilde{\Delta}^a_{\ell_1}(\chi) \tilde{\Delta}^b_{\ell_1}(\chi) \tilde{\Delta}^c_{\ell_2}(\chi) \tilde{\Delta}^d_{\ell_2}(\chi)\, \bar{T}_{abcd}\left[\frac{\ell_1+1/2}{\chi}, \frac{\ell_2+1/2}{\chi}, a(\chi)\right]\]

where \(\Delta^x_\ell(\chi)\) is the transfer function for tracer \(x\) (see Eq. 39 in the CCL note), and \(\bar{T}_{abcd}(k_1,k_2,a)\) is the isotropized connected trispectrum of the four tracers (see the documentation of the Tk3D class for details).

Parameters:

cosmo (Cosmology) – A Cosmology object.
tracer1 (Tracer) – a Tracer object.
tracer2 (Tracer) – a second Tracer object.
ell (float or array) – Angular wavenumber(s) at which to evaluate the first dimension of the angular power spectrum covariance.
t_of_kk_a (Tk3D) – 3D connected trispectrum.
tracer3 (Tracer) – a Tracer object. If None, tracer1 will be used instead.
tracer4 (Tracer) – a Tracer object. If None, tracer2 will be used instead.
ell2 (float or array) – Angular wavenumber(s) at which to evaluate the second dimension of the angular power spectrum covariance. If None, ell will be used instead.
sigma2_B (tuple or None) – A tuple of arrays (a, sigma2_B(a)) containing the variance of the projected matter overdensity over the footprint as a function of the scale factor. If None, a compact circular footprint will be assumed covering a sky fraction fsky.
fsky (float) – sky fraction.
integration_method (str) – integration method to be used for the Limber integrals. Possibilities: 'qag_quad' (GSL’s qag method backed up by quad when it fails) and 'spline' (the integrand is splined and then integrated analytically).

Returns:

2D array containing the super-sample Angular power spectrum covariance \({\rm Cov}_{\rm SSC}(\ell_1,\ell_2)\), for the four input tracers, as a function of \(\ell_1\) and \(\ell_2\). The ordering is such that out[i2, i1] = Cov(ell2[i2], ell[i1]).

Return type:

(float or array)

angular_cl_cov_cNG(tracer1, tracer2, *, ell, t_of_kk_a, tracer3=None, tracer4=None, ell2=None, fsky=1.0, integration_method='qag_quad')

Calculate the connected non-Gaussian covariance for a pair of angular power spectra \(C_{\ell_1}^{ab}\) and \(C_{\ell_2}^{cd}\), between two pairs of tracers (\((a,b)\) and \((c,d)\)).

Specifically, it computes:

\[{\rm Cov}_{\rm cNG}(\ell_1,\ell_2)=\frac{1}{4\pi f_{\rm sky}} \int \frac{d\chi}{\chi^6} \tilde{\Delta}^a_{\ell_1}(\chi) \tilde{\Delta}^b_{\ell_1}(\chi) \tilde{\Delta}^c_{\ell_2}(\chi) \tilde{\Delta}^d_{\ell_2}(\chi)\, \bar{T}_{abcd}\left[\frac{\ell_1+1/2}{\chi}, \frac{\ell_2+1/2}{\chi}, a(\chi)\right]\]

where \(\Delta^x_\ell(\chi)\) is the transfer function for tracer \(x\) (see Eq. 39 in the CCL note), and \(\bar{T}_{abcd}(k_1,k_2,a)\) is the isotropized connected trispectrum of the four tracers (see the documentation of the Tk3D class for details).

Parameters:

cosmo (Cosmology) – A Cosmology object.
tracer1 (Tracer) – a Tracer object.
tracer2 (Tracer) – a second Tracer object.
ell (float or array) – Angular wavenumber(s) at which to evaluate the first dimension of the angular power spectrum covariance.
t_of_kk_a (Tk3D) – 3D connected trispectrum.
tracer3 (Tracer) – a Tracer object. If None, tracer1 will be used instead.
tracer4 (Tracer) – a Tracer object. If None, tracer2 will be used instead.
ell2 (float or array) – Angular wavenumber(s) at which to evaluate the second dimension of the angular power spectrum covariance. If None, ell will be used instead.
fsky (float) – sky fraction.
integration_method (str) – integration method to be used for the Limber integrals. Possibilities: 'qag_quad' (GSL’s qag method backed up by quad when it fails) and 'spline' (the integrand is splined and then integrated analytically).

Returns:

2D array containing the connected non-Gaussian Angular power spectrum covariance \({\rm Cov}_{\rm cNG}(\ell_1,\ell_2)\), for the four input tracers, as a function of \(\ell_1\) and \(\ell_2\). The ordering is such that out[i2, i1] = Cov(ell2[i2], ell[i1]).

Return type:

(float or array)

angular_diameter_distance(a1, a2=None)

Angular diameter distance.

The angular diameter distance in Mpc from scale factor a1 to scale factor a2. If a2 is not provided, it is assumed that the distance will be calculated between 1 and a1.

Note

a2 has to be smaller than a1 (i.e. a source at a2 is behind one at a1). You can compute the distance between a single lens at a1 and multiple sources at a2 by passing a scalar a1.

Parameters:

cosmo (Cosmology) – Cosmological parameters.
a1 (float or array) – Scale factor(s), normalized to 1 today.
a2 (float or array) – Scale factor(s), normalized to 1 today, optional.

Returns:

angular diameter distance; Mpc.

Return type:

(float or array)

comoving_angular_distance(a)

Comoving angular distance.

Note

This quantity is otherwise known as the transverse comoving distance, and is NOT angular diameter distance or angular separation. The comoving angular distance is defined such that the comoving distance between two objects at a fixed scale factor separated by an angle \(\theta\) is \(\theta r_{A}(a)\) where \(r_{A}(a)\) is the comoving angular distance.

Parameters:

cosmo (Cosmology) – Cosmological parameters.
a (float or array) – Scale factor(s), normalized to 1 today.

Returns:

Comoving angular distance; Mpc.

Return type:

(float or array)

comoving_radial_distance(a)

Comoving radial distance.

Parameters:

cosmo (Cosmology) – Cosmological parameters.
a (float or array) – Scale factor(s), normalized to 1 today.

Returns:

Comoving radial distance; Mpc.

Return type:

(float or array)

comoving_volume(a, *, solid_angle=12.566370614359172)

Comoving volume, in \(\rm Mpc^3\).

\[V_{\rm C} = \int_{\Omega} \mathrm{{d}}\Omega \int_z \mathrm{d}z D_{\rm H} \frac{(1+z)^2 D_{\mathrm{A}}^2}{E(z)}\]

See Eq. 29 in Hogg 2000.

Parameters:

cosmo (Cosmology) – Cosmological parameters.
a (float or (na,) array_like) – Scale factor(s) normalized to 1 today.
solid_angle (float) – Solid angle subtended in the sky for which the comoving volume is calculated.

Returns:

V_C – Comoving volume at a.

Return type:

float or (na,) ndarray