Notation, Models and Other Cosmological Conventions

The documentation here provides a brief description of CCL and its contents. For a more thorough description of the underlying equations CCL implements, see the CCL note and the CCL paper.

Units

The following conventions are used by CCL:

  • All units are non-h-inverse (e.g., Mpc as opposed to Mpc/h).

  • Distances are in units of Mpc, masses are in \(M_\odot\) units.

Cosmological Parameters

CCL uses the following parameters to define the cosmological model.

Background Parameters

  • Omega_c: the density fraction at z=0 of CDM.

  • Omega_b: the density fraction at z=0 of baryons.

  • h: the Hubble constant in units of 100 \({\rm km}/{\rm s}/{\rm Mpc}\).

  • Omega_k: the curvature density fraction at \(z=0\).

  • Omega_g: the density of radiation (not including massless neutrinos).

  • w0: first order term of the dark energy equation of state.

  • wa: second order term of the dark energy equation of state.

Power Spectrum Normalization

The power spectrum normalization is given either as A_s (i.e., the primordial amplitude) or as sigma8 (i.e., a measure of the amplitude today). Note that not all transfer functions support specifying a primordial amplitude.

  • sigma8: the normalization of the power spectrum today, given by the RMS variance in spheres of 8 \({\rm Mpc}/h\).

  • A_s: the primordial normalization of the power spectrum at \(k_p=0.05\,{\rm Mpc}^{-1}\).

Relativistic Species

  • Neff: effective number of massless+massive neutrinos present at recombination.

  • m_nu: the total mass of massive neutrinos or the masses of the massive neutrinos in eV.

  • mass_split: how to interpret the m_nu argument, see the options below.

  • T_CMB: the temperature of the CMB today.

  • T_ncdm: non-CDM temperature in units of the photon temperature.

Supported Models for Power Spectra and other options

pyccl accepts strings indicating which model to use for various physical quantities (e.g., the transfer function). The various options are as follows.

transfer_function options

  • None : do not compute a linear power spectrum.

  • ‘eisenstein_hu’: the Eisenstein and Hu (1998) fitting function.

  • ‘bbks’: the BBKS approximation.

  • ‘boltzmann_class’: use CLASS to compute the transfer function.

  • ‘boltzmann_camb’: use CAMB to compute the transfer function (default).

  • ‘boltzmann_isitgr’: use ISiTGR to compute the transfer function.

  • An EmulatorPk object.

matter_power_spectrum options

  • ‘halofit’: use HALOFIT (default).

  • ‘linear’: neglect non-linear power spectrum contributions.

  • An EmulatorPk object.

baryonic_effects: a Baryons object.

mg_parametrization: a ModifiedGravity object.

mass_split options

This parameter specifies the model for massive neutrinos.

  • ‘list’: specify each mass yourself in eV

  • ‘normal’: use the normal hierarchy to convert total mass to individual masses (default)

  • ‘inverted’: use the inverted hierarchy to convert total mass to individual masses

  • ‘equal’: assume equal masses when converting the total mass to individual masses

The Calculator Mode

Although pyccl aspires to support a wide variety of models, there will always be more models out there (e.g. specific modified-gravity models, different prescriptions for baryonic effects, etc.), which are not implemented, but which you might like to use, together with CCL in order to calculate specific observables (e.g. weak lensing power spectra). To enable this, CCL can create cosmologies in “calculator mode”.

CosmologyCalculator objects are versions of the standard Cosmology class that can be constructed from building blocks calculated by external libraries. The core building blocks are the distance-redshift relation \(\chi(z)\), the expansion history \(H(z)\), the growth factor and growth rate \(D(z)\), \(f(z)\), the linear matter power spectrum, and the non-linear matter power spectrum. CCL can then use these building blocks to construct observer-level predictions (angular power spectra, cluster counts, correlation functions, etc.). Power spectra can be generated and passed to the CosmologyCalculator in the form of Pk2D objects.

Controlling Splines and Numerical Accuracy

The internal splines and integration accuracy are controlled by the global instances pyccl.spline_params and pyccl.gsl_params. Upon instantiation, the Cosmology object assumes the accuracy parameters from these instances. For example, you can set the generic relative accuracy for integration by executing pyccl.gsl_params["INTEGRATION_EPSREL"] = 1e-5. To reset the accuracy parameters to their default valus listed in src/ccl_core.c, you may run pyccl.gsl_params.reload() or pyccl.spline_params.reload().

The internal splines are controlled by the following parameters.

  • A_SPLINE_NLOG: the number of logarithmically spaced bins between A_SPLINE_MINLOG and A_SPLINE_MIN.

  • A_SPLINE_NA: the number of linearly spaced bins between A_SPLINE_MIN and A_SPLINE_MAX.

  • A_SPLINE_MINLOG: the minimum value of the scale factor splines used for distances, etc.

  • A_SPLINE_MIN: the transition scale factor between logarithmically spaced spline points and linearly spaced spline points.

  • A_SPLINE_MAX: the the maximum value of the scale factor splines used for distances, etc.

  • LOGM_SPLINE_NM: the number of logarithmically spaced values in mass for splines used in the computation of the halo mass function.

  • LOGM_SPLINE_MIN: the base-10 logarithm of the minimum halo mass for splines used in the computation of the halo mass function.

  • LOGM_SPLINE_MAX: the base-10 logarithm of the maximum halo mass for splines used in the computation of the halo mass function.

  • LOGM_SPLINE_DELTA: the step in base-10 logarithmic units for computing finite difference derivatives in the computation of the mass function.

  • A_SPLINE_NLOG_PK: the number of logarithmically spaced bins between A_SPLINE_MINLOG_PK and A_SPLINE_MIN_PK.

  • A_SPLINE_NA_PK: the number of linearly spaced bins between A_SPLINE_MIN_PK and A_SPLINE_MAX.

  • A_SPLINE_MINLOG_PK: the minimum value of the scale factor used for the power spectrum splines.

  • A_SPLINE_MIN_PK: the transition scale factor between logarithmically spaced spline points and linearly spaced spline points for the power spectrum.

  • K_MIN: the minimum wavenumber for the power spectrum splines for analytic models (e.g., BBKS, Eisenstein & Hu, etc.).

  • K_MAX: the maximum wavenumber for the power spectrum splines for analytic models (e.g., BBKS, Eisenstein & Hu, etc.).

  • K_MAX_SPLINE: the maximum wavenumber for the power spectrum splines for numerical models (e.g., CLASS).

  • N_K: the number of spline nodes per decade for the power spectrum splines.

  • N_K_3DCOR: the number of spline points in wavenumber per decade used for computing the 3D correlation function.

  • ELL_MIN_CORR: the minimum value of the spline in angular wavenumber for correlation function computations with FFTlog.

  • ELL_MAX_CORR: the maximum value of the spline in angular wavenumber for correlation function computations with FFTlog.

  • N_ELL_CORR: the number of logarithmically spaced bins in angular wavenumber between ELL_MIN_CORR and ELL_MAX_CORR.

The numerical accuracy of GSL computations is controlled by the following parameters.

  • N_ITERATION: the size of the GSL workspace for numerical integration.

  • INTEGRATION_GAUSS_KRONROD_POINTS: the Gauss-Kronrod quadrature rule used for adaptive integrations.

  • INTEGRATION_EPSREL: the relative error tolerance for numerical integration; used if not specified by a more specific parameter.

  • INTEGRATION_LIMBER_GAUSS_KRONROD_POINTS: the Gauss-Kronrod quadrature rule used for adaptive integrations on subintervals for Limber integrals.

  • INTEGRATION_LIMBER_EPSREL: the relative error tolerance for numerical integration of Limber integrals.

  • INTEGRATION_DISTANCE_EPSREL: the relative error tolerance for numerical integration of distance integrals.

  • INTEGRATION_SIGMAR_EPSREL: the relative error tolerance for numerical integration of power spectrum variance intrgals for the mass function.

  • ROOT_EPSREL: the relative error tolerance for root finding used to invert the relationship between comoving distance and scale factor.

  • ROOT_N_ITERATION: the maximum number of iterations used to for root finding to invert the relationship between comoving distance and scale factor.

  • ODE_GROWTH_EPSREL: the relative error tolerance for integrating the linear growth ODEs.

  • EPS_SCALEFAC_GROWTH: 10x the starting step size for integrating the linear growth ODEs and the scale factor of the initial condition for the linear growth ODEs.

  • NZ_NORM_SPLINE_INTEGRATION: Use spline integration for the normalization of the n(z).

  • LENSING_KERNEL_SPLINE_INTEGRATION: Use spline integration for the lensing kernel integral.

Specifying Physical Constants

The values of physical constants are set globally and are frozen. We do not recommend changing them, as some constants derive from others (such as Newton’s gravitational constant and the solar mass). However, if you know what you are doing, you can unfreeze with pyccl.physical_constants.unfreeze() and then set your desired value to the parameter you would like to change. The following constants are defined and their default values are located in src/ccl_core.c. Note that the neutrino mass splittings are taken from Lesgourgues & Pastor (2012). Also, see the CCL note for a discussion of the values of these constants from different sources.

basic physical constants

  • CLIGHT_HMPC: speed of light divided by \(H_0\) in units of \({\rm Mpc}/h\).

  • GNEWT: Newton’s gravitational constant in units of \({\rm m}^3{\rm kg}^{-1}{\rm s}^{-2}\).

  • SOLAR_MASS: solar mass in units of \({\rm kg}\).

  • MPC_TO_METER: conversion factor for Mpc to meters.

  • RHO_CRITICAL: critical density in units of \(M_\odot/h/({\rm Mpc}/h)^3\).

  • KBOLTZ: Boltzmann constant in units of J/K.

  • STBOLTZ: Stefan-Boltzmann constant in units of \({\rm kg}/{\rm s}^3 / {\rm K}^4\).

  • HPLANCK: Planck’s constant in units \({\rm kg}\,{\rm m}^2 {\rm s}^{-1}\).

  • CLIGHT: speed of light in m/s.

  • EV_IN_J: conversion factor between electron volts and Joules.

neutrino mass splittings

  • DELTAM12_sq: squared mass difference between eigenstates 2 and 1.

  • DELTAM13_sq_pos: squared mass difference between eigenstates 3 and 1 for the normal hierarchy.

  • DELTAM13_sq_neg: squared mass difference between eigenstates 3 and 1 for the inverted hierarchy.