Cosmological Calculations (astropy.cosmology)

Introduction

The astropy.cosmology subpackage contains classes for representing cosmologies, and utility functions for calculating commonly used quantities that depend on a cosmological model. This includes distances, ages and lookback times corresponding to a measured redshift or the transverse separation corresponding to a measured angular separation.

Getting Started

There are many functions available to calculate cosmological quantities. They generally take a redshift as input. For example, the two cases below give you the value of the hubble constant at z=0 (i.e., H0), and the number of transverse proper kpc corresponding to an arcminute at z=3:

>>> from astropy import cosmology
>>> cosmology.H(0)
70.4
>>> cosmology.kpc_proper_per_arcmin(3)
472.8071851564037

All the functions available are listed in the Reference/API section. These will use the “current” cosmology to calculate the values (see The Current Cosmology section below for more details). If you haven’t set this explicitly, they will use the 7-year WMAP cosmological parameters and print a warning message.

There are also several standard cosmologies already defined. These are objects with methods and attributes that calculate cosmological values. For example, the comoving distance in Mpc to redshift 4 using the 5-year WMAP parameters:

>>> from astropy.cosmology import WMAP5
>>> WMAP5.comoving_distance(4)
7329.328120760829

A full list of the pre-defined cosmologies is given by cosmology.parameters.available.

An important point is that the cosmological parameters of each instance are immutable – that is, if you want to change, say, Om, you need to make a new instance of the class.

Using cosmology

Most of the functionality is enabled by the FLRW object. This represents a homogenous and isotropic cosmology (a cosmology characterized by the Friedmann-Lemaitre-Robertson-Walker metric, named after the people who solved Einstein’s field equation for this special case). However, you can’t work with this class directly, as you must specify a dark energy model by using one of its subclasses instead, such as FlatLambdaCDM.

You can create a new FlatLambdaCDM object with arguments giving the hubble parameter and omega matter (both at z=0):

>>> from astropy.cosmology import FlatLambdaCDM
>>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3)
>>> cosmo
LambdaCDM(H0=70, Om0=0.3, Ode0=0.7)

A number of additional dark energy models are provided (described below). Note that photons and neutrinos are included in these models, so Om0 + Ode0 is not quite one.

The pre-defined cosmologies described in the Getting Started section are instances of FlatLambdaCDM, and have the same methods. So we can find the luminosity distance in Mpc to redshift 4 by:

>>> cosmo.luminosity_distance(4)
35842.35374316948

or the age of the universe at z = 0 in Gyr:

>>> cosmo.age(0)
13.461701807287566

They also accept arrays of redshifts:

>>> cosmo.age([0.5, 1, 1.5])
array([ 8.42128059,  5.74698062,  4.1964541 ])

See the FLRW and FlatLambdaCDM object docstring for all the methods and attributes available. In addition to flat Universes, non-flat varieties are supported such as LambdaCDM. There are also a variety of standard cosmologies with the parameters already defined:

>>> from astropy.cosmology import WMAP7   # WMAP 7-year cosmology
>>> WMAP7.critical_density(0)       # critical density at z = 0 in g/cm^3
9.31000313202047e-30
>>> from astropy.cosmology import WMAP5   # WMAP 5-year
>>> WMAP5.H(3)                    # Hubble parameter at z = 3 in km/s/Mpc
301.71804314602889

You can see how the density parameters evolve with redshift as well

>>> from astropy.cosmology import WMAP7   # WMAP 7-year cosmology
>>> WMAP7.Om([0,1.0,2.0]), WMAP7.Ode([0.,1.0,2.0])
(array([ 0.272     ,  0.74898525,  0.9090524 ]),
 array([ 0.72791572,  0.25055062,  0.09010261]))

Note that these don’t quite add up to one even though WMAP7 assumes a flat Universe because photons and neutrinos are included.

In addition to the LambdaCDM object, there are convenience functions that calculate some of these quantities without needing to explicitly give a cosmology - but there are more methods available if you work directly with the cosmology object.

>>> from astropy import cosmology
>>> cosmology.kpc_proper_per_arcmin(3)
472.8071851564037
>>> cosmology.arcsec_per_kpc_proper(3)
0.12690162477152736

These functions will perform calculations using the “current” cosmology. This is a specific cosmology that is currently active in astropy and it’s described further in the following section. They can also be explicitly given a cosmology using the cosmo keyword argument. A full list of convenience functions is included below, in the Reference/API section.

The Current Cosmology

Sometimes it’s useful for Astropy functions to assume a default cosmology so that the desired cosmology doesn’t have to be specified every time the function is called – the convenience functions described in the previous section are one example. For these cases it’s possible to specify a “current” cosmology.

You can set the current cosmology to a pre-defined value by using the “default_cosmology” option in the [cosmology.core] section of the configuration file (see Configuration system (astropy.config)). Alternatively, you can use the set_current function to set a cosmology for the current Python session.

If you haven’t set a current cosmology using one of the methods described above, then the cosmology module will use the 7-year WMAP parameters and print a warning message letting you know this. For example, if you call a convenience function without setting the current cosmology or using the cosmo= keyword you see the following message:

>>> from astropy import cosmology
>>> cosmology.lookback_time(1)          # lookback time in Gyr at z=1
WARNING: No default cosmology has been specified, using 7-year WMAP.
[astropy.cosmology.core]
7.787767002228743

The 9-year WMAP cosmology is also available

>>> from astropy.cosmology import WMAP9   # WMAP 9-year
>>> WMAP9.lookback_time(2)                # lookback time in Gyr at z=2
10.444367272683863

Note

In general it’s better to use an explicit cosmology (for example WMAP7.H(0) instead of cosmology.H(0)). The motivation for this is that when you go back to use the code at a later date or share your scripts with someone else, the default cosmology may have changed. Use of the convenience functions should generally be reserved for interactive work or cases where the flexibility of quickly changing between different cosmologies is for some reason useful. Alternatively, putting (for example) cosmology.set_current(WMAP7) at the top of your code will ensure that the right cosmology is always used.

Using cosmology inside Astropy

If you are writing code for the astropy core or an affiliated package, it is strongly recommended that you use the current cosmology through the get_current function. It is also recommended that you provide an override option something like the following:

def myfunc(..., cosmo=None):
    from astropy.cosmology import get_current

    if cosmo is None:
        cosmo = get_current()

    ... your code here ...

This ensures that all code consistently uses the current cosmology unless explicitly overridden.

Specifying a dark energy model

In addition to the standard FlatLambdaCDM model described above, a number of additional dark energy models are provided. FlatLambdaCDM and FlatLambdaCDM assume that dark energy is a cosmological constant, and should be the most commonly used case. wCDM assumes a constant dark energy equation of state parameterized by w_0. Two forms of a variable dark energy equation of state are provided: the simple first order linear expansion w(z) = w_0 + w_z z by w0wzCDM, as well as the common CPL form by w0waCDM: w(z) = w_0 + w_a (1 - a) =
w_0 + w_a z / (1 + z) and its generalization to include a pivot redshift by wpwaCDM: w(z) = w_p + w_a
(a_p - a).

Users can specify their own equation of state by sub-classing FLRW. See the provided subclasses for examples.

Relativistic Species

The cosmology classes include the contribution to the energy density from both photons and massless neutrinos. The two parameters controlling the proporties of these species are Tcmb0 (the temperature of the CMB at z=0) and Neff, the effective number of neutrino species. Both have standard default values (2.725 and 3.04, respectively; the reason that Neff is not 3 has to do with a small bump in the neutrino energy spectrum due to electron-positron annihilation).

>>> from astropy.cosmology import WMAP7   # WMAP 7-year cosmology
>>> z = [0,1.0,2.0]
>>> WMAP7.Ogamma(z), WMAP7.Onu(z)
(array([  4.98569503e-05,   2.74574414e-04]),
 array([  3.44204408e-05,   1.89561782e-04]),
 array([  8.42773911e-05,   4.64136197e-04]))

If you want to exclude photons and neutrinos from your calculations, simply set the CMB Temperature to 0:

>>> from astropy.cosmology import FlatLambdaCDM
>>> cos = FlatLambdaCDM(70.4, 0.272, Tcmb0 = 0.0)
>>> cos.Ogamma0, cos.Onu0
(0.0, 0.0)

Neutrinos can be removed (while leaving photons) by setting Neff=0:

>>> from astropy.cosmology import FlatLambdaCDM
>>> cos = FlatLambdaCDM(70.4, 0.272, Neff=0)
>>> cos.Ogamma([0,1,2]),cos.Onu([0,1,2])
(array([  4.98569503e-05,   2.74623219e-04,   5.00051845e-04]),
 array([ 0.,  0.,  0.]))

While these examples used FlatLambdaCDM, the above examples also apply for all of the other cosmology classes.

See Also

Range of validity and reliability

The code in this sub-package is tested against several widely-used online cosmology calculators, and has been used to perform calculations in refereed papers. You can check the range of redshifts over which the code is regularly tested in the module astropy.cosmology.tests.test_cosmology. If you find any bugs, please let us know by opening an issue at the github repository!

Reference/API

astropy.cosmology Module

astropy.cosmology contains classes and functions for cosmological distance measures and other cosmology-related calculations.

See the Astropy documentation for more detailed usage examples and references.

Functions

H(z[, cosmo]) Hubble parameter (km/s/Mpc) at redshift z.
angular_diameter_distance(z[, cosmo]) Angular diameter distance in Mpc at a given redshift.
arcsec_per_kpc_comoving(z[, cosmo]) Angular separation in arcsec corresponding to a comoving kpc at redshift z.
arcsec_per_kpc_proper(z[, cosmo]) Angular separation in arcsec corresponding to a proper kpc at redshift z.
comoving_distance(z[, cosmo]) Comoving distance in Mpc at redshift z.
critical_density(z[, cosmo]) Critical density in grams per cubic cm at redshift z.
distmod(z[, cosmo]) Distance modulus at redshift z.
get_current() Get the current cosmology.
kpc_comoving_per_arcmin(z[, cosmo]) Separation in transverse comoving kpc corresponding to an arcminute at redshift z.
kpc_proper_per_arcmin(z[, cosmo]) Separation in transverse proper kpc corresponding to an arcminute at redshift z.
lookback_time(z[, cosmo]) Lookback time in Gyr to redshift z.
luminosity_distance(z[, cosmo]) Luminosity distance in Mpc at redshift z.
scale_factor(z[, cosmo]) Scale factor at redshift z.
set_current(cosmo) Set the current cosmology.

Classes

FLRW(H0, Om0, Ode0[, Tcmb0, Neff, name]) A class describing an isotropic and homogeneous (Friedmann-Lemaitre-Robertson-Walker) cosmology.
FlatLambdaCDM(H0, Om0[, Tcmb0, Neff, name]) FLRW cosmology with a cosmological constant and no curvature.
Flatw0waCDM(H0, Om0[, w0, wa, Tcmb0, Neff, name]) FLRW cosmology with a CPL dark energy equation of state and no curvature.
FlatwCDM(H0, Om0[, w0, Tcmb0, Neff, name]) FLRW cosmology with a constant dark energy equation of state and no spatial curvature.
LambdaCDM(H0, Om0, Ode0[, Tcmb0, Neff, name]) FLRW cosmology with a cosmological constant and curvature.
w0waCDM(H0, Om0, Ode0[, w0, wa, Tcmb0, ...]) FLRW cosmology with a CPL dark energy equation of state and curvature.
w0wzCDM(H0, Om0, Ode0[, w0, wz, Tcmb0, ...]) FLRW cosmology with a variable dark energy equation of state and curvature.
wCDM(H0, Om0, Ode0[, w0, Tcmb0, Neff, name]) FLRW cosmology with a constant dark energy equation of state and curvature.
wpwaCDM(H0, Om0, Ode0[, wp, wa, zp, Tcmb0, ...]) FLRW cosmology with a CPL dark energy equation of state, a pivot redshift, and curvature.

Class Inheritance Diagram

Inheritance diagram of astropy.cosmology.core.FLRW, astropy.cosmology.core.FlatLambdaCDM, astropy.cosmology.core.Flatw0waCDM, astropy.cosmology.core.FlatwCDM, astropy.cosmology.core.LambdaCDM, astropy.cosmology.core.w0waCDM, astropy.cosmology.core.w0wzCDM, astropy.cosmology.core.wCDM, astropy.cosmology.core.wpwaCDM