Cosmological Calculations (astropy.cosmology)¶

Introduction¶

The astropy.cosmology sub-package 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.

astropy.cosmology.units extends the astropy.units sub-package, adding and collecting cosmological units and equivalencies, like $$h$$ for keeping track of (dimensionless) factors of the Hubble constant.

For details on reading and writing cosmologies from files, see Read, Write, and Convert Cosmology Objects.

For notes on building custom Cosmology classes and interfacing astropy.cosmology with 3rd-party packages, see Cosmology For Developers.

Getting Started¶

Cosmological quantities are calculated using methods of a Cosmology object.

Examples¶

To calculate the Hubble constant at z=0 (i.e., H0) and the number of transverse proper kiloparsecs (kpc) corresponding to an arcminute at z=3:

>>> from astropy.cosmology import WMAP9 as cosmo
>>> cosmo.H(0)
<Quantity 69.32 km / (Mpc s)>

>>> cosmo.kpc_proper_per_arcmin(3)
<Quantity 472.97709620405266 kpc / arcmin>


Here WMAP9 is a built-in object describing a cosmology with the parameters from the nine-year WMAP results. Several other built-in cosmologies are also available (see Built-in Cosmologies). The available methods of the cosmology object are listed in the methods summary for the FLRW class.

All of these methods also accept an arbitrarily-shaped array of redshifts as input:

>>> import numpy as np
>>> from astropy.cosmology import WMAP9 as cosmo
>>> cosmo.comoving_distance(np.array([0.5, 1.0, 1.5]))
<Quantity [1916.06941724, 3363.07062107, 4451.7475201 ] Mpc>


You can create your own FLRW-like cosmology using one of the cosmology classes:

>>> from astropy.cosmology import FlatLambdaCDM
>>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3, Tcmb0=2.725)
>>> cosmo
FlatLambdaCDM(H0=70.0 km / (Mpc s), Om0=0.3, Tcmb0=2.725 K,
Neff=3.04, m_nu=[0. 0. 0.] eV, Ob0=None)


Note the presence of additional cosmological parameters (e.g., Neff, the number of effective neutrino species) with default values; these can also be specified explicitly in the call to the constructor.

The cosmology sub-package makes use of units, so in many cases returns values with units attached. Consult the documentation for that sub-package for more details, but briefly here we will show how to access the floating point or array values:

>>> from astropy.cosmology import WMAP9 as cosmo
>>> H0 = cosmo.H(0)
>>> H0.value, H0.unit
(69.32, Unit("km / (Mpc s)"))


Using astropy.cosmology¶

More detailed information on using the package is provided on separate pages, listed below.

Most of the functionality is enabled by the FLRW object. This represents a homogeneous and isotropic 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 cannot work with this class directly, as you must specify a dark energy model by using one of its subclasses instead, such as FlatLambdaCDM.

Examples¶

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
FlatLambdaCDM(H0=70.0 km / (Mpc s), Om0=0.3, Tcmb0=0.0 K,
Neff=3.04, m_nu=None, Ob0=None)


This can also be done more explicitly using units, which is recommended:

>>> from astropy.cosmology import FlatLambdaCDM
>>> import astropy.units as u
>>> cosmo = FlatLambdaCDM(H0=70 * u.km / u.s / u.Mpc, Tcmb0=2.725 * u.K, Om0=0.3)


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

>>> cosmo.luminosity_distance(4)
<Quantity 35842.353618623194 Mpc>


Or the age of the universe at z = 0:

>>> cosmo.age(0)
<Quantity 13.461701658024014 Gyr>


They also accept arrays of redshifts:

>>> import astropy.cosmology.units as cu
>>> cosmo.age([0.5, 1, 1.5] * cu.redshift)
<Quantity [8.42128013, 5.74698021, 4.19645373] Gyr>


See the FLRW and FlatLambdaCDM object docstring for all of the methods and attributes available.

In addition to flat universes, non-flat varieties are supported, such as LambdaCDM. A variety of standard cosmologies with the parameters already defined are also available (see Built-in Cosmologies)

>>> from astropy.cosmology import WMAP7   # WMAP 7-year cosmology
>>> WMAP7.critical_density(0)  # critical density at z = 0
<Quantity 9.31000324385361e-30 g / cm3>


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

>>> import numpy as np
>>> from astropy.cosmology import WMAP7   # WMAP 7-year cosmology
>>> WMAP7.Om(np.array([0, 1.0, 2.0]))
array([0.272     , 0.74898522, 0.90905234])
>>> WMAP7.Ode(np.array([0., 1.0, 2.0]))
array([0.72791572, 0.2505506 , 0.0901026 ])


Note that these do not quite add up to one, even though WMAP7 assumes a flat universe, because photons and neutrinos are included. Also note that the density parameters are unitless and so are not Quantity objects.

It is possible to specify the baryonic matter density at redshift zero at class instantiation by passing the keyword argument Ob0:

>>> from astropy.cosmology import FlatLambdaCDM
>>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3, Ob0=0.05)
>>> cosmo
FlatLambdaCDM(H0=70.0 km / (Mpc s), Om0=0.3, Tcmb0=0.0 K,
Neff=3.04, m_nu=None, Ob0=0.05)


In this case the dark matter-only density at redshift 0 is available as class attribute Odm0 and the redshift evolution of dark and baryonic matter densities can be computed using the methods Odm and Ob, respectively. If Ob0 is not specified at class instantiation, it defaults to None and any method relying on it being specified will raise a ValueError:

>>> from astropy.cosmology import FlatLambdaCDM
>>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3)
>>> cosmo.Odm(1)
Traceback (most recent call last):
...
ValueError: Baryonic density not set for this cosmology, unclear
meaning of dark matter density


Cosmological instances have an optional name attribute which can be used to describe the cosmology:

>>> from astropy.cosmology import FlatwCDM
>>> cosmo = FlatwCDM(name='SNLS3+WMAP7', H0=71.58, Om0=0.262, w0=-1.016)
>>> cosmo
FlatwCDM(name="SNLS3+WMAP7", H0=71.58 km / (Mpc s), Om0=0.262,
w0=-1.016, Tcmb0=0.0 K, Neff=3.04, m_nu=None, Ob0=None)


This is also an example with a different model for dark energy: a flat universe with a constant dark energy equation of state, but not necessarily a cosmological constant. A variety of additional dark energy models are also supported (see Specifying a dark energy model).

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. To make this more convenient, a clone() operation is provided, which allows you to make a copy with specified values changed. Note that you cannot change the type of cosmology with this operation (e.g., flat to non-flat).

To make a copy of a cosmological instance using the clone operation:

>>> from astropy.cosmology import WMAP9
>>> newcosmo = WMAP9.clone(name='WMAP9 modified', Om0=0.3141)
>>> WMAP9.H0, newcosmo.H0  # some values unchanged
(<Quantity 69.32 km / (Mpc s)>, <Quantity 69.32 km / (Mpc s)>)
>>> WMAP9.Om0, newcosmo.Om0  # some changed
(0.2865, 0.3141)
>>> WMAP9.Ode0, newcosmo.Ode0  # Indirectly changed since this is flat
(0.7134130719051658, 0.6858130719051657)


Finding the Redshift at a Given Value of a Cosmological Quantity¶

If you know a cosmological quantity and you want to know the redshift which it corresponds to, you can use z_at_value().

Example¶

To find the redshift using z_at_value:

>>> import astropy.units as u
>>> from astropy.cosmology import Planck13, z_at_value
>>> z_at_value(Planck13.age, 2 * u.Gyr)
<Quantity 3.19812061 redshift>


For some quantities, there can be more than one redshift that satisfies a value. In this case you can use the zmin and zmax keywords to restrict the search range or set bracket to initialize it in the desired domain. See the z_at_value() docstring for more detailed usage examples.

Built-in Cosmologies¶

A number of preloaded cosmologies are available from analyses using the WMAP and Planck satellite data. For example:

>>> from astropy.cosmology import Planck13  # Planck 2013
>>> Planck13.lookback_time(2)  # lookback time in Gyr at z=2
<Quantity 10.51184138 Gyr>


A full list of the predefined cosmologies is given by cosmology.realizations.available and summarized below:

Name

Source

H0

Om

Flat

WMAP1

Spergel et al. 2003

72.0

0.257

Yes

WMAP3

Spergel et al. 2007

70.1

0.276

Yes

WMAP5

Komatsu et al. 2009

70.2

0.277

Yes

WMAP7

Komatsu et al. 2011

70.4

0.272

Yes

WMAP9

Hinshaw et al. 2013

69.3

0.287

Yes

Planck13

Planck Collab 2013, Paper XVI

67.8

0.307

Yes

Planck15

Planck Collab 2015, Paper XIII

67.7

0.307

Yes

Planck18

Planck Collab 2018, Paper VI

67.7

0.310

Yes

Note

Unlike the Planck 2015 paper, the Planck 2018 paper includes massive neutrinos in Om0 but the Planck18 object includes them in m_nu instead for consistency. Hence, the Om0 value in Planck18 differs slightly from the Planck 2018 paper but represents the same cosmological model.

Currently, all are instances of FlatLambdaCDM. More details about exactly where each set of parameters comes from are available in the docstring for each object:

>>> from astropy.cosmology import WMAP7
>>> print(WMAP7.__doc__)
WMAP7 instance of FlatLambdaCDM cosmology
(from Komatsu et al. 2011, ApJS, 192, 18, doi: 10.1088/0067-0049/192/2/18.
Table 1 (WMAP + BAO + H0 ML).)


Specifying a Dark Energy Model¶

Along with the standard FlatLambdaCDM model described above, a number of additional dark energy models are provided. FlatLambdaCDM and LambdaCDM assume that dark energy is a cosmological constant, and should be the most commonly used cases; the former assumes a flat universe, the latter allows for spatial curvature. FlatwCDM and wCDM assume 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 subclassing FLRW. See the provided subclasses for examples. It is advisable to stick to subclassing FLRW rather than one of its subclasses, since some of them use internal optimizations that also need to be propagated to any subclasses. Users wishing to use similar tricks (which can make distance calculations much faster) should consult the cosmology module source code for details.

Photons and Neutrinos¶

The cosmology classes (can) include the contribution to the energy density from both photons and neutrinos. By default, the latter are assumed massless. The three parameters controlling the properties of these species, which are arguments to the initializers of all of the cosmological classes, are Tcmb0 (the temperature of the cosmic microwave background at z=0), Neff (the effective number of neutrino species), and m_nu (the rest mass of the neutrino species). Tcmb0 and m_nu should be expressed as unit Quantities. All three have standard default values — 0 K, 3.04, and 0 eV, respectively. (The reason that Neff is not 3 has to do primarily with a small bump in the neutrino energy spectrum due to electron-positron annihilation, but is also affected by weak interaction physics.) Setting the CMB temperature to 0 removes the contribution of both neutrinos and photons. This is the default to ensure these components are excluded unless the user explicitly requests them.

Massive neutrinos are treated using the approach described in the WMAP seven-year cosmology paper (Komatsu et al. 2011, ApJS, 192, 18, section 3.3). This is not the simple $$\Omega_{\nu 0} h^2 = \sum_i m_{\nu\, i} / 93.04\,\mathrm{eV}$$ approximation. Also note that the values of $$\Omega_{\nu}(z)$$ include both the kinetic energy and the rest mass energy components, and that the Planck13 and Planck15 cosmologies include a single species of neutrinos with non-zero mass (which is not included in $$\Omega_{m0}$$).

Adding massive neutrinos can have significant performance implications. In particular, the computation of distance measures and lookback times are factors of three to four times slower than in the massless neutrino case. Therefore, if you need to compute many distances in such a cosmology and performance is critical, it is particularly useful to calculate them on a grid and use interpolation.

Examples¶

The contribution of photons and neutrinos to the total mass-energy density can be found as a function of redshift:

>>> from astropy.cosmology import WMAP7   # WMAP 7-year cosmology
>>> WMAP7.Ogamma0, WMAP7.Onu0  # Current epoch values
(4.985694972799396e-05, 3.442154948307989e-05)
>>> z = np.array([0, 1.0, 2.0])
>>> WMAP7.Ogamma(z), WMAP7.Onu(z)
(array([4.98603986e-05, 2.74593395e-04, 4.99915942e-04]),
array([3.44239306e-05, 1.89580995e-04, 3.45145089e-04]))


If you want to exclude photons and neutrinos from your calculations, you can set Tcmb0 to 0 (which is also the default):

>>> from astropy.cosmology import FlatLambdaCDM
>>> import astropy.units as u
>>> cos = FlatLambdaCDM(70.4 * u.km / u.s / u.Mpc, 0.272, Tcmb0 = 0.0 * u.K)
>>> cos.Ogamma0, cos.Onu0
(0.0, 0.0)


You can include photons but exclude any contributions from neutrinos by setting Tcmb0 to be non-zero (2.725 K is the standard value for our Universe) but setting Neff to 0:

>>> from astropy.cosmology import FlatLambdaCDM
>>> cos = FlatLambdaCDM(70.4, 0.272, Tcmb0=2.725, Neff=0)
>>> cos.Ogamma(np.array([0, 1, 2]))  # Photons are still present
array([4.98603986e-05, 2.74642208e-04, 5.00086413e-04])
>>> cos.Onu(np.array([0, 1, 2]))  # But not neutrinos
array([0., 0., 0.])


The number of neutrino species is assumed to be the floor of Neff, which in the default case is Neff=3. Therefore, if non-zero neutrino masses are desired, then three masses should be provided. However, if only one value is provided, all of the species are assumed to have the same mass. Neff is assumed to be shared equally between each species.

>>> from astropy.cosmology import FlatLambdaCDM
>>> import astropy.units as u
>>> H0 = 70.4 * u.km / u.s / u.Mpc
>>> m_nu = 0 * u.eV
>>> cosmo = FlatLambdaCDM(H0, 0.272, Tcmb0=2.725, m_nu=m_nu)
>>> cosmo.has_massive_nu
False
>>> cosmo.m_nu
<Quantity [0., 0., 0.] eV>
>>> m_nu = [0.0, 0.05, 0.10] * u.eV
>>> cosmo = FlatLambdaCDM(H0, 0.272, Tcmb0=2.725, m_nu=m_nu)
>>> cosmo.has_massive_nu
True
>>> cosmo.m_nu
<Quantity [0.  , 0.05, 0.1 ] eV>
>>> cosmo.Onu(np.array([0, 1.0, 15.0]))
array([0.00327011, 0.00896845, 0.01257946])
>>> cosmo.Onu(1) * cosmo.critical_density(1)
<Quantity 2.444380380370406e-31 g / cm3>


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

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 many 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!

A more difficult question is the range of redshifts over which the code is expected to return valid results. This is necessarily model-dependent, but in general you should not expect the numeric results to be well behaved for redshifts more than a few times larger than the epoch of matter-radiation equality (so, for typical models, not above z = 5-6,000, but for some models much lower redshifts may be ill-behaved). In particular, you should pay attention to warnings from the scipy.integrate package about integrals failing to converge (which may only be issued once per session).

The built-in cosmologies use the parameters as listed in the respective papers. These provide only a limited range of precision, and so you should not expect derived quantities to match beyond that precision. For example, the Planck 2013 and 2015 results only provide the Hubble constant to four digits. Therefore, they should not be expected to match the age quoted by the Planck team to better than that, despite the fact that five digits are quoted in the papers.

Reference/API¶

More detailed information on using the package is provided on separate pages, listed below.

astropy.cosmology Package¶

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¶

 z_at_value(func, fval[, zmin, zmax, ztol, …]) Find the redshift z at which func(z) = fval.

Classes¶

 Cosmology([name, meta]) Base-class for all Cosmologies. CosmologyError Mixin class for flat cosmologies. FLRW(H0, Om0, Ode0[, Tcmb0, Neff, m_nu, …]) A class describing an isotropic and homogeneous (Friedmann-Lemaitre-Robertson-Walker) cosmology. FlatFLRWMixin(*args, **kw) Mixin class for flat FLRW cosmologies. LambdaCDM(H0, Om0, Ode0[, Tcmb0, Neff, …]) FLRW cosmology with a cosmological constant and curvature. FlatLambdaCDM(H0, Om0[, Tcmb0, Neff, m_nu, …]) FLRW cosmology with a cosmological constant and no curvature. wCDM(H0, Om0, Ode0[, w0, Tcmb0, Neff, m_nu, …]) FLRW cosmology with a constant dark energy equation of state and curvature. FlatwCDM(H0, Om0[, w0, Tcmb0, Neff, m_nu, …]) FLRW cosmology with a constant dark energy equation of state and no spatial curvature. w0waCDM(H0, Om0, Ode0[, w0, wa, Tcmb0, …]) FLRW cosmology with a CPL dark energy equation of state and curvature. Flatw0waCDM(H0, Om0[, w0, wa, Tcmb0, Neff, …]) FLRW cosmology with a CPL dark energy equation of state and no curvature. wpwaCDM(H0, Om0, Ode0[, wp, wa, zp, Tcmb0, …]) FLRW cosmology with a CPL dark energy equation of state, a pivot redshift, and curvature. w0wzCDM(H0, Om0, Ode0[, w0, wz, Tcmb0, …]) FLRW cosmology with a variable dark energy equation of state and curvature. The default cosmology to use. Parameter(*[, derived, unit, equivalencies, …]) Cosmological parameter (descriptor).