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- Cosmological Calculations (`astropy.cosmology`)

# 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

Cosmological quantities are calculated using methods of a `Cosmology` object. For example, to calculate 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.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 9-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. If you’re using IPython you can also use tab completion to print a list of the available methods. To do this, after importing the cosmology as in the above example, type `cosmo.` at the IPython prompt and then press the tab key.

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

    >>> from astropy.cosmology import WMAP9 as cosmo
    >>> cosmo.comoving_distance([0.5, 1.0, 1.5])
    <Quantity [ 1916.0694236 , 3363.07064333, 4451.74756242] 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)
    >>> cosmo
    FlatLambdaCDM(H0=70 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 subpackage makes use of `units`, so in many cases returns values with units attached. Consult the documentation for that subpackage for more details, but briefly, 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`

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 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
    FlatLambdaCDM(H0=70 km / (Mpc s), Om0=0.3, Tcmb0=2.725 K,
                  Neff=3.04, m_nu=[ 0.  0.  0.] eV, 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, Om0=0.3)

However, most of the parameters that accept units (`H0`, `Tcmb0`) have default units, so unit quantities do not have to be used. The exception are neutrino masses, where you must supply a units if you want massive neutrinos.

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 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:

    >>> cosmo.age([0.5, 1, 1.5]).value
    array([ 8.42128047,  5.74698053,  4.19645402])

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 (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:

    >>> 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     ,  0.74898524,  0.90905239]),
     array([ 0.72791572,  0.25055061,  0.0901026 ]))

Note that these don’t quite add up to one even though WMAP7 assumes a flat Universe because photons and neutrinos are included. Also note that they 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 km / (Mpc s), Om0=0.3, Tcmb0=2.725 K,
                  Neff=3.04, m_nu=[ 0.  0.  0.] eV, Ob0=0.05)

In this case the dark matter only density at redshift zero 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=71.6 km / (Mpc s), Om0=0.262,
             w0=-1.02, Tcmb0=2.725 K, Neff=3.04, m_nu=[ 0.  0.  0.] eV,
             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.

A 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 can’t change the type of cosmology with this operation (e.g., flat to non-flat). For example:

    >>> from astropy.cosmology import WMAP9
    >>> newcosmo = WMAP9.clone(name='WMAP9 modified', Om0=0.3141)
    >>> WMAP9.H0, newcosmo.H0  # some values unchanged
    (<Quantity 69.3... km / (Mpc s)>, <Quantity 69.3... km / (Mpc s)>)
    >>> WMAP9.Om0, newcosmo.Om0  # some changed
    (0.286..., 0.314...)
    >>> WMAP9.Ode0, newcosmo.Ode0  # Indirectly changed since this is flat
    (0.713..., 0.685...)

### 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`:

    >>> import astropy.units as u
    >>> from astropy.cosmology import Planck13, z_at_value
    >>> z_at_value(Planck13.age, 2 * u.Gyr)
    3.1981226843560968

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. See the `z_at_value` docstring for more detailed usage examples.

### Built-in Cosmologies

A number of pre-loaded 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 10.511841788576083 Gyr>

A full list of the pre-defined cosmologies is given by `cosmology.parameters.available`, and summarized below:

Name | Source | H0 | Om | Flat
--- | --- | --- | --- | ---
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

Currently, all are instances of `FlatLambdaCDM`. More details about exactly where each set of parameters come 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

In addition to 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 sub-classing `FLRW`. See the provided subclasses for examples. It is recommended, but not required, that all arguments to the constructor of a new subclass be available as properties, since the `clone` method assumes this is the case. It is also 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 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 the cosmological classes, are `Tcmb0` (the temperature of the CMB 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 (2.725 K, 3.04, and 0 eV respectively; the reason that `Neff` is not 3 primarily has to do with a small bump in the neutrino energy spectrum due to electron-positron annihilation, but is also affected by weak interaction physics).

Massive neutrinos are treated using the approach described in the WMAP 7-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 includes 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 3-4 slower than in the massless neutrino case. Therefore, if you need to compute a lot of distances in such a cosmology and performance is critical, it is particularly useful to calculate them on a grid and use interpolation.

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 = [0, 1.0, 2.0]
    >>> WMAP7.Ogamma(z), WMAP7.Onu(z)
    (array([  4.98569497e-05,   2.74574409e-04,   4.99881391e-04]),
     array([  3.44215495e-05,   1.89567887e-04,   3.45121234e-04]))

If you want to exclude photons and neutrinos from your calculations, simply set `Tcmb0` to 0:

    >>> 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)

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

    >>> from astropy.cosmology import FlatLambdaCDM
    >>> cos = FlatLambdaCDM(70.4, 0.272, Neff=0)
    >>> cos.Ogamma([0, 1, 2])  # Photons are still present
    array([  4.98569497e-05,   2.74623215e-04,   5.00051839e-04])
    >>> cos.Onu([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 3. Therefore, if non-zero neutrino masses are desired, then 3 masses should be provided. However, if only one value is provided, all 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, 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, m_nu=m_nu)
    >>> cosmo.has_massive_nu
    True
    >>> cosmo.m_nu
    <Quantity [ 0.  , 0.05, 0.1 ] eV>
    >>> cosmo.Onu([0, 1.0, 15.0])
    array([ 0.00326988,  0.00896783,  0.0125786 ])
    >>> 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.

## For Developers: Using `astropy.cosmology` inside Astropy

If you are writing code for the Astropy core or an affiliated package, it’s often useful to assume a default cosmology, so that the exact cosmology doesn’t have to be specified every time a function or method is called. In this case it’s possible to specify a “default” cosmology.

You can set the default 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` function of `default_cosmology` to set a cosmology for the current Python session. If you haven’t set a default cosmology using one of the methods described above, then the cosmology module will default to using the 9-year WMAP parameters.

It is strongly recommended that you use the default cosmology through the `default_cosmology` science state object. An override option can then be provided using something like the following:

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

        if cosmo is None:
            cosmo = default_cosmology.get()

        ... your code here ...

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

Note

In general it’s better to use an explicit cosmology (for example `WMAP9.H(0)` instead of `cosmology.default_cosmology.get().H(0)`). Use of the default cosmology should generally be reserved for code that will be included in the Astropy core or an affiliated package.

## See Also

- Hogg, “Distance measures in cosmology”, http://arxiv.org/abs/astroph/9905116
- Linder, “Exploring the Expansion History of the Universe”, http://arxiv.org/abs/astro-ph/0208512
- NASA’s Legacy Archive for Microwave Background Data Analysis, http://lambda.gsfc.nasa.gov/

## 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!

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 4 digits. Therefore, they shouldn’t be expected to match the age quoted by the Planck team to better than that, despite the fact that 5 digits are quoted in the papers.

## Reference/API

### 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

`FLRW`(H0, Om0, Ode0[, Tcmb0, Neff, m_nu, ...]) | A class describing an isotropic and homogeneous (Friedmann-Lemaitre-Robertson-Walker) cosmology.
`FlatLambdaCDM`(H0, Om0[, Tcmb0, Neff, m_nu, ...]) | FLRW cosmology with a cosmological constant and no curvature.
`Flatw0waCDM`(H0, Om0[, w0, wa, Tcmb0, Neff, ...]) | FLRW cosmology with a CPL dark energy equation of state and no curvature.
`FlatwCDM`(H0, Om0[, w0, Tcmb0, Neff, m_nu, ...]) | FLRW cosmology with a constant dark energy equation of state and no spatial curvature.
`LambdaCDM`(H0, Om0, Ode0[, Tcmb0, Neff, ...]) | FLRW cosmology with a cosmological constant and curvature.
`default_cosmology`() | The default cosmology to use.
`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, m_nu, ...]) | 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