FlatLambdaCDM¶

class astropy.cosmology.FlatLambdaCDM(H0, Om0, Tcmb0=<Quantity 0. K>, Neff=3.04, m_nu=<Quantity 0. eV>, Ob0=None, *, name=None, meta=None)[source]¶

Bases: FlatFLRWMixin, LambdaCDM

FLRW cosmology with a cosmological constant and no curvature.

This has no additional attributes beyond those of FLRW.

Parameters:

H0python:float or scalar astropy:quantity-like [:ref: ‘frequency’]: Hubble constant at z = 0. If a float, must be in [km/sec/Mpc].
Om0python:float: Omega matter: density of non-relativistic matter in units of the critical density at z=0.
Tcmb0python:float or scalar astropy:quantity-like [:ref: ‘temperature’], optional: Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones).
Neffpython:float, optional: Effective number of Neutrino species. Default 3.04.
m_nuastropy:quantity-like [:ref: ‘energy’, :ref: ‘mass’] or numpy:array_like, optional: Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino.
Ob0python:float or python:None, optional: Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception.
namepython:str or python:None (optional, keyword-only): Name for this cosmological object.
metapython:mapping or python:None (optional, keyword-only): Metadata for the cosmology, e.g., a reference.

Examples

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

The comoving distance in Mpc at redshift z:

>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)

To get an equivalent cosmology, but of type astropy.cosmology.LambdaCDM, use astropy.cosmology.FlatFLRWMixin.nonflat.

>>> cosmo.nonflat
LambdaCDM(H0=70.0 km / (Mpc s), Om0=0.3, ...

Attributes Summary

`H0`	Hubble constant as an `Quantity` at z=0.
`Neff`	Number of effective neutrino species.
`Ob0`	Omega baryon; baryonic matter density/critical density at z=0.
`Ode0`	Omega dark energy; dark energy density/critical density at z=0.
`Odm0`	Omega dark matter; dark matter density/critical density at z=0.
`Ogamma0`	Omega gamma; the density/critical density of photons at z=0.
`Ok0`	Omega curvature; the effective curvature density/critical density at z=0.
`Om0`	Omega matter; matter density/critical density at z=0.
`Onu0`	Omega nu; the density/critical density of neutrinos at z=0.
`Otot0`	Omega total; the total density/critical density at z=0.
`Tcmb0`	Temperature of the CMB as `Quantity` at z=0.
`Tnu0`	Temperature of the neutrino background as `Quantity` at z=0.
`critical_density0`	Critical density as `Quantity` at z=0.
`from_format`	Transform object to a `Cosmology`.
`h`	Dimensionless Hubble constant: h = H_0 / 100 [km/sec/Mpc].
`has_massive_nu`	Does this cosmology have at least one massive neutrino species?
`hubble_distance`	Hubble distance as `Quantity`.
`hubble_time`	Hubble time as `Quantity`.
`is_flat`	Return `True`, the cosmology is flat.
`m_nu`	Mass of neutrino species.
`meta`
`name`	The name of the Cosmology instance.
`nonflat`	Return the equivalent non-flat-class instance of this cosmology.
`read`	Read and parse data to a `Cosmology`.
`to_format`	Transform this Cosmology to another format.
`write`	Write this Cosmology object out in the specified format.

Methods Summary

`H`(z)	Hubble parameter (km/s/Mpc) at redshift `z`.
`Ob`(z)	Return the density parameter for baryonic matter at redshift `z`.
`Ode`(z)	Return the density parameter for dark energy at redshift `z`.
`Odm`(z)	Return the density parameter for dark matter at redshift `z`.
`Ogamma`(z)	Return the density parameter for photons at redshift `z`.
`Ok`(z)	Return the equivalent density parameter for curvature at redshift `z`.
`Om`(z)	Return the density parameter for non-relativistic matter at redshift `z`.
`Onu`(z)	Return the density parameter for neutrinos at redshift `z`.
`Otot`(z)	The total density parameter at redshift `z`.
`Tcmb`(z)	Return the CMB temperature at redshift `z`.
`Tnu`(z)	Return the neutrino temperature at redshift `z`.
`abs_distance_integrand`(z)	Integrand of the absorption distance [1].
`absorption_distance`(z, /)	Absorption distance at redshift `z`.
`age`(z)	Age of the universe in Gyr at redshift `z`.
`angular_diameter_distance`(z)	Angular diameter distance in Mpc at a given redshift.
`angular_diameter_distance_z1z2`(z1, z2)	Angular diameter distance between objects at 2 redshifts.
`arcsec_per_kpc_comoving`(z)	Angular separation in arcsec corresponding to a comoving kpc at redshift `z`.
`arcsec_per_kpc_proper`(z)	Angular separation in arcsec corresponding to a proper kpc at redshift `z`.
`clone`(*[, meta, to_nonflat])	Returns a copy of this object with updated parameters, as specified.
`comoving_distance`(z)	Comoving line-of-sight distance in Mpc at a given redshift.
`comoving_transverse_distance`(z)	Comoving transverse distance in Mpc at a given redshift.
`comoving_volume`(z)	Comoving volume in cubic Mpc at redshift `z`.
`critical_density`(z)	Critical density in grams per cubic cm at redshift `z`.
`de_density_scale`(z)	Evaluates the redshift dependence of the dark energy density.
`differential_comoving_volume`(z)	Differential comoving volume at redshift z.
`distmod`(z)	Distance modulus at redshift `z`.
`efunc`(z)	Function used to calculate H(z), the Hubble parameter.
`inv_efunc`(z)	Function used to calculate \(\frac{1}{H_z}\).
`is_equivalent`(other, /, *[, format])	Check equivalence between Cosmologies.
`kpc_comoving_per_arcmin`(z)	Separation in transverse comoving kpc corresponding to an arcminute at redshift `z`.
`kpc_proper_per_arcmin`(z)	Separation in transverse proper kpc corresponding to an arcminute at redshift `z`.
`lookback_distance`(z)	The lookback distance is the light travel time distance to a given redshift.
`lookback_time`(z)	Lookback time in Gyr to redshift `z`.
`lookback_time_integrand`(z)	Integrand of the lookback time (equation 30 of [1]).
`luminosity_distance`(z)	Luminosity distance in Mpc at redshift `z`.
`nu_relative_density`(z)	Neutrino density function relative to the energy density in photons.
`scale_factor`(z)	Scale factor at redshift `z`.
`w`(z)	Returns dark energy equation of state at redshift `z`.

Attributes Documentation

H0¶: Hubble constant as an Quantity at z=0.

Neff¶: Number of effective neutrino species.

Ob0¶: Omega baryon; baryonic matter density/critical density at z=0.

Ode0¶: Omega dark energy; dark energy density/critical density at z=0.

Odm0¶: Omega dark matter; dark matter density/critical density at z=0.

Ogamma0¶: Omega gamma; the density/critical density of photons at z=0.

Ok0¶: Omega curvature; the effective curvature density/critical density at z=0.

Om0¶: Omega matter; matter density/critical density at z=0.

Onu0¶: Omega nu; the density/critical density of neutrinos at z=0.

Otot0¶: Omega total; the total density/critical density at z=0.

Tcmb0¶: Temperature of the CMB as Quantity at z=0.

Tnu0¶: Temperature of the neutrino background as Quantity at z=0.

critical_density0¶: Critical density as Quantity at z=0.

from_format¶

Transform object to a Cosmology.

This function provides the Cosmology interface to the Astropy unified I/O layer. This allows easily parsing supported data formats using syntax such as:

>>> from astropy.cosmology import Cosmology
>>> cosmo1 = Cosmology.from_format(cosmo_mapping, format='mapping')

When the from_format method is called from a subclass the subclass will provide a keyword argument cosmology=<class> to the registered parser. The method uses this cosmology class, regardless of the class indicated in the data, and sets parameters’ default values from the class’ signature.

Get help on the available readers using the help() method:

>>> Cosmology.from_format.help()  # Get help and list supported formats
>>> Cosmology.from_format.help('<format>')  # Get detailed help on a format
>>> Cosmology.from_format.list_formats()  # Print list of available formats

Parameters:

objobject

The object to parse according to ‘format’

*args

Positional arguments passed through to data parser.

formatpython:str or python:None, optional keyword-only

Object format specifier. For None (default) CosmologyFromFormat tries to identify the correct format.

**kwargs

Keyword arguments passed through to data parser. Parsers should accept the following keyword arguments:

cosmologythe class (or string name thereof) to use / check when
constructing the cosmology instance.

Returns:

outCosmology subclass instance: Cosmology corresponding to obj contents.

h¶: Dimensionless Hubble constant: h = H_0 / 100 [km/sec/Mpc].

has_massive_nu¶: Does this cosmology have at least one massive neutrino species?

hubble_distance¶: Hubble distance as Quantity.

hubble_time¶: Hubble time as Quantity.

is_flat¶: Return True, the cosmology is flat.

m_nu¶: Mass of neutrino species.

meta¶

name¶: The name of the Cosmology instance.

nonflat¶

read¶

Read and parse data to a Cosmology.

This function provides the Cosmology interface to the Astropy unified I/O layer. This allows easily reading a file in supported data formats using syntax such as:

>>> from astropy.cosmology import Cosmology
>>> cosmo1 = Cosmology.read('<file name>')

When the read method is called from a subclass the subclass will provide a keyword argument cosmology=<class> to the registered read method. The method uses this cosmology class, regardless of the class indicated in the file, and sets parameters’ default values from the class’ signature.

Get help on the available readers using the help() method:

>>> Cosmology.read.help()  # Get help reading and list supported formats
>>> Cosmology.read.help(format='<format>')  # Get detailed help on a format
>>> Cosmology.read.list_formats()  # Print list of available formats

Parameters:

*args: Positional arguments passed through to data reader. If supplied the first argument is typically the input filename.
formatpython:str (optional, keyword-only): File format specifier.
**kwargs: Keyword arguments passed through to data reader.

Returns:

outCosmology subclass instance: Cosmology corresponding to file contents.

to_format¶

Transform this Cosmology to another format.

This function provides the Cosmology interface to the astropy unified I/O layer. This allows easily transforming to supported data formats using syntax such as:

>>> from astropy.cosmology import Planck18
>>> Planck18.to_format("mapping")
{'cosmology': astropy.cosmology.core.FlatLambdaCDM,
 'name': 'Planck18',
 'H0': <Quantity 67.66 km / (Mpc s)>,
 'Om0': 0.30966,
 ...

Get help on the available representations for Cosmology using the help() method:

>>> Cosmology.to_format.help()  # Get help and list supported formats
>>> Cosmology.to_format.help('<format>')  # Get detailed help on format
>>> Cosmology.to_format.list_formats()  # Print list of available formats

Parameters:

formatpython:str: Format specifier.
*args: Positional arguments passed through to data writer. If supplied the first argument is the output filename.
**kwargs: Keyword arguments passed through to data writer.

write¶

Write this Cosmology object out in the specified format.

This function provides the Cosmology interface to the astropy unified I/O layer. This allows easily writing a file in supported data formats using syntax such as:

>>> from astropy.cosmology import Planck18
>>> Planck18.write('<file name>')

Get help on the available writers for Cosmology using the help() method:

>>> Cosmology.write.help()  # Get help writing and list supported formats
>>> Cosmology.write.help(format='<format>')  # Get detailed help on format
>>> Cosmology.write.list_formats()  # Print list of available formats

Parameters:

*args: Positional arguments passed through to data writer. If supplied the first argument is the output filename.
formatpython:str (optional, keyword-only): File format specifier.
**kwargs: Keyword arguments passed through to data writer.

Methods Documentation

H(z)¶

Hubble parameter (km/s/Mpc) at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

HQuantity [:ref: ‘frequency’]: Hubble parameter at each input redshift.

Ob(z)¶

Return the density parameter for baryonic matter at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Obndarray or python:float: The density of baryonic matter relative to the critical density at each redshift. Returns float if the input is scalar.

Raises:

ValueError: If Ob0 is None.

Ode(z)¶

Return the density parameter for dark energy at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Odendarray or python:float: The density of non-relativistic matter relative to the critical density at each redshift. Returns float if the input is scalar.

Odm(z)¶

Return the density parameter for dark matter at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Odmndarray or python:float: The density of non-relativistic dark matter relative to the critical density at each redshift. Returns float if the input is scalar.

Raises:

ValueError: If Ob0 is None.

Notes

This does not include neutrinos, even if non-relativistic at the redshift of interest.

Ogamma(z)¶

Return the density parameter for photons at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Ogammandarray or python:float: The energy density of photons relative to the critical density at each redshift. Returns float if the input is scalar.

Ok(z)¶

Return the equivalent density parameter for curvature at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Okndarray or python:float: The equivalent density parameter for curvature at each redshift. Returns float if the input is scalar.

Om(z)¶

Return the density parameter for non-relativistic matter at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Omndarray or python:float: The density of non-relativistic matter relative to the critical density at each redshift. Returns float if the input is scalar.

Notes

This does not include neutrinos, even if non-relativistic at the redshift of interest; see Onu.

Onu(z)¶

Return the density parameter for neutrinos at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Onundarray or python:float: The energy density of neutrinos relative to the critical density at each redshift. Note that this includes their kinetic energy (if they have mass), so it is not equal to the commonly used \(\sum \frac{m_{\nu}}{94 eV}\), which does not include kinetic energy. Returns float if the input is scalar.

Otot(z)¶

The total density parameter at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshifts.

Returns:

Ototndarray or python:float: Returns float if input scalar. Value of 1.

Tcmb(z)¶

Return the CMB temperature at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

TcmbQuantity [:ref: ‘temperature’]: The temperature of the CMB in K.

Tnu(z)¶

Return the neutrino temperature at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

TnuQuantity [:ref: ‘temperature’]: The temperature of the cosmic neutrino background in K.

abs_distance_integrand(z)¶

Integrand of the absorption distance [1].

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Xpython:float or array: The integrand for the absorption distance.

References

[1] (1,2)

Hogg, D. (1999). Distance measures in cosmology, section 11. arXiv e-prints, astro-ph/9905116.

absorption_distance(z, /)¶

Absorption distance at redshift z.

This is used to calculate the number of objects with some cross section of absorption and number density intersecting a sightline per unit redshift path ([1], [2]).

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dpython:float or ndarray: Absorption distance (dimensionless) at each input redshift. Returns float if input scalar, ndarray otherwise.

References

[1]

Hogg, D. (1999). Distance measures in cosmology, section 11. arXiv e-prints, astro-ph/9905116.

[2]

Bahcall, John N. and Peebles, P.J.E. 1969, ApJ, 156L, 7B

age(z)¶

Age of the universe in Gyr at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

tQuantity [:ref: ‘time’]: The age of the universe in Gyr at each input redshift.

See also

z_at_value: Find the redshift corresponding to an age.

angular_diameter_distance(z)¶

Angular diameter distance in Mpc at a given redshift.

This gives the proper (sometimes called ‘physical’) transverse distance corresponding to an angle of 1 radian for an object at redshift z ([1], [2], [3]).

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dQuantity [:ref: ‘length’]: Angular diameter distance in Mpc at each input redshift.

References

[1]

Weinberg, 1972, pp 420-424; Weedman, 1986, pp 421-424.

[2]

Weedman, D. (1986). Quasar astronomy, pp 65-67.

[3]

Peebles, P. (1993). Principles of Physical Cosmology, pp 325-327.

angular_diameter_distance_z1z2(z1, z2)¶

Angular diameter distance between objects at 2 redshifts.

Useful for gravitational lensing, for example computing the angular diameter distance between a lensed galaxy and the foreground lens.

Parameters:

z1, z2Quantity-like [‘redshift’], numpy:array_like, or Number: Input redshifts. For most practical applications such as gravitational lensing, z2 should be larger than z1. The method will work for z2 < z1; however, this will return negative distances.

Returns:

dQuantity: The angular diameter distance between each input redshift pair. Returns scalar if input is scalar, array else-wise.

arcsec_per_kpc_comoving(z)¶

Angular separation in arcsec corresponding to a comoving kpc at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

thetaQuantity [:ref: ‘angle’]: The angular separation in arcsec corresponding to a comoving kpc at each input redshift.

arcsec_per_kpc_proper(z)¶

Angular separation in arcsec corresponding to a proper kpc at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

thetaQuantity [:ref: ‘angle’]: The angular separation in arcsec corresponding to a proper kpc at each input redshift.

clone(*, meta: Optional[Mapping] = None, to_nonflat: bool = None, **kwargs: Any)¶

Returns a copy of this object with updated parameters, as specified.

This cannot be used to change the type of the cosmology, except for changing to the non-flat version of this cosmology.

Parameters:

metapython:mapping or python:None (optional, keyword-only): Metadata that will update the current metadata.
to_nonflatbool or python:None, optional keyword-only: Whether to change to the non-flat version of this cosmology.
**kwargs: Cosmology parameter (and name) modifications. If any parameter is changed and a new name is not given, the name will be set to “[old name] (modified)”.

Returns:

newcosmoCosmology subclass instance: A new instance of this class with updated parameters as specified. If no arguments are given, then a reference to this object is returned instead of copy.

Examples

To make a copy of the Planck13 cosmology with a different matter density (Om0), and a new name:

>>> from astropy.cosmology import Planck13
>>> Planck13.clone(name="Modified Planck 2013", Om0=0.35)
FlatLambdaCDM(name="Modified Planck 2013", H0=67.77 km / (Mpc s),
  Om0=0.35, ...

If no name is specified, the new name will note the modification.

>>> Planck13.clone(Om0=0.35).name
'Planck13 (modified)'

The keyword ‘to_nonflat’ can be used to clone on the non-flat equivalent cosmology.

>>> Planck13.clone(to_nonflat=True)
LambdaCDM(name="Planck13", ...

>>> Planck13.clone(H0=70, to_nonflat=True)
LambdaCDM(name="Planck13 (modified)", H0=70.0 km / (Mpc s), ...

With ‘to_nonflat’ True, Ode0 can be modified.

>>> Planck13.clone(to_nonflat=True, Ode0=1)
LambdaCDM(name="Planck13 (modified)", H0=67.77 km / (Mpc s),
          Om0=0.30712, Ode0=1.0, ...

comoving_distance(z)¶

Comoving line-of-sight distance in Mpc at a given redshift.

The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dQuantity [:ref: ‘length’]: Comoving distance in Mpc to each input redshift.

comoving_transverse_distance(z)¶

Comoving transverse distance in Mpc at a given redshift.

This value is the transverse comoving distance at redshift z corresponding to an angular separation of 1 radian. This is the same as the comoving distance if \(\Omega_k\) is zero (as in the current concordance Lambda-CDM model).

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dQuantity [:ref: ‘length’]: Comoving transverse distance in Mpc at each input redshift.

Notes

This quantity is also called the ‘proper motion distance’ in some texts.

comoving_volume(z)¶

Comoving volume in cubic Mpc at redshift z.

This is the volume of the universe encompassed by redshifts less than z. For the case of \(\Omega_k = 0\) it is a sphere of radius comoving_distance but it is less intuitive if \(\Omega_k\) is not.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

VQuantity: Comoving volume in \(Mpc^3\) at each input redshift.

critical_density(z)¶

Critical density in grams per cubic cm at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

rhoQuantity: Critical density in g/cm^3 at each input redshift.

de_density_scale(z)¶

Evaluates the redshift dependence of the dark energy density.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Indarray or python:float: The scaling of the energy density of dark energy with redshift. Returns float if the input is scalar.

Notes

The scaling factor, I, is defined by \(\rho(z) = \rho_0 I\), and in this case is given by \(I = 1\).

differential_comoving_volume(z)¶

Differential comoving volume at redshift z.

Useful for calculating the effective comoving volume. For example, allows for integration over a comoving volume that has a sensitivity function that changes with redshift. The total comoving volume is given by integrating differential_comoving_volume to redshift z and multiplying by a solid angle.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dVQuantity: Differential comoving volume per redshift per steradian at each input redshift.

distmod(z)¶

Distance modulus at redshift z.

The distance modulus is defined as the (apparent magnitude - absolute magnitude) for an object at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

distmodQuantity [:ref: ‘length’]: Distance modulus at each input redshift, in magnitudes.

See also

z_at_value: Find the redshift corresponding to a distance modulus.

efunc(z)[source]¶

Function used to calculate H(z), the Hubble parameter.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Endarray or python:float: The redshift scaling of the Hubble constant. Returns float if the input is scalar. Defined such that \(H(z) = H_0 E(z)\).

inv_efunc(z)[source]¶

Function used to calculate \(\frac{1}{H_z}\).

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Endarray or python:float: The inverse redshift scaling of the Hubble constant. Returns float if the input is scalar. Defined such that \(H_z = H_0 / E\).

is_equivalent(other: Any, /, *, format: _FormatType = False) → bool¶

Check equivalence between Cosmologies.

Two cosmologies may be equivalent even if not the same class. For example, an instance of LambdaCDM might have \(\Omega_0=1\) and \(\Omega_k=0\) and therefore be flat, like FlatLambdaCDM.

Parameters:

otherCosmology subclass instance, positional-only: The object to which to compare.
formatbool or python:None or python:str, optional keyword-only: Whether to allow, before equivalence is checked, the object to be converted to a Cosmology. This allows, e.g. a Table to be equivalent to a Cosmology. False (default) will not allow conversion. True or None will, and will use the auto-identification to try to infer the correct format. A str is assumed to be the correct format to use when converting. format is broadcast to match the shape of other. Note that the cosmology arguments are not broadcast against format, so it cannot determine the output shape.

Returns:

bool: True if cosmologies are equivalent, False otherwise.

Examples

Two cosmologies may be equivalent even if not of the same class. In this examples the LambdaCDM has Ode0 set to the same value calculated in FlatLambdaCDM.

>>> import astropy.units as u
>>> from astropy.cosmology import LambdaCDM, FlatLambdaCDM
>>> cosmo1 = LambdaCDM(70 * (u.km/u.s/u.Mpc), 0.3, 0.7)
>>> cosmo2 = FlatLambdaCDM(70 * (u.km/u.s/u.Mpc), 0.3)
>>> cosmo1.is_equivalent(cosmo2)
True

While in this example, the cosmologies are not equivalent.

>>> cosmo3 = FlatLambdaCDM(70 * (u.km/u.s/u.Mpc), 0.3, Tcmb0=3 * u.K)
>>> cosmo3.is_equivalent(cosmo2)
False

Also, using the keyword argument, the notion of equivalence is extended to any Python object that can be converted to a Cosmology.

>>> from astropy.cosmology import Planck18
>>> tbl = Planck18.to_format("astropy.table")
>>> Planck18.is_equivalent(tbl, format=True)
True

The list of valid formats, e.g. the Table in this example, may be checked with Cosmology.from_format.list_formats().

As can be seen in the list of formats, not all formats can be auto-identified by Cosmology.from_format.registry. Objects of these kinds can still be checked for equivalence, but the correct format string must be used.

>>> tbl = Planck18.to_format("yaml")
>>> Planck18.is_equivalent(tbl, format="yaml")
True

kpc_comoving_per_arcmin(z)¶

Separation in transverse comoving kpc corresponding to an arcminute at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dQuantity [:ref: ‘length’]: The distance in comoving kpc corresponding to an arcmin at each input redshift.

kpc_proper_per_arcmin(z)¶

Separation in transverse proper kpc corresponding to an arcminute at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dQuantity [:ref: ‘length’]: The distance in proper kpc corresponding to an arcmin at each input redshift.

lookback_distance(z)¶

The lookback distance is the light travel time distance to a given redshift. It is simply c * lookback_time. It may be used to calculate the proper distance between two redshifts, e.g. for the mean free path to ionizing radiation.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dQuantity [:ref: ‘length’]: Lookback distance in Mpc

lookback_time(z)¶

Lookback time in Gyr to redshift z.

The lookback time is the difference between the age of the Universe now and the age at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

tQuantity [:ref: ‘time’]: Lookback time in Gyr to each input redshift.

See also

z_at_value: Find the redshift corresponding to a lookback time.

lookback_time_integrand(z)¶

Integrand of the lookback time (equation 30 of [1]).

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

Ipython:float or array: The integrand for the lookback time.

References

[1] (1,2)

Hogg, D. (1999). Distance measures in cosmology, section 11. arXiv e-prints, astro-ph/9905116.

luminosity_distance(z)¶

Luminosity distance in Mpc at redshift z.

This is the distance to use when converting between the bolometric flux from an object at redshift z and its bolometric luminosity [1].

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

dQuantity [:ref: ‘length’]: Luminosity distance in Mpc at each input redshift.

See also

z_at_value: Find the redshift corresponding to a luminosity distance.

References

[1]

Weinberg, 1972, pp 420-424; Weedman, 1986, pp 60-62.

nu_relative_density(z)¶

Neutrino density function relative to the energy density in photons.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

fndarray or python:float: The neutrino density scaling factor relative to the density in photons at each redshift. Only returns float if z is scalar.

Notes

The density in neutrinos is given by

\[\rho_{\nu} \left(a\right) = 0.2271 \, N_{eff} \, f\left(m_{\nu} a / T_{\nu 0} \right) \, \rho_{\gamma} \left( a \right)\]

where

\[f \left(y\right) = \frac{120}{7 \pi^4} \int_0^{\infty} \, dx \frac{x^2 \sqrt{x^2 + y^2}} {e^x + 1}\]

assuming that all neutrino species have the same mass. If they have different masses, a similar term is calculated for each one. Note that f has the asymptotic behavior \(f(0) = 1\). This method returns \(0.2271 f\) using an analytical fitting formula given in Komatsu et al. 2011, ApJS 192, 18.

scale_factor(z)¶

Scale factor at redshift z.

The scale factor is defined as \(a = 1 / (1 + z)\).

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

andarray or python:float: Scale factor at each input redshift. Returns float if the input is scalar.

w(z)¶

Returns dark energy equation of state at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like, or Number: Input redshift.

Returns:

wndarray or python:float: The dark energy equation of state. Returns float if the input is scalar.

Notes

The dark energy equation of state is defined as \(w(z) = P(z)/\rho(z)\), where \(P(z)\) is the pressure at redshift z and \(\rho(z)\) is the density at redshift z, both in units where c=1. Here this is \(w(z) = -1\).

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