redback.utils.UserCosmology

class redback.utils.UserCosmology(H0: Parameter, Om0: Parameter, Tcmb0: Parameter = Parameter(default=<Quantity 0. K>, derived=False, unit=Unit("K"), equivalencies=[], fvalidate='scalar', doc='Temperature of the CMB at z=0.'), Neff: Parameter = Parameter(default=3.04, derived=False, unit=None, equivalencies=[], fvalidate='non-negative', doc='Number of effective neutrino species.'), m_nu: Parameter = Parameter(default=<Quantity 0. eV>, derived=False, unit=Unit("eV"), equivalencies=[(Unit("kg"), Unit("J"), <function mass_energy.<locals>.<lambda>>, <function mass_energy.<locals>.<lambda>>), (Unit("kg / m2"), Unit("J / m2"), <function mass_energy.<locals>.<lambda>>, <function mass_energy.<locals>.<lambda>>), (Unit("kg / m3"), Unit("J / m3"), <function mass_energy.<locals>.<lambda>>, <function mass_energy.<locals>.<lambda>>), (Unit("kg / s"), Unit("J / s"), <function mass_energy.<locals>.<lambda>>, <function mass_energy.<locals>.<lambda>>)], fvalidate=<function FLRW.m_nu>, doc='Mass of neutrino species.'), Ob0: Parameter = Parameter(default=0.0, derived=False, unit=None, equivalencies=[], fvalidate=<function FLRW.Ob0>, doc='Omega baryon; baryonic matter density/critical density at z=0.'), *, name: _NameField = None, meta: MetaData = None)

Bases: FlatLambdaCDM

Dummy cosmology class that behaves like an Astropy cosmology, except that the luminosity distance is fixed to the user‐specified value.

Parameters:

• dl (astropy.units.Quantity) – The luminosity distance to return (e.g., 100 * u.Mpc).

• **kwargs – Additional keyword arguments for FlatLambdaCDM (e.g. H0, Om0) if needed.

__init__(**kwargs)

__call__(**kwargs)

Call self as a function.

Methods

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.
__init__(**kwargs)
abs_distance_integrand(z)	Integrand of the absorption distance (eq.
absorption_distance(z, /)	Absorption distance at redshift z (eq.
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 equal 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 equal to an arcmin at redshift z.
kpc_proper_per_arcmin(z)	Separation in transverse proper kpc equal 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(redshift)	Return the user-specified luminosity distance, ignoring the redshift.
nu_relative_density(z)	Neutrino density function relative to the energy density in photons.
scale_factor(z)	Scale factor at redshift z.
set_luminosity_distance(dl)	Set (or update) the user-specified luminosity distance.
w(z)	Returns dark energy equation of state at redshift z.

Attributes

H0	Hubble constant 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 at z=0.
Tnu0	Temperature of the neutrino background as |Quantity| at z=0.
critical_density0	Critical density at z=0.
from_format
h	h = H_0 / 100 [km/sec/Mpc].
has_massive_nu	Does this cosmology have at least one massive neutrino species?
hubble_distance	Hubble distance.
hubble_time	Hubble time.
is_flat	Return True, the cosmology is flat.
m_nu	Mass of neutrino species.
meta
name	The name of the cosmology realization, e.g. 'Planck2018' or None.
nonflat	Return the equivalent non-flat-class instance of this cosmology.
parameters	Immutable mapping of the Parameters.
read
scale_factor0	Scale factor at redshift 0.
to_format
write

H(z)

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

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

H – Hubble parameter at each input redshift.

Return type:

Quantity [‘frequency’]

H0: Parameter

Hubble constant at z=0.

Neff: Parameter

Number of effective neutrino species.

Ob(z)

Return the density parameter for baryonic matter at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Ob – The density of baryonic matter relative to the critical density at each redshift. Returns float if the input is scalar.

Return type:

ndarray or float

Ob0: Parameter

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

Ode(z)

Return the density parameter for dark energy at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Ode – The density of non-relativistic matter relative to the critical density at each redshift. Returns float if the input is scalar.

Return type:

ndarray or float

Ode0: Parameter

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

Odm(z)

Return the density parameter for dark matter at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Odm – The density of non-relativistic dark matter relative to the critical density at each redshift. Returns float if the input is scalar.

Return type:

ndarray or float

Notes

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

property Odm0: float

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

Ogamma(z)

Return the density parameter for photons at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Ogamma – The energy density of photons relative to the critical density at each redshift. Returns float if the input is scalar.

Return type:

ndarray or float

property Ogamma0: float

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

Ok(z)

Return the equivalent density parameter for curvature at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Ok – The equivalent density parameter for curvature at each redshift. Returns float if the input is scalar.

Return type:

ndarray or float

property Ok0: float

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

Om(z)

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

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Om – The density of non-relativistic matter relative to the critical density at each redshift. Returns float if the input is scalar.

Return type:

ndarray or float

Notes

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

Om0: Parameter

Omega matter; matter density/critical density at z=0.

Onu(z)

Return the density parameter for neutrinos at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Onu – 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.

Return type:

ndarray or float

property Onu0: float

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

Otot(z)

The total density parameter at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Otot – Returns float if input scalar. Value of 1.

Return type:

ndarray or float

property Otot0

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

Tcmb(z)

Return the CMB temperature at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Tcmb – The temperature of the CMB in K.

Return type:

Quantity [‘temperature’]

Tcmb0: Parameter

Temperature of the CMB at z=0.

Tnu(z)

Return the neutrino temperature at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Tnu – The temperature of the cosmic neutrino background in K.

Return type:

Quantity [‘temperature’]

property Tnu0: Quantity

Temperature of the neutrino background as |Quantity| at z=0.

abs_distance_integrand(z)

Integrand of the absorption distance (eq. 4, [1]_).

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

dX – The integrand for the absorption distance (dimensionless).

Return type:

float or array

References

[1]

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

absorption_distance(z, /)

Absorption distance at redshift z (eq. 4, [1]_).

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]_.

Parameters:

z (Quantity-like ['redshift'], array-like, positional-only) – Input redshift.

Returns:

X – Absorption distance (dimensionless) at each input redshift. Returns float if input scalar, ~numpy.ndarray otherwise.

Return type:

float or ndarray

References

[1]

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

age(z)

Age of the universe in Gyr at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

t – The age of the universe in Gyr at each input redshift.

Return type:

Quantity [‘time’]

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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

d – Angular diameter distance in Mpc at each input redshift.

Return type:

Quantity [‘length’]

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 (Quantity-like ['redshift'], array-like) – 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.

• z2 (Quantity-like ['redshift'], array-like) – 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:

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

Return type:

Quantity [‘length’]

arcsec_per_kpc_comoving(z)

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

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

theta – The angular separation in arcsec corresponding to a comoving kpc at each input redshift.

Return type:

Quantity [‘angle’]

arcsec_per_kpc_proper(z)

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

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

theta – The angular separation in arcsec corresponding to a proper kpc at each input redshift.

Return type:

Quantity [‘angle’]

clone(*, meta: Mapping | None = None, to_nonflat: bool = False, **kwargs) → Self

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:

• meta (mapping or None (optional, keyword-only)) – Metadata that will update the current metadata.

• to_nonflat (bool, 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:

newcosmo – 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.

Return type:

~astropy.cosmology.Cosmology subclass instance

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=<Quantity 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. For FLRW cosmologies this means Ode0 can be modified:

>>> Planck13.clone(to_nonflat=True, Ode0=1)
LambdaCDM(name='Planck13 (modified)', H0=<Quantity 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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

d – Comoving distance in Mpc to each input redshift.

Return type:

Quantity [‘length’]

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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

d – Comoving transverse distance in Mpc at each input redshift.

Return type:

Quantity [‘length’]

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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

V – Comoving volume in \(Mpc^3\) at each input redshift.

Return type:

Quantity [‘volume’]

critical_density(z)

Critical density in grams per cubic cm at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

rho – Critical density at each input redshift.

Return type:

Quantity [‘mass density’]

property critical_density0: Quantity

Critical density at z=0.

de_density_scale(z)

Evaluates the redshift dependence of the dark energy density.

Parameters:

z (Quantity-like ['redshift'] or array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

I – The scaling of the energy density of dark energy with redshift. Returns float if the input is scalar.

Return type:

ndarray or float

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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

dV – Differential comoving volume per redshift per steradian at each input redshift.

Return type:

Quantity

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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

distmod – Distance modulus at each input redshift, in magnitudes.

Return type:

Quantity [‘length’]

See also

z_at_value

Find the redshift corresponding to a distance modulus.

efunc(z)

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

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

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

Return type:

ndarray or float

property h: float

h = H_0 / 100 [km/sec/Mpc].

Type:

Dimensionless Hubble constant

property has_massive_nu: bool

Does this cosmology have at least one massive neutrino species?

property hubble_distance: Quantity

Hubble distance.

property hubble_time: Quantity

Hubble time.

inv_efunc(z)

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

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

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

Return type:

ndarray or float

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:

• other (~astropy.cosmology.Cosmology subclass instance, positional-only) – The object to which to compare.

• format (bool or None or 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:

True if cosmologies are equivalent, False otherwise.

Return type:

bool

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

property is_flat

Return True, the cosmology is flat.

kpc_comoving_per_arcmin(z)

Separation in transverse comoving kpc equal to an arcmin at redshift z.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

d – The distance in comoving kpc corresponding to an arcmin at each input redshift.

Return type:

Quantity [‘length’]

kpc_proper_per_arcmin(z)

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

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

d – The distance in proper kpc corresponding to an arcmin at each input redshift.

Return type:

Quantity [‘length’]

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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

d – Lookback distance in Mpc

Return type:

Quantity [‘length’]

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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

t – Lookback time in Gyr to each input redshift.

Return type:

Quantity [‘time’]

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:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

I – The integrand for the lookback time.

Return type:

float or array

References

[1]

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

luminosity_distance(redshift)

Return the user-specified luminosity distance, ignoring the redshift.

Parameters:

redshift (float or array-like) – Redshift (ignored).

Returns:

The user-specified luminosity distance.

Return type:

astropy.units.Quantity

m_nu: Parameter

Mass of neutrino species.

name: _NameField = None

The name of the cosmology realization, e.g. ‘Planck2018’ or None.

property nonflat: _FLRWT

Return the equivalent non-flat-class instance of this cosmology.

nu_relative_density(z)

Neutrino density function relative to the energy density in photons.

Parameters:

z (Quantity-like ['redshift'], array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

f – The neutrino density scaling factor relative to the density in photons at each redshift. Only returns float if z is scalar.

Return type:

ndarray or float

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.

parameters = mappingproxy({'H0': Parameter(derived=False, unit=Unit("km / (Mpc s)"), equivalencies=[], fvalidate='scalar', doc='Hubble constant at z=0.'), 'Om0': Parameter(derived=False, unit=None, equivalencies=[], fvalidate='non-negative', doc='Omega matter; matter density/critical density at z=0.'), 'Tcmb0': Parameter(default=<Quantity 0. K>, derived=False, unit=Unit("K"), equivalencies=[], fvalidate='scalar', doc='Temperature of the CMB at z=0.'), 'Neff': Parameter(default=3.04, derived=False, unit=None, equivalencies=[], fvalidate='non-negative', doc='Number of effective neutrino species.'), 'm_nu': Parameter(default=<Quantity 0. eV>, derived=False, unit=Unit("eV"), equivalencies=[(Unit("kg"), Unit("J"), <function mass_energy.<locals>.<lambda>>, <function mass_energy.<locals>.<lambda>>), (Unit("kg / m2"), Unit("J / m2"), <function mass_energy.<locals>.<lambda>>, <function mass_energy.<locals>.<lambda>>), (Unit("kg / m3"), Unit("J / m3"), <function mass_energy.<locals>.<lambda>>, <function mass_energy.<locals>.<lambda>>), (Unit("kg / s"), Unit("J / s"), <function mass_energy.<locals>.<lambda>>, <function mass_energy.<locals>.<lambda>>)], fvalidate=<function FLRW.m_nu>, doc='Mass of neutrino species.'), 'Ob0': Parameter(default=0.0, derived=False, unit=None, equivalencies=[], fvalidate=<function FLRW.Ob0>, doc='Omega baryon; baryonic matter density/critical density at z=0.')})

Immutable mapping of the Parameters.

If accessed from the class, this returns a mapping of the Parameter objects themselves. If accessed from an instance, this returns a mapping of the values of the Parameters.

scale_factor(z: Quantity | ArrayLike) → Quantity | NDArray[Any] | float

Scale factor at redshift z.

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

Parameters:

z (Quantity-like ['redshift'] | array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Scale factor at each input redshift. Returns float if the input is scalar.

Return type:

|Quantity| | ndarray | float

property scale_factor0: Quantity

Scale factor at redshift 0.

The scale factor is defined as \(a = \frac{a_0}{1 + z}\). The common convention is to set \(a_0 = 1\). However, in some cases, e.g. in some old CMB papers, \(a_0\) is used to normalize a to be a convenient number at the redshift of interest for that paper. Explicitly using \(a_0\) in both calculation and code avoids ambiguity.

set_luminosity_distance(dl)

Set (or update) the user-specified luminosity distance.

Parameters:

dl (astropy.units.Quantity) – The new luminosity distance.

w(z)

Returns dark energy equation of state at redshift z.

Parameters:

z (Quantity-like ['redshift'] or array-like) –

Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

w – The dark energy equation of state. Returns float if the input is scalar.

Return type:

ndarray or float

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 \(w(z) = -1\).