# FLRW¶

class astropy.cosmology.FLRW(H0, Om0, Ode0, Tcmb0=0, Neff=3.04, m_nu=<Quantity 0. eV>, Ob0=None, name=None)[source]

Bases: astropy.cosmology.core.Cosmology

A class describing an isotropic and homogeneous (Friedmann-Lemaitre-Robertson-Walker) cosmology.

This is an abstract base class – you can’t instantiate examples of this class, but must work with one of its subclasses such as LambdaCDM or wCDM.

Parameters
H0float or scalar Quantity

Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]

Om0float

Omega matter: density of non-relativistic matter in units of the critical density at z=0. Note that this does not include massive neutrinos.

Ode0float

Omega dark energy: density of dark energy in units of the critical density at z=0.

Tcmb0float or scalar Quantity, 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).

Nefffloat, optional

Effective number of Neutrino species. Default 3.04.

m_nuQuantity, optional

Mass of each neutrino species. 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.

Ob0float or 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.

namestr, optional

Name for this cosmological object.

Notes

Class instances are static – you can’t change the values of the parameters. That is, all of the attributes above are read only.

Attributes Summary

 H0 Return the 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 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 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 m_nu Mass of neutrino species

Methods Summary

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

Attributes Documentation

H0

Return the 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

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

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

m_nu

Mass of neutrino species

Methods Documentation

H(self, z)[source]

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

Parameters
zarray-like

Input redshifts.

Returns
HQuantity

Hubble parameter at each input redshift.

Ob(self, z)[source]

Return the density parameter for baryonic matter at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
Obndarray, or float if input scalar

The density of baryonic matter relative to the critical density at each redshift.

Raises
ValueError

If Ob0 is None.

Ode(self, z)[source]

Return the density parameter for dark energy at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
Odendarray, or float if input scalar

The density of non-relativistic matter relative to the critical density at each redshift.

Odm(self, z)[source]

Return the density parameter for dark matter at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
Odmndarray, or float if input scalar

The density of non-relativistic dark matter relative to the critical density at each redshift.

Raises
ValueError

If Ob0 is None.

Notes

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

Ogamma(self, z)[source]

Return the density parameter for photons at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
Ogammandarray, or float if input scalar

The energy density of photons relative to the critical density at each redshift.

Ok(self, z)[source]

Return the equivalent density parameter for curvature at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
Okndarray, or float if input scalar

The equivalent density parameter for curvature at each redshift.

Om(self, z)[source]

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

Parameters
zarray-like

Input redshifts.

Returns
Omndarray, or float if input scalar

The density of non-relativistic matter relative to the critical density at each redshift.

Notes

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

Onu(self, z)[source]

Return the density parameter for neutrinos at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
Onundarray, or float if input scalar

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.

Tcmb(self, z)[source]

Return the CMB temperature at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
TcmbQuantity

The temperature of the CMB in K.

Tnu(self, z)[source]

Return the neutrino temperature at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
TnuQuantity

The temperature of the cosmic neutrino background in K.

abs_distance_integrand(self, z)[source]

Integrand of the absorption distance.

Parameters
zfloat or array

Input redshift.

Returns
Xfloat or array

The integrand for the absorption distance

References

See Hogg 1999 section 11.

absorption_distance(self, z)[source]

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.

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
dfloat or ndarray

Absorption distance (dimensionless) at each input redshift.

References

Hogg 1999 Section 11. (astro-ph/9905116) Bahcall, John N. and Peebles, P.J.E. 1969, ApJ, 156L, 7B

age(self, z)[source]

Age of the universe in Gyr at redshift z.

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
tQuantity

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

z_at_value

Find the redshift corresponding to an age.

angular_diameter_distance(self, z)[source]

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.

Weinberg, 1972, pp 421-424; Weedman, 1986, pp 65-67; Peebles, 1993, pp 325-327.

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
dQuantity

Angular diameter distance in Mpc at each input redshift.

angular_diameter_distance_z1z2(self, z1, z2)[source]

Angular diameter distance between objects at 2 redshifts. Useful for gravitational lensing.

Parameters
z1, z2array-like, shape (N,)

Input redshifts. z2 must be large than z1.

Returns
dQuantity, shape (N,) or single if input scalar

The angular diameter distance between each input redshift pair.

arcsec_per_kpc_comoving(self, z)[source]

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

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
thetaQuantity

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

arcsec_per_kpc_proper(self, z)[source]

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

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
thetaQuantity

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

clone(self, **kwargs)[source]

Returns a copy of this object, potentially with some changes.

Returns
newcosSubclass of FLRW
A new instance of this class with the specified changes.

Notes

This assumes that the values of all constructor arguments are available as properties, which is true of all the provided subclasses but may not be true of user-provided ones. You can’t change the type of class, so this can’t be used to change between flat and non-flat. If no modifications are requested, then a reference to this object is returned.

Examples

To make a copy of the Planck13 cosmology with a different Omega_m and a new name:

>>> from astropy.cosmology import Planck13
>>> newcos = Planck13.clone(name="Modified Planck 2013", Om0=0.35)

comoving_distance(self, z)[source]

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
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
dQuantity

Comoving distance in Mpc to each input redshift.

comoving_transverse_distance(self, z)[source]

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
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
dQuantity

Comoving transverse distance in Mpc at each input redshift.

Notes

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

comoving_volume(self, z)[source]

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 0.

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
VQuantity

Comoving volume in $$Mpc^3$$ at each input redshift.

critical_density(self, z)[source]

Critical density in grams per cubic cm at redshift z.

Parameters
zarray-like

Input redshifts.

Returns
rhoQuantity

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

de_density_scale(self, z)[source]

Evaluates the redshift dependence of the dark energy density.

Parameters
zarray-like

Input redshifts.

Returns
Indarray, or float if input scalar

The scaling of the energy density of dark energy with redshift.

Notes

The scaling factor, I, is defined by $$\rho(z) = \rho_0 I$$, and is given by

$I = \exp \left( 3 \int_{a}^1 \frac{ da^{\prime} }{ a^{\prime} } \left[ 1 + w\left( a^{\prime} \right) \right] \right)$

It will generally helpful for subclasses to overload this method if the integral can be done analytically for the particular dark energy equation of state that they implement.

differential_comoving_volume(self, z)[source]

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
zarray-like

Input redshifts.

Returns
dVQuantity

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

distmod(self, z)[source]

Distance modulus at redshift z.

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

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
distmodQuantity

Distance modulus at each input redshift, in magnitudes

z_at_value

Find the redshift corresponding to a distance modulus.

efunc(self, z)[source]

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

Parameters
zarray-like

Input redshifts.

Returns
Endarray, or float if input scalar

The redshift scaling of the Hubble constant.

Notes

The return value, E, is defined such that $$H(z) = H_0 E$$.

It is not necessary to override this method, but if de_density_scale takes a particularly simple form, it may be advantageous to.

inv_efunc(self, z)[source]

Inverse of efunc.

Parameters
zarray-like

Input redshifts.

Returns
Endarray, or float if input scalar

The redshift scaling of the inverse Hubble constant.

kpc_comoving_per_arcmin(self, z)[source]

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

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
dQuantity

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

kpc_proper_per_arcmin(self, z)[source]

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

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
dQuantity

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

lookback_distance(self, z)[source]

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
zarray-like

Input redshifts. Must be 1D or scalar

Returns
dQuantity

Lookback distance in Mpc

lookback_time(self, z)[source]

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
zarray-like

Input redshifts. Must be 1D or scalar

Returns
tQuantity

Lookback time in Gyr to each input redshift.

z_at_value

Find the redshift corresponding to a lookback time.

lookback_time_integrand(self, z)[source]

Integrand of the lookback time.

Parameters
zfloat or array-like

Input redshift.

Returns
Ifloat or array

The integrand for the lookback time

References

Eqn 30 from Hogg 1999.

luminosity_distance(self, z)[source]

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.

Parameters
zarray-like

Input redshifts. Must be 1D or scalar.

Returns
dQuantity

Luminosity distance in Mpc at each input redshift.

z_at_value

Find the redshift corresponding to a luminosity distance.

References

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

nu_relative_density(self, z)[source]

Neutrino density function relative to the energy density in photons.

Parameters
zarray like

Redshift

Returns
fndarray, or float if z is scalar

The neutrino density scaling factor relative to the density in photons at each redshift

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(self, z)[source]

Scale factor at redshift z.

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

Parameters
zarray-like

Input redshifts.

Returns
andarray, or float if input scalar

Scale factor at each input redshift.

abstract w(self, z)[source]

The dark energy equation of state.

Parameters
zarray-like

Input redshifts.

Returns
wndarray, or float if input scalar

The dark energy equation of state

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.

This must be overridden by subclasses.