# Licensed under a 3-clause BSD style license - see LICENSE.rst
from __future__ import annotations
import warnings
from abc import abstractmethod
from math import exp, floor, log, pi, sqrt
from numbers import Number
from typing import TypeVar
import numpy as np
from numpy import inf, sin
import astropy.constants as const
import astropy.units as u
from astropy.cosmology._utils import aszarr, vectorize_redshift_method
from astropy.cosmology.core import Cosmology, FlatCosmologyMixin
from astropy.cosmology.parameter import Parameter
from astropy.cosmology.parameter._converter import (
_validate_non_negative,
_validate_with_unit,
)
from astropy.utils.compat.optional_deps import HAS_SCIPY
from astropy.utils.decorators import lazyproperty
from astropy.utils.exceptions import AstropyUserWarning
__all__ = ["FLRW", "FlatFLRWMixin"]
__doctest_requires__ = {"*": ["scipy"]}
# isort: split
if HAS_SCIPY:
from scipy.integrate import quad
else:
def quad(*args, **kwargs):
raise ModuleNotFoundError("No module named 'scipy.integrate'")
##############################################################################
# Parameters
# Some conversion constants -- useful to compute them once here and reuse in
# the initialization rather than have every object do them.
_H0units_to_invs = (u.km / (u.s * u.Mpc)).to(1.0 / u.s)
_sec_to_Gyr = u.s.to(u.Gyr)
# const in critical density in cgs units (g cm^-3)
_critdens_const = (3 / (8 * pi * const.G)).cgs.value
# angle conversions
_radian_in_arcsec = (1 * u.rad).to(u.arcsec)
_radian_in_arcmin = (1 * u.rad).to(u.arcmin)
# Radiation parameter over c^2 in cgs (g cm^-3 K^-4)
_a_B_c2 = (4 * const.sigma_sb / const.c**3).cgs.value
# Boltzmann constant in eV / K
_kB_evK = const.k_B.to(u.eV / u.K)
# typing
_FLRWT = TypeVar("_FLRWT", bound="FLRW")
_FlatFLRWMixinT = TypeVar("_FlatFLRWMixinT", bound="FlatFLRWMixin")
##############################################################################
class _ScaleFactorMixin:
@property
def scale_factor0(self):
r"""Scale factor at redshift 0.
The scale factor is defined as :math:`a = \frac{a_0}{1 + z}`. The common
convention is to set :math:`a_0 = 1`. However, in some cases, e.g. in some old
CMB papers, :math:`a_0` is used to normalize `a` to be a convenient number at
the redshift of interest for that paper. Explicitly using :math:`a_0` in both
calculation and code avoids ambiguity.
"""
return u.Quantity(self.scale_factor(0), unit=u.one)
def scale_factor(self, z):
"""Scale factor at redshift ``z``.
The scale factor is defined as :math:`a = 1 / (1 + z)`.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
a : ndarray or float
Scale factor at each input redshift.
Returns `float` if the input is scalar.
"""
return 1.0 / (aszarr(z) + 1.0)
ParameterOde0 = Parameter(
doc="Omega dark energy; dark energy density/critical density at z=0.",
fvalidate="float",
)
[文档]
class FLRW(Cosmology, _ScaleFactorMixin):
"""An isotropic and homogeneous (Friedmann-Lemaitre-Robertson-Walker) cosmology.
This is an abstract base class -- you cannot instantiate examples of this
class, but must work with one of its subclasses, such as
:class:`~astropy.cosmology.LambdaCDM` or :class:`~astropy.cosmology.wCDM`.
Parameters
----------
H0 : float or scalar quantity-like ['frequency']
Hubble constant at z = 0. If a float, must be in [km/sec/Mpc].
Om0 : float
Omega matter: density of non-relativistic matter in units of the
critical density at z=0. Note that this does not include massive
neutrinos.
Ode0 : float
Omega dark energy: density of dark energy in units of the critical
density at z=0.
Tcmb0 : float or scalar quantity-like ['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).
Neff : float, optional
Effective number of Neutrino species. Default 3.04.
m_nu : quantity-like ['energy', 'mass'] or 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.
Ob0 : float 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.
name : str or None (optional, keyword-only)
Name for this cosmological object.
meta : mapping or None (optional, keyword-only)
Metadata for the cosmology, e.g., a reference.
Notes
-----
Class instances are immutable -- you cannot change the parameters' values.
That is, all of the above attributes (except meta) are read only.
For details on how to create performant custom subclasses, see the
documentation on :ref:`astropy-cosmology-fast-integrals`.
"""
H0 = Parameter(
doc="Hubble constant as an `~astropy.units.Quantity` at z=0.",
unit="km/(s Mpc)",
fvalidate="scalar",
)
Om0 = Parameter(
doc="Omega matter; matter density/critical density at z=0.",
fvalidate="non-negative",
)
Ode0 = ParameterOde0
Tcmb0 = Parameter(
default=0.0 * u.K,
doc="Temperature of the CMB as `~astropy.units.Quantity` at z=0.",
unit="Kelvin",
fvalidate="scalar",
)
Neff = Parameter(
default=3.04,
doc="Number of effective neutrino species.",
fvalidate="non-negative",
)
m_nu = Parameter(
default=0.0 * u.eV,
doc="Mass of neutrino species.",
unit="eV",
equivalencies=u.mass_energy(),
)
Ob0 = Parameter(
default=None,
doc="Omega baryon; baryonic matter density/critical density at z=0.",
)
def __init__(
self,
H0,
Om0,
Ode0,
Tcmb0=0.0 * u.K,
Neff=3.04,
m_nu=0.0 * u.eV,
Ob0=None,
*,
name=None,
meta=None,
):
super().__init__(name=name, meta=meta)
# Assign (and validate) Parameters
self.H0 = H0
self.Om0 = Om0
self.Ode0 = Ode0
self.Tcmb0 = Tcmb0
self.Neff = Neff
self.m_nu = m_nu # (reset later, this is just for unit validation)
self.Ob0 = Ob0 # (must be after Om0)
# Derived quantities:
# Dark matter density; matter - baryons, if latter is not None.
self._Odm0 = None if Ob0 is None else (self._Om0 - self._Ob0)
# 100 km/s/Mpc * h = H0 (so h is dimensionless)
self._h = self._H0.value / 100.0
# Hubble distance
self._hubble_distance = (const.c / self._H0).to(u.Mpc)
# H0 in s^-1
H0_s = self._H0.value * _H0units_to_invs
# Hubble time
self._hubble_time = (_sec_to_Gyr / H0_s) << u.Gyr
# Critical density at z=0 (grams per cubic cm)
cd0value = _critdens_const * H0_s**2
self._critical_density0 = cd0value << u.g / u.cm**3
# Compute photon density from Tcmb
self._Ogamma0 = _a_B_c2 * self._Tcmb0.value**4 / self._critical_density0.value
# Compute Neutrino temperature:
# The constant in front is (4/11)^1/3 -- see any cosmology book for an
# explanation -- for example, Weinberg 'Cosmology' p 154 eq (3.1.21).
self._Tnu0 = 0.7137658555036082 * self._Tcmb0
# Compute neutrino parameters:
if self._m_nu is None:
self._nneutrinos = 0
self._neff_per_nu = None
self._massivenu = False
self._massivenu_mass = None
self._nmassivenu = self._nmasslessnu = None
else:
self._nneutrinos = floor(self._Neff)
# We are going to share Neff between the neutrinos equally. In
# detail this is not correct, but it is a standard assumption
# because properly calculating it is a) complicated b) depends on
# the details of the massive neutrinos (e.g., their weak
# interactions, which could be unusual if one is considering
# sterile neutrinos).
self._neff_per_nu = self._Neff / self._nneutrinos
# Now figure out if we have massive neutrinos to deal with, and if
# so, get the right number of masses. It is worth keeping track of
# massless ones separately (since they are easy to deal with, and a
# common use case is to have only one massive neutrino).
massive = np.nonzero(self._m_nu.value > 0)[0]
self._massivenu = massive.size > 0
self._nmassivenu = len(massive)
self._massivenu_mass = (
self._m_nu[massive].value if self._massivenu else None
)
self._nmasslessnu = self._nneutrinos - self._nmassivenu
# Compute Neutrino Omega and total relativistic component for massive
# neutrinos. We also store a list version, since that is more efficient
# to do integrals with (perhaps surprisingly! But small python lists
# are more efficient than small NumPy arrays).
if self._massivenu: # (`_massivenu` set in `m_nu`)
nu_y = self._massivenu_mass / (_kB_evK * self._Tnu0)
self._nu_y = nu_y.value
self._nu_y_list = self._nu_y.tolist()
self._Onu0 = self._Ogamma0 * self.nu_relative_density(0)
else:
# This case is particularly simple, so do it directly The 0.2271...
# is 7/8 (4/11)^(4/3) -- the temperature bit ^4 (blackbody energy
# density) times 7/8 for FD vs. BE statistics.
self._Onu0 = 0.22710731766 * self._Neff * self._Ogamma0
self._nu_y = self._nu_y_list = None
# Compute curvature density
self._Ok0 = 1.0 - self._Om0 - self._Ode0 - self._Ogamma0 - self._Onu0
# Subclasses should override this reference if they provide
# more efficient scalar versions of inv_efunc.
self._inv_efunc_scalar = self.inv_efunc
self._inv_efunc_scalar_args = ()
# ---------------------------------------------------------------
# Parameter details
@Ob0.validator
def Ob0(self, param, value):
"""Validate baryon density to None or positive float > matter density."""
if value is None:
return value
value = _validate_non_negative(self, param, value)
if value > self.Om0:
raise ValueError(
"baryonic density can not be larger than total matter density."
)
return value
@m_nu.validator
def m_nu(self, param, value):
"""Validate neutrino masses to right value, units, and shape.
There are no neutrinos if floor(Neff) or Tcmb0 are 0. The number of
neutrinos must match floor(Neff). Neutrino masses cannot be
negative.
"""
# Check if there are any neutrinos
if (nneutrinos := floor(self._Neff)) == 0 or self._Tcmb0.value == 0:
return None # None, regardless of input
# Validate / set units
value = _validate_with_unit(self, param, value)
# Check values and data shapes
if value.shape not in ((), (nneutrinos,)):
raise ValueError(
"unexpected number of neutrino masses — "
f"expected {nneutrinos}, got {len(value)}."
)
elif np.any(value.value < 0):
raise ValueError("invalid (negative) neutrino mass encountered.")
# scalar -> array
if value.isscalar:
value = np.full_like(value, value, shape=nneutrinos)
return value
# ---------------------------------------------------------------
# properties
@property
def is_flat(self):
"""Return bool; `True` if the cosmology is flat."""
return bool((self._Ok0 == 0.0) and (self.Otot0 == 1.0))
@property
def Otot0(self):
"""Omega total; the total density/critical density at z=0."""
return self._Om0 + self._Ogamma0 + self._Onu0 + self._Ode0 + self._Ok0
@property
def Odm0(self):
"""Omega dark matter; dark matter density/critical density at z=0."""
return self._Odm0
@property
def Ok0(self):
"""Omega curvature; the effective curvature density/critical density at z=0."""
return self._Ok0
@property
def Tnu0(self):
"""Temperature of the neutrino background as |Quantity| at z=0."""
return self._Tnu0
@property
def has_massive_nu(self):
"""Does this cosmology have at least one massive neutrino species?"""
if self._Tnu0.value == 0:
return False
return self._massivenu
@property
def h(self):
"""Dimensionless Hubble constant: h = H_0 / 100 [km/sec/Mpc]."""
return self._h
@property
def hubble_time(self):
"""Hubble time as `~astropy.units.Quantity`."""
return self._hubble_time
@property
def hubble_distance(self):
"""Hubble distance as `~astropy.units.Quantity`."""
return self._hubble_distance
@property
def critical_density0(self):
"""Critical density as `~astropy.units.Quantity` at z=0."""
return self._critical_density0
@property
def Ogamma0(self):
"""Omega gamma; the density/critical density of photons at z=0."""
return self._Ogamma0
@property
def Onu0(self):
"""Omega nu; the density/critical density of neutrinos at z=0."""
return self._Onu0
# ---------------------------------------------------------------
[文档]
@abstractmethod
def w(self, z):
r"""The dark energy equation of state.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
w : ndarray or float
The dark energy equation of state.
`float` if scalar input.
Notes
-----
The dark energy equation of state is defined as
:math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at
redshift z and :math:`\rho(z)` is the density at redshift z, both in
units where c=1.
This must be overridden by subclasses.
"""
raise NotImplementedError("w(z) is not implemented")
[文档]
def Otot(self, z):
"""The total density parameter at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshifts.
Returns
-------
Otot : ndarray or float
The total density relative to the critical density at each redshift.
Returns float if input scalar.
"""
return self.Om(z) + self.Ogamma(z) + self.Onu(z) + self.Ode(z) + self.Ok(z)
[文档]
def Om(self, z):
"""Return the density parameter for non-relativistic matter at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Om : ndarray or 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`.
"""
z = aszarr(z)
return self._Om0 * (z + 1.0) ** 3 * self.inv_efunc(z) ** 2
[文档]
def Ob(self, z):
"""Return the density parameter for baryonic matter at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Ob : ndarray or 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`.
"""
if self._Ob0 is None:
raise ValueError("Baryon density not set for this cosmology")
z = aszarr(z)
return self._Ob0 * (z + 1.0) ** 3 * self.inv_efunc(z) ** 2
[文档]
def Odm(self, z):
"""Return the density parameter for dark matter at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Odm : ndarray or 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.
"""
if self._Odm0 is None:
raise ValueError(
"Baryonic density not set for this cosmology, "
"unclear meaning of dark matter density"
)
z = aszarr(z)
return self._Odm0 * (z + 1.0) ** 3 * self.inv_efunc(z) ** 2
[文档]
def Ok(self, z):
"""Return the equivalent density parameter for curvature at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Ok : ndarray or float
The equivalent density parameter for curvature at each redshift.
Returns `float` if the input is scalar.
"""
z = aszarr(z)
if self._Ok0 == 0: # Common enough to be worth checking explicitly
return np.zeros(z.shape) if hasattr(z, "shape") else 0.0
return self._Ok0 * (z + 1.0) ** 2 * self.inv_efunc(z) ** 2
[文档]
def Ode(self, z):
"""Return the density parameter for dark energy at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Ode : ndarray or float
The density of non-relativistic matter relative to the critical
density at each redshift.
Returns `float` if the input is scalar.
"""
z = aszarr(z)
if self._Ode0 == 0: # Common enough to be worth checking explicitly
return np.zeros(z.shape) if hasattr(z, "shape") else 0.0
return self._Ode0 * self.de_density_scale(z) * self.inv_efunc(z) ** 2
[文档]
def Ogamma(self, z):
"""Return the density parameter for photons at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Ogamma : ndarray or float
The energy density of photons relative to the critical density at
each redshift.
Returns `float` if the input is scalar.
"""
z = aszarr(z)
return self._Ogamma0 * (z + 1.0) ** 4 * self.inv_efunc(z) ** 2
[文档]
def Onu(self, z):
r"""Return the density parameter for neutrinos at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Onu : ndarray or 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
:math:`\sum \frac{m_{\nu}}{94 eV}`, which does not include
kinetic energy.
Returns `float` if the input is scalar.
"""
z = aszarr(z)
if self._Onu0 == 0: # Common enough to be worth checking explicitly
return np.zeros(z.shape) if hasattr(z, "shape") else 0.0
return self.Ogamma(z) * self.nu_relative_density(z)
[文档]
def Tcmb(self, z):
"""Return the CMB temperature at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Tcmb : `~astropy.units.Quantity` ['temperature']
The temperature of the CMB in K.
"""
return self._Tcmb0 * (aszarr(z) + 1.0)
[文档]
def Tnu(self, z):
"""Return the neutrino temperature at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
Tnu : `~astropy.units.Quantity` ['temperature']
The temperature of the cosmic neutrino background in K.
"""
return self._Tnu0 * (aszarr(z) + 1.0)
[文档]
def nu_relative_density(self, z):
r"""Neutrino density function relative to the energy density in photons.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
f : ndarray or 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
.. math::
\rho_{\nu} \left(a\right) = 0.2271 \, N_{eff} \,
f\left(m_{\nu} a / T_{\nu 0} \right) \,
\rho_{\gamma} \left( a \right)
where
.. math::
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 :math:`f(0) = 1`. This
method returns :math:`0.2271 f` using an analytical fitting formula
given in Komatsu et al. 2011, ApJS 192, 18.
"""
# Note that there is also a scalar-z-only cython implementation of
# this in scalar_inv_efuncs.pyx, so if you find a problem in this
# you need to update there too.
# See Komatsu et al. 2011, eq 26 and the surrounding discussion
# for an explanation of what we are doing here.
# However, this is modified to handle multiple neutrino masses
# by computing the above for each mass, then summing
prefac = 0.22710731766 # 7/8 (4/11)^4/3 -- see any cosmo book
# The massive and massless contribution must be handled separately
# But check for common cases first
z = aszarr(z)
if not self._massivenu:
return (
prefac * self._Neff * (np.ones(z.shape) if hasattr(z, "shape") else 1.0)
)
# These are purely fitting constants -- see the Komatsu paper
p = 1.83
invp = 0.54644808743 # 1.0 / p
k = 0.3173
curr_nu_y = self._nu_y / (1.0 + np.expand_dims(z, axis=-1))
rel_mass_per = (1.0 + (k * curr_nu_y) ** p) ** invp
rel_mass = rel_mass_per.sum(-1) + self._nmasslessnu
return prefac * self._neff_per_nu * rel_mass
def _w_integrand(self, ln1pz):
"""Internal convenience function for w(z) integral (eq. 5 of [1]_).
Parameters
----------
ln1pz : `~numbers.Number` or scalar ndarray
Assumes scalar input, since this should only be called inside an
integral.
References
----------
.. [1] Linder, E. (2003). Exploring the Expansion History of the
Universe. Phys. Rev. Lett., 90, 091301.
"""
return 1.0 + self.w(exp(ln1pz) - 1.0)
[文档]
def de_density_scale(self, z):
r"""Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
I : ndarray or 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 :math:`\rho(z) = \rho_0 I`,
and is given by
.. math::
I = \exp \left( 3 \int_{a}^1 \frac{ da^{\prime} }{ a^{\prime} }
\left[ 1 + w\left( a^{\prime} \right) \right] \right)
The actual integral used is rewritten from [1]_ to be in terms of z.
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.
References
----------
.. [1] Linder, E. (2003). Exploring the Expansion History of the
Universe. Phys. Rev. Lett., 90, 091301.
"""
# This allows for an arbitrary w(z) following eq (5) of
# Linder 2003, PRL 90, 91301. The code here evaluates
# the integral numerically. However, most popular
# forms of w(z) are designed to make this integral analytic,
# so it is probably a good idea for subclasses to overload this
# method if an analytic form is available.
z = aszarr(z)
if not isinstance(z, (Number, np.generic)): # array/Quantity
ival = np.array(
[quad(self._w_integrand, 0, log(1 + redshift))[0] for redshift in z]
)
return np.exp(3 * ival)
else: # scalar
ival = quad(self._w_integrand, 0, log(z + 1.0))[0]
return exp(3 * ival)
[文档]
def efunc(self, z):
"""Function used to calculate H(z), the Hubble parameter.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
E : ndarray or float
The redshift scaling of the Hubble constant.
Returns `float` if the input is scalar.
Defined such that :math:`H(z) = H_0 E(z)`.
Notes
-----
It is not necessary to override this method, but if de_density_scale
takes a particularly simple form, it may be advantageous to.
"""
Or = self._Ogamma0 + (
self._Onu0
if not self._massivenu
else self._Ogamma0 * self.nu_relative_density(z)
)
zp1 = aszarr(z) + 1.0 # (converts z [unit] -> z [dimensionless])
return np.sqrt(
zp1**2 * ((Or * zp1 + self._Om0) * zp1 + self._Ok0)
+ self._Ode0 * self.de_density_scale(z)
)
[文档]
def inv_efunc(self, z):
"""Inverse of ``efunc``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
E : ndarray or float
The redshift scaling of the inverse Hubble constant.
Returns `float` if the input is scalar.
"""
# Avoid the function overhead by repeating code
Or = self._Ogamma0 + (
self._Onu0
if not self._massivenu
else self._Ogamma0 * self.nu_relative_density(z)
)
zp1 = aszarr(z) + 1.0 # (converts z [unit] -> z [dimensionless])
return (
zp1**2 * ((Or * zp1 + self._Om0) * zp1 + self._Ok0)
+ self._Ode0 * self.de_density_scale(z)
) ** (-0.5)
def _lookback_time_integrand_scalar(self, z):
"""Integrand of the lookback time (equation 30 of [1]_).
Parameters
----------
z : float
Input redshift.
Returns
-------
I : float
The integrand for the lookback time.
References
----------
.. [1] Hogg, D. (1999). Distance measures in cosmology, section 11.
arXiv e-prints, astro-ph/9905116.
"""
return self._inv_efunc_scalar(z, *self._inv_efunc_scalar_args) / (z + 1.0)
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def lookback_time_integrand(self, z):
"""Integrand of the lookback time (equation 30 of [1]_).
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
I : float or array
The integrand for the lookback time.
References
----------
.. [1] Hogg, D. (1999). Distance measures in cosmology, section 11.
arXiv e-prints, astro-ph/9905116.
"""
z = aszarr(z)
return self.inv_efunc(z) / (z + 1.0)
def _abs_distance_integrand_scalar(self, z):
"""Integrand of the absorption distance [1]_.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
X : float
The integrand for the absorption distance.
References
----------
.. [1] Hogg, D. (1999). Distance measures in cosmology, section 11.
arXiv e-prints, astro-ph/9905116.
"""
args = self._inv_efunc_scalar_args
return (z + 1.0) ** 2 * self._inv_efunc_scalar(z, *args)
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def abs_distance_integrand(self, z):
"""Integrand of the absorption distance [1]_.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
X : float or array
The integrand for the absorption distance.
References
----------
.. [1] Hogg, D. (1999). Distance measures in cosmology, section 11.
arXiv e-prints, astro-ph/9905116.
"""
z = aszarr(z)
return (z + 1.0) ** 2 * self.inv_efunc(z)
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def H(self, z):
"""Hubble parameter (km/s/Mpc) at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
H : `~astropy.units.Quantity` ['frequency']
Hubble parameter at each input redshift.
"""
return self._H0 * self.efunc(z)
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def lookback_time(self, 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, or `~numbers.Number`
Input redshift.
Returns
-------
t : `~astropy.units.Quantity` ['time']
Lookback time in Gyr to each input redshift.
See Also
--------
z_at_value : Find the redshift corresponding to a lookback time.
"""
return self._lookback_time(z)
def _lookback_time(self, 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, or `~numbers.Number`
Input redshift.
Returns
-------
t : `~astropy.units.Quantity` ['time']
Lookback time in Gyr to each input redshift.
"""
return self._hubble_time * self._integral_lookback_time(z)
@vectorize_redshift_method
def _integral_lookback_time(self, z, /):
"""Lookback time to redshift ``z``. Value in units of Hubble time.
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, or `~numbers.Number`
Input redshift.
Returns
-------
t : float or ndarray
Lookback time to each input redshift in Hubble time units.
Returns `float` if input scalar, `~numpy.ndarray` otherwise.
"""
return quad(self._lookback_time_integrand_scalar, 0, z)[0]
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def lookback_distance(self, 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, or `~numbers.Number`
Input redshift.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Lookback distance in Mpc
"""
return (self.lookback_time(z) * const.c).to(u.Mpc)
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def age(self, z):
"""Age of the universe in Gyr at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
t : `~astropy.units.Quantity` ['time']
The age of the universe in Gyr at each input redshift.
See Also
--------
z_at_value : Find the redshift corresponding to an age.
"""
return self._age(z)
def _age(self, z):
"""Age of the universe in Gyr at redshift ``z``.
This internal function exists to be re-defined for optimizations.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
t : `~astropy.units.Quantity` ['time']
The age of the universe in Gyr at each input redshift.
"""
return self._hubble_time * self._integral_age(z)
@vectorize_redshift_method
def _integral_age(self, z, /):
"""Age of the universe at redshift ``z``. Value in units of Hubble time.
Calculated using explicit integration.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
t : float or ndarray
The age of the universe at each input redshift in Hubble time units.
Returns `float` if input scalar, `~numpy.ndarray` otherwise.
See Also
--------
z_at_value : Find the redshift corresponding to an age.
"""
return quad(self._lookback_time_integrand_scalar, z, inf)[0]
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def critical_density(self, z):
"""Critical density in grams per cubic cm at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
rho : `~astropy.units.Quantity`
Critical density in g/cm^3 at each input redshift.
"""
return self._critical_density0 * (self.efunc(z)) ** 2
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def comoving_distance(self, 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, or `~numbers.Number`
Input redshift.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc to each input redshift.
"""
return self._comoving_distance_z1z2(0, z)
def _comoving_distance_z1z2(self, z1, z2):
"""Comoving line-of-sight distance in Mpc between redshifts ``z1`` and ``z2``.
The comoving distance along the line-of-sight between two objects
remains constant with time for objects in the Hubble flow.
Parameters
----------
z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshifts.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc between each input redshift.
"""
return self._integral_comoving_distance_z1z2(z1, z2)
@vectorize_redshift_method(nin=2)
def _integral_comoving_distance_z1z2_scalar(self, z1, z2, /):
"""Comoving line-of-sight distance in Mpc between objects at redshifts ``z1`` and ``z2``.
The comoving distance along the line-of-sight between two objects
remains constant with time for objects in the Hubble flow.
Parameters
----------
z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshifts.
Returns
-------
d : float or ndarray
Comoving distance in Mpc between each input redshift.
Returns `float` if input scalar, `~numpy.ndarray` otherwise.
"""
return quad(self._inv_efunc_scalar, z1, z2, args=self._inv_efunc_scalar_args)[0]
def _integral_comoving_distance_z1z2(self, z1, z2):
"""Comoving line-of-sight distance in Mpc between objects at redshifts ``z1`` and ``z2``.
The comoving distance along the line-of-sight between two objects remains
constant with time for objects in the Hubble flow.
Parameters
----------
z1, z2 : Quantity-like ['redshift'] or array-like
Input redshifts.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc between each input redshift.
"""
return self._hubble_distance * self._integral_comoving_distance_z1z2_scalar(z1, z2) # fmt: skip
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def comoving_transverse_distance(self, z):
r"""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 :math:`\Omega_k` is zero (as in the current
concordance Lambda-CDM model).
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving transverse distance in Mpc at each input redshift.
Notes
-----
This quantity is also called the 'proper motion distance' in some texts.
"""
return self._comoving_transverse_distance_z1z2(0, z)
def _comoving_transverse_distance_z1z2(self, z1, z2):
r"""Comoving transverse distance in Mpc between two redshifts.
This value is the transverse comoving distance at redshift ``z2`` as
seen from redshift ``z1`` corresponding to an angular separation of
1 radian. This is the same as the comoving distance if :math:`\Omega_k`
is zero (as in the current concordance Lambda-CDM model).
Parameters
----------
z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshifts.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving transverse distance in Mpc between input redshift.
Notes
-----
This quantity is also called the 'proper motion distance' in some texts.
"""
Ok0 = self._Ok0
dc = self._comoving_distance_z1z2(z1, z2)
if Ok0 == 0:
return dc
sqrtOk0 = sqrt(abs(Ok0))
dh = self._hubble_distance
if Ok0 > 0:
return dh / sqrtOk0 * np.sinh(sqrtOk0 * dc.value / dh.value)
else:
return dh / sqrtOk0 * sin(sqrtOk0 * dc.value / dh.value)
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def angular_diameter_distance(self, 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, or `~numbers.Number`
Input redshift.
Returns
-------
d : `~astropy.units.Quantity` ['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.
"""
z = aszarr(z)
return self.comoving_transverse_distance(z) / (z + 1.0)
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def luminosity_distance(self, 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
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
d : `~astropy.units.Quantity` ['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.
"""
z = aszarr(z)
return (z + 1.0) * self.comoving_transverse_distance(z)
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def angular_diameter_distance_z1z2(self, 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, z2 : Quantity-like ['redshift'], array-like, or `~numbers.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
-------
d : `~astropy.units.Quantity`
The angular diameter distance between each input redshift pair.
Returns scalar if input is scalar, array else-wise.
"""
z1, z2 = aszarr(z1), aszarr(z2)
if np.any(z2 < z1):
warnings.warn(
f"Second redshift(s) z2 ({z2}) is less than first "
f"redshift(s) z1 ({z1}).",
AstropyUserWarning,
)
return self._comoving_transverse_distance_z1z2(z1, z2) / (z2 + 1.0)
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@vectorize_redshift_method
def absorption_distance(self, 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
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
d : float or ndarray
Absorption distance (dimensionless) at each input redshift.
Returns `float` if input scalar, `~numpy.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
"""
return quad(self._abs_distance_integrand_scalar, 0, z)[0]
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def distmod(self, 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, or `~numbers.Number`
Input redshift.
Returns
-------
distmod : `~astropy.units.Quantity` ['length']
Distance modulus at each input redshift, in magnitudes.
See Also
--------
z_at_value : Find the redshift corresponding to a distance modulus.
"""
# Remember that the luminosity distance is in Mpc
# Abs is necessary because in certain obscure closed cosmologies
# the distance modulus can be negative -- which is okay because
# it enters as the square.
val = 5.0 * np.log10(abs(self.luminosity_distance(z).value)) + 25.0
return u.Quantity(val, u.mag)
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def comoving_volume(self, z):
r"""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 :math:`\Omega_k = 0` it is a sphere of radius
`comoving_distance` but it is less intuitive if :math:`\Omega_k` is not.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
V : `~astropy.units.Quantity`
Comoving volume in :math:`Mpc^3` at each input redshift.
"""
Ok0 = self._Ok0
if Ok0 == 0:
return 4.0 / 3.0 * pi * self.comoving_distance(z) ** 3
dh = self._hubble_distance.value # .value for speed
dm = self.comoving_transverse_distance(z).value
term1 = 4.0 * pi * dh**3 / (2.0 * Ok0) * u.Mpc**3
term2 = dm / dh * np.sqrt(1 + Ok0 * (dm / dh) ** 2)
term3 = sqrt(abs(Ok0)) * dm / dh
if Ok0 > 0:
return term1 * (term2 - 1.0 / sqrt(abs(Ok0)) * np.arcsinh(term3))
else:
return term1 * (term2 - 1.0 / sqrt(abs(Ok0)) * np.arcsin(term3))
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def differential_comoving_volume(self, 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, or `~numbers.Number`
Input redshift.
Returns
-------
dV : `~astropy.units.Quantity`
Differential comoving volume per redshift per steradian at each
input redshift.
"""
dm = self.comoving_transverse_distance(z)
return self._hubble_distance * (dm**2.0) / (self.efunc(z) << u.steradian)
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def kpc_comoving_per_arcmin(self, z):
"""Separation in transverse comoving kpc equal to an arcmin at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
d : `~astropy.units.Quantity` ['length']
The distance in comoving kpc corresponding to an arcmin at each
input redshift.
"""
return self.comoving_transverse_distance(z).to(u.kpc) / _radian_in_arcmin
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def kpc_proper_per_arcmin(self, z):
"""Separation in transverse proper kpc equal to an arcminute at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
d : `~astropy.units.Quantity` ['length']
The distance in proper kpc corresponding to an arcmin at each input
redshift.
"""
return self.angular_diameter_distance(z).to(u.kpc) / _radian_in_arcmin
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def arcsec_per_kpc_comoving(self, z):
"""Angular separation in arcsec equal to a comoving kpc at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
theta : `~astropy.units.Quantity` ['angle']
The angular separation in arcsec corresponding to a comoving kpc at
each input redshift.
"""
return _radian_in_arcsec / self.comoving_transverse_distance(z).to(u.kpc)
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def arcsec_per_kpc_proper(self, z):
"""Angular separation in arcsec corresponding to a proper kpc at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
theta : `~astropy.units.Quantity` ['angle']
The angular separation in arcsec corresponding to a proper kpc at
each input redshift.
"""
return _radian_in_arcsec / self.angular_diameter_distance(z).to(u.kpc)
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class FlatFLRWMixin(FlatCosmologyMixin):
"""Mixin class for flat FLRW cosmologies.
Do NOT instantiate directly. Must precede the base class in the
multiple-inheritance so that this mixin's ``__init__`` proceeds the
base class'. Note that all instances of ``FlatFLRWMixin`` are flat, but
not all flat cosmologies are instances of ``FlatFLRWMixin``. As
example, ``LambdaCDM`` **may** be flat (for the a specific set of
parameter values), but ``FlatLambdaCDM`` **will** be flat.
"""
Ode0 = ParameterOde0.clone(derived=True) # same as FLRW, but derived.
def __init_subclass__(cls):
super().__init_subclass__()
if "Ode0" in cls._init_signature.parameters:
raise TypeError(
"subclasses of `FlatFLRWMixin` cannot have `Ode0` in `__init__`"
)
def __init__(self, *args, **kw):
super().__init__(*args, **kw) # guaranteed not to have `Ode0`
# Do some twiddling after the fact to get flatness
self._Ok0 = 0.0
self._Ode0 = 1.0 - (self._Om0 + self._Ogamma0 + self._Onu0 + self._Ok0)
@lazyproperty
def nonflat(self: _FlatFLRWMixinT) -> _FLRWT:
# Create BoundArgument to handle args versus kwargs.
# This also handles all errors from mismatched arguments
ba = self.__nonflatclass__._init_signature.bind_partial(
**self.parameters, Ode0=self.Ode0, name=self.name
)
# Make new instance, respecting args vs kwargs
inst = self.__nonflatclass__(*ba.args, **ba.kwargs)
# Because of machine precision, make sure parameters exactly match
for n in (*inst._parameters_all, "Ok0"):
setattr(inst, "_" + n, getattr(self, n))
return inst
@property
def Otot0(self):
"""Omega total; the total density/critical density at z=0."""
return 1.0
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def Otot(self, z):
"""The total density parameter at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshifts.
Returns
-------
Otot : ndarray or float
Returns float if input scalar. Value of 1.
"""
return (
1.0 if isinstance(z, (Number, np.generic)) else np.ones_like(z, subok=False)
)
