# Licensed under a 3-clause BSD style license - see LICENSE.rst
import sys
from math import acos, sin, cos, sqrt, pi, exp, log, floor
from abc import ABCMeta, abstractmethod
from inspect import signature
import warnings
import numpy as np
from . import scalar_inv_efuncs
from astropy import constants as const
from astropy import units as u
from astropy.utils import isiterable
from astropy.utils.state import ScienceState
from astropy.utils.exceptions import AstropyDeprecationWarning, AstropyUserWarning
from . import parameters
# Originally authored by Andrew Becker (becker@astro.washington.edu),
# and modified by Neil Crighton (neilcrighton@gmail.com) and Roban
# Kramer (robanhk@gmail.com).
# Many of these adapted from Hogg 1999, astro-ph/9905116
# and Linder 2003, PRL 90, 91301
__all__ = ["Cosmology", "FLRW", "LambdaCDM", "FlatLambdaCDM", "wCDM",
"FlatwCDM", "Flatw0waCDM", "w0waCDM", "wpwaCDM", "w0wzCDM"]
__doctest_requires__ = {'*': ['scipy']}
# Notes about speeding up integrals:
# ---------------------------------
# The supplied cosmology classes use a few tricks to speed
# up distance and time integrals. It is not necessary for
# anyone subclassing FLRW to use these tricks -- but if they
# do, such calculations may be a lot faster.
# The first, more basic, idea is that, in many cases, it's a big deal to
# provide explicit formulae for inv_efunc rather than simply
# setting up de_energy_scale -- assuming there is a nice expression.
# As noted above, almost all of the provided classes do this, and
# that template can pretty much be followed directly with the appropriate
# formula changes.
# The second, and more advanced, option is to also explicitly
# provide a scalar only version of inv_efunc. This results in a fairly
# large speedup (>10x in most cases) in the distance and age integrals,
# even if only done in python, because testing whether the inputs are
# iterable or pure scalars turns out to be rather expensive. To take
# advantage of this, the key thing is to explicitly set the
# instance variables self._inv_efunc_scalar and self._inv_efunc_scalar_args
# in the constructor for the subclass, where the latter are all the
# arguments except z to _inv_efunc_scalar.
#
# The provided classes do use this optimization, and in fact go
# even further and provide optimizations for no radiation, and for radiation
# with massless neutrinos coded in cython. Consult the subclasses for
# details, and scalar_inv_efuncs for the details.
#
# However, the important point is that it is -not- necessary to do this.
# Some conversion constants -- useful to compute them once here
# and reuse in the initialization rather than have every object do them
# Note that the call to cgs is actually extremely expensive,
# so we actually skip using the units package directly, and
# hardwire the conversion from mks to cgs. This assumes that constants
# will always return mks by default -- if this is made faster for simple
# cases like this, it should be changed back.
# Note that the unit tests should catch it if this happens
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.value * 1000)
arcsec_in_radians = pi / (3600. * 180)
arcmin_in_radians = pi / (60. * 180)
# Radiation parameter over c^2 in cgs (g cm^-3 K^-4)
a_B_c2 = 4e-3 * const.sigma_sb.value / const.c.value ** 3
# Boltzmann constant in eV / K
kB_evK = const.k_B.to(u.eV / u.K)
class CosmologyError(Exception):
pass
[ドキュメント]class Cosmology(metaclass=ABCMeta):
"""Base-class for all Cosmologies.
Parameters
----------
*args
Arguments into the cosmology; used by subclasses, not this base class.
name : str or None (optional, keyword-only)
The name of the cosmology.
**kwargs
Arguments into the cosmology; used by subclasses, not this base class.
Notes
-----
Class instances are static -- you cannot (and should not) change the values
of the parameters. That is, all of the attributes above are
read only.
"""
def __init_subclass__(cls):
super().__init_subclass__()
# get signature, dropping "self" by taking arguments [1:]
sig = signature(cls.__init__)
sig = sig.replace(parameters=list(sig.parameters.values())[1:])
cls._init_signature = sig # store (immutable) initialization signature
def __new__(cls, *args, **kwargs):
# bundle initialization argument
if not args:
parameters = kwargs
else: # have to merge args and kwargs.
# bind any arguments passed
ba = cls._init_signature.bind_partial(*args, **kwargs)
ba.apply_defaults() # and fill in the defaults
# get dictionary of arguments
parameters = ba.arguments
self = super().__new__(cls)
self._init_arguments = parameters # store arguments from initialization
return self
def __init__(self, *args, name=None, **kwargs):
self._name = name
@property
def name(self):
return self._name
[ドキュメント] def clone(self, **kwargs):
"""Returns a copy of this object with updated parameters, as specified.
This cannot be used to change the type of the cosmology, so ``clone()``
cannot be used to change between flat and non-flat cosmologies. If no
modifications are requested, then a reference to this object is
returned.
Returns
-------
newcosmo : `~astropy.cosmology.Cosmology` subclass instance
A new instance of this class with updated parameters as specified.
Examples
--------
To make a copy of the ``Planck13`` cosmology with a different matter
density (``Om0``):
and a new name:
>>> from astropy.cosmology import Planck13
>>> newcosmo = Planck13.clone(name="Modified Planck 2013", Om0=0.35)
"""
# Quick return check, taking advantage of the
# immutability of cosmological objects
if len(kwargs) == 0:
return self
# There are changed parameter values.
# The name needs to be changed accordingly, if it wasn't already.
kwargs.setdefault("name", (self.name + " (modified)"
if self.name is not None else None))
# Mix kwargs into initial arguments, preferring the former.
full_kwargs = {**self._init_arguments, **kwargs}
# Create BoundArgument to handle args versus kwargs.
# This also handles all errors from mismatched arguments
try:
ba = self._init_signature.bind_partial(**full_kwargs)
except TypeError as e:
# for backward compatibility, map TypeError to AttributeError
warnings.warn("Starting in Astropy v5.0, passing an unrecognized "
"argument will instead raise a TypeError.",
category=AstropyDeprecationWarning)
raise AttributeError(e)
# Return new instance
return self.__class__(*ba.args, **ba.kwargs)
[ドキュメント]class FLRW(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
----------
H0 : float or scalar `~astropy.units.Quantity` ['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 `~astropy.units.Quantity` ['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
Name for this cosmological object.
Notes
-----
Class instances are static -- you cannot change the values
of the parameters. That is, all of the attributes above are
read only.
"""
def __init__(self, H0, Om0, Ode0, Tcmb0=0, Neff=3.04,
m_nu=u.Quantity(0.0, u.eV), Ob0=None, name=None):
super().__init__(name=name)
# all densities are in units of the critical density
self._Om0 = float(Om0)
if self._Om0 < 0.0:
raise ValueError("Matter density can not be negative")
self._Ode0 = float(Ode0)
if Ob0 is not None:
self._Ob0 = float(Ob0)
if self._Ob0 < 0.0:
raise ValueError("Baryonic density can not be negative")
if self._Ob0 > self._Om0:
raise ValueError("Baryonic density can not be larger than "
"total matter density")
self._Odm0 = self._Om0 - self._Ob0
else:
self._Ob0 = None
self._Odm0 = None
self._Neff = float(Neff)
if self._Neff < 0.0:
raise ValueError("Effective number of neutrinos can "
"not be negative")
# Tcmb may have units
self._Tcmb0 = u.Quantity(Tcmb0, unit=u.K)
if not self._Tcmb0.isscalar:
raise ValueError("Tcmb0 is a non-scalar quantity")
# Hubble parameter at z=0, km/s/Mpc
self._H0 = u.Quantity(H0, unit=u.km / u.s / u.Mpc)
if not self._H0.isscalar:
raise ValueError("H0 is a non-scalar quantity")
# 100 km/s/Mpc * h = H0 (so h is dimensionless)
self._h = self._H0.value / 100.
# Hubble distance
self._hubble_distance = (const.c / self._H0).to(u.Mpc)
# H0 in s^-1; don't use units for speed
H0_s = self._H0.value * H0units_to_invs
# Hubble time; again, avoiding units package for speed
self._hubble_time = u.Quantity(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 = u.Quantity(cd0value, u.g / u.cm ** 3)
# Load up neutrino masses. Note: in Py2.x, floor is floating
self._nneutrinos = int(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._massivenu = False
if self._nneutrinos > 0 and self._Tcmb0.value > 0:
self._neff_per_nu = self._Neff / self._nneutrinos
with u.add_enabled_equivalencies(u.mass_energy()):
m_nu = u.Quantity(m_nu, u.eV)
# Now, figure out if we have massive neutrinos to deal with,
# and, if so, get the right number of masses
# It is worth the effort to keep track of massless ones separately
# (since they are quite easy to deal with, and a common use case
# is to set only one neutrino to have mass)
if m_nu.isscalar:
# Assume all neutrinos have the same mass
if m_nu.value == 0:
self._nmasslessnu = self._nneutrinos
self._nmassivenu = 0
else:
self._massivenu = True
self._nmasslessnu = 0
self._nmassivenu = self._nneutrinos
self._massivenu_mass = (m_nu.value *
np.ones(self._nneutrinos))
else:
# Make sure we have the right number of masses
# -unless- they are massless, in which case we cheat a little
if m_nu.value.min() < 0:
raise ValueError("Invalid (negative) neutrino mass"
" encountered")
if m_nu.value.max() == 0:
self._nmasslessnu = self._nneutrinos
self._nmassivenu = 0
else:
self._massivenu = True
if len(m_nu) != self._nneutrinos:
errstr = "Unexpected number of neutrino masses"
raise ValueError(errstr)
# Segregate out the massless ones
self._nmasslessnu = len(np.nonzero(m_nu.value == 0)[0])
self._nmassivenu = self._nneutrinos - self._nmasslessnu
w = np.nonzero(m_nu.value > 0)[0]
self._massivenu_mass = m_nu[w]
# Compute photon density, Tcmb, neutrino parameters
# Tcmb0=0 removes both photons and neutrinos, is handled
# as a special case for efficiency
if self._Tcmb0.value > 0:
# 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 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:
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
else:
self._Ogamma0 = 0.0
self._Tnu0 = u.Quantity(0.0, u.K)
self._Onu0 = 0.0
# 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 = ()
def _namelead(self):
""" Helper function for constructing __repr__"""
if self.name is None:
return f"{self.__class__.__name__}("
else:
return f"{self.__class__.__name__}(name=\"{self.name}\", "
def __repr__(self):
retstr = "{0}H0={1:.3g}, Om0={2:.3g}, Ode0={3:.3g}, "\
"Tcmb0={4:.4g}, Neff={5:.3g}, m_nu={6}, "\
"Ob0={7:s})"
return retstr.format(self._namelead(), self._H0, self._Om0, self._Ode0,
self._Tcmb0, self._Neff, self.m_nu,
_float_or_none(self._Ob0))
# Set up a set of properties for H0, Om0, Ode0, Ok0, etc. for user access.
# Note that we don't let these be set (so, obj.Om0 = value fails)
@property
def H0(self):
""" Return the Hubble constant as an `~astropy.units.Quantity` at z=0"""
return self._H0
@property
def Om0(self):
""" Omega matter; matter density/critical density at z=0"""
return self._Om0
@property
def Ode0(self):
""" Omega dark energy; dark energy density/critical density at z=0"""
return self._Ode0
@property
def Ob0(self):
""" Omega baryon; baryonic matter density/critical density at z=0"""
return self._Ob0
@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 Tcmb0(self):
""" Temperature of the CMB as `~astropy.units.Quantity` at z=0"""
return self._Tcmb0
@property
def Tnu0(self):
""" Temperature of the neutrino background as `~astropy.units.Quantity` at z=0"""
return self._Tnu0
@property
def Neff(self):
""" Number of effective neutrino species"""
return self._Neff
@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 m_nu(self):
""" Mass of neutrino species"""
if self._Tnu0.value == 0:
return None
if not self._massivenu:
# Only massless
return u.Quantity(np.zeros(self._nmasslessnu), u.eV)
if self._nmasslessnu == 0:
# Only massive
return u.Quantity(self._massivenu_mass, u.eV)
# A mix -- the most complicated case
numass = np.append(np.zeros(self._nmasslessnu),
self._massivenu_mass.value)
return u.Quantity(numass, u.eV)
@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
[ドキュメント] @abstractmethod
def w(self, z):
""" The dark energy equation of state.
Parameters
----------
z : array-like
Input redshifts.
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")
[ドキュメント] def Om(self, z):
""" Return the density parameter for non-relativistic matter
at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
Om : ndarray or float
The density of non-relativistic matter relative to the critical
density at each redshift.
Returns float if input scalar.
Notes
-----
This does not include neutrinos, even if non-relativistic
at the redshift of interest; see `Onu`.
"""
if isiterable(z):
z = np.asarray(z)
return self._Om0 * (1. + z) ** 3 * self.inv_efunc(z) ** 2
[ドキュメント] def Ob(self, z):
""" Return the density parameter for baryonic matter at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
Ob : ndarray or float
The density of baryonic matter relative to the critical density at
each redshift.
Returns float if input scalar.
Raises
------
ValueError
If Ob0 is None.
"""
if self._Ob0 is None:
raise ValueError("Baryon density not set for this cosmology")
if isiterable(z):
z = np.asarray(z)
return self._Ob0 * (1. + z) ** 3 * self.inv_efunc(z) ** 2
[ドキュメント] def Odm(self, z):
""" Return the density parameter for dark matter at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
Odm : ndarray or float
The density of non-relativistic dark matter relative to the critical
density at each redshift.
Returns float if input 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")
if isiterable(z):
z = np.asarray(z)
return self._Odm0 * (1. + z) ** 3 * self.inv_efunc(z) ** 2
[ドキュメント] def Ok(self, z):
""" Return the equivalent density parameter for curvature
at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
Ok : ndarray or float
The equivalent density parameter for curvature at each redshift.
Returns float if input scalar.
"""
if isiterable(z):
z = np.asarray(z)
# Common enough case to be worth checking explicitly
if self._Ok0 == 0:
return np.zeros(np.asanyarray(z).shape)
else:
if self._Ok0 == 0:
return 0.0
return self._Ok0 * (1. + z) ** 2 * self.inv_efunc(z) ** 2
[ドキュメント] def Ode(self, z):
""" Return the density parameter for dark energy at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
Ode : ndarray or float
The density of non-relativistic matter relative to the critical
density at each redshift.
Returns float if input scalar.
"""
if isiterable(z):
z = np.asarray(z)
# Common case worth checking
if self._Ode0 == 0:
return np.zeros(np.asanyarray(z).shape)
else:
if self._Ode0 == 0:
return 0.0
return self._Ode0 * self.de_density_scale(z) * self.inv_efunc(z) ** 2
[ドキュメント] def Ogamma(self, z):
""" Return the density parameter for photons at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
Ogamma : ndarray or float
The energy density of photons relative to the critical
density at each redshift.
Returns float if input scalar.
"""
if isiterable(z):
z = np.asarray(z)
return self._Ogamma0 * (1. + z) ** 4 * self.inv_efunc(z) ** 2
[ドキュメント] def Onu(self, z):
""" Return the density parameter for neutrinos at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
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 input scalar.
"""
if isiterable(z):
z = np.asarray(z)
if self._Onu0 == 0:
return np.zeros(np.asanyarray(z).shape)
else:
if self._Onu0 == 0:
return 0.0
return self.Ogamma(z) * self.nu_relative_density(z)
[ドキュメント] def Tcmb(self, z):
""" Return the CMB temperature at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
Tcmb : `~astropy.units.Quantity` ['temperature']
The temperature of the CMB in K.
"""
if isiterable(z):
z = np.asarray(z)
return self._Tcmb0 * (1. + z)
[ドキュメント] def Tnu(self, z):
""" Return the neutrino temperature at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
Tnu : `~astropy.units.Quantity` ['temperature']
The temperature of the cosmic neutrino background in K.
"""
if isiterable(z):
z = np.asarray(z)
return self._Tnu0 * (1. + z)
[ドキュメント] def nu_relative_density(self, z):
""" Neutrino density function relative to the energy density in
photons.
Parameters
----------
z : array-like
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
if not self._massivenu:
if np.isscalar(z):
return prefac * self._Neff
else:
return prefac * self._Neff * np.ones(np.asanyarray(z).shape)
# These are purely fitting constants -- see the Komatsu paper
p = 1.83
invp = 0.54644808743 # 1.0 / p
k = 0.3173
z = np.asarray(z)
curr_nu_y = self._nu_y / (1. + 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."""
# See Linder 2003, PRL 90, 91301 eq (5)
# Assumes scalar input, since this should only be called
# inside an integral
z = exp(ln1pz) - 1.0
return 1.0 + self.w(z)
[ドキュメント] def de_density_scale(self, z):
r""" Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns float if input 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)
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.
"""
# 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.
#
# The integral we actually use (the one given in Linder)
# is rewritten in terms of z, so looks slightly different than the
# one in the documentation string, but it's the same thing.
from scipy.integrate import quad
if isiterable(z):
z = np.asarray(z)
ival = np.array([quad(self._w_integrand, 0, log(1 + redshift))[0]
for redshift in z])
return np.exp(3 * ival)
else:
ival = quad(self._w_integrand, 0, log(1 + z))[0]
return exp(3 * ival)
[ドキュメント] def efunc(self, z):
""" Function used to calculate H(z), the Hubble parameter.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`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.
"""
if isiterable(z):
z = np.asarray(z)
Om0, Ode0, Ok0 = self._Om0, self._Ode0, self._Ok0
if self._massivenu:
Or = self._Ogamma0 * (1 + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1.0 + z
return np.sqrt(zp1 ** 2 * ((Or * zp1 + Om0) * zp1 + Ok0) +
Ode0 * self.de_density_scale(z))
[ドキュメント] def inv_efunc(self, z):
"""Inverse of efunc.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The redshift scaling of the inverse Hubble constant.
Returns float if input scalar.
"""
# Avoid the function overhead by repeating code
if isiterable(z):
z = np.asarray(z)
Om0, Ode0, Ok0 = self._Om0, self._Ode0, self._Ok0
if self._massivenu:
Or = self._Ogamma0 * (1 + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1.0 + z
return (zp1 ** 2 * ((Or * zp1 + Om0) * zp1 + Ok0) +
Ode0 * self.de_density_scale(z))**(-0.5)
def _lookback_time_integrand_scalar(self, z):
""" Integrand of the lookback time.
Parameters
----------
z : float
Input redshift.
Returns
-------
I : float
The integrand for the lookback time
References
----------
Eqn 30 from Hogg 1999.
"""
args = self._inv_efunc_scalar_args
return self._inv_efunc_scalar(z, *args) / (1.0 + z)
[ドキュメント] def lookback_time_integrand(self, z):
""" Integrand of the lookback time.
Parameters
----------
z : float or array-like
Input redshift.
Returns
-------
I : float or array
The integrand for the lookback time
References
----------
Eqn 30 from Hogg 1999.
"""
if isiterable(z):
zp1 = 1.0 + np.asarray(z)
else:
zp1 = 1. + z
return self.inv_efunc(z) / zp1
def _abs_distance_integrand_scalar(self, z):
""" Integrand of the absorption distance.
Parameters
----------
z : float
Input redshift.
Returns
-------
X : float
The integrand for the absorption distance
References
----------
See Hogg 1999 section 11.
"""
args = self._inv_efunc_scalar_args
return (1.0 + z) ** 2 * self._inv_efunc_scalar(z, *args)
[ドキュメント] def abs_distance_integrand(self, z):
""" Integrand of the absorption distance.
Parameters
----------
z : float or array
Input redshift.
Returns
-------
X : float or array
The integrand for the absorption distance
References
----------
See Hogg 1999 section 11.
"""
if isiterable(z):
zp1 = 1.0 + np.asarray(z)
else:
zp1 = 1. + z
return zp1 ** 2 * self.inv_efunc(z)
[ドキュメント] def H(self, z):
""" Hubble parameter (km/s/Mpc) at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
H : `~astropy.units.Quantity` ['frequency']
Hubble parameter at each input redshift.
"""
return self._H0 * self.efunc(z)
[ドキュメント] def scale_factor(self, z):
""" Scale factor at redshift ``z``.
The scale factor is defined as :math:`a = 1 / (1 + z)`.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
a : ndarray or float
Scale factor at each input redshift.
Returns float if input scalar.
"""
if isiterable(z):
z = np.asarray(z)
return 1. / (1. + z)
[ドキュメント] 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 : array-like
Input redshifts. Must be 1D or scalar
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 : array-like
Input redshifts. Must be 1D or scalar
Returns
-------
t : `~astropy.units.Quantity` ['time']
Lookback time in Gyr to each input redshift.
"""
return self._integral_lookback_time(z)
def _integral_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 : array-like
Input redshifts. Must be 1D or scalar
Returns
-------
t : `~astropy.units.Quantity` ['time']
Lookback time in Gyr to each input redshift.
"""
from scipy.integrate import quad
f = lambda red: quad(self._lookback_time_integrand_scalar, 0, red)[0]
return self._hubble_time * vectorize_if_needed(f, z)
[ドキュメント] 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 : array-like
Input redshifts. Must be 1D or scalar
Returns
-------
d : `~astropy.units.Quantity` ['length']
Lookback distance in Mpc
"""
return (self.lookback_time(z) * const.c).to(u.Mpc)
[ドキュメント] def age(self, z):
""" Age of the universe in Gyr at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
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 : array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
t : `~astropy.units.Quantity` ['time']
The age of the universe in Gyr at each input redshift.
"""
return self._integral_age(z)
def _integral_age(self, z):
""" Age of the universe in Gyr at redshift ``z``.
Calculated using explicit integration.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
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.
"""
from scipy.integrate import quad
f = lambda red: quad(self._lookback_time_integrand_scalar,
red, np.inf)[0]
return self._hubble_time * vectorize_if_needed(f, z)
[ドキュメント] def critical_density(self, z):
""" Critical density in grams per cubic cm at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
rho : `~astropy.units.Quantity`
Critical density in g/cm^3 at each input redshift.
"""
return self._critical_density0 * (self.efunc(z)) ** 2
[ドキュメント] 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 : array-like
Input redshifts. Must be 1D or scalar.
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 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 : (N,) array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc between each input redshift.
"""
return self._integral_comoving_distance_z1z2(z1, z2)
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 : (N,) array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc between each input redshift.
"""
from scipy.integrate import quad
f = lambda z1, z2: quad(self._inv_efunc_scalar, z1, z2,
args=self._inv_efunc_scalar_args)[0]
return self._hubble_distance * vectorize_if_needed(f, z1, z2)
[ドキュメント] def comoving_transverse_distance(self, 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 : array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving transverse distance in Mpc at each input redshift.
Notes
-----
This quantity 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):
"""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 omega_k is zero (as in the current
concordance lambda CDM model).
Parameters
----------
z1, z2 : (N,) array-like
Input redshifts. Must be 1D or scalar.
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 * np.sin(sqrtOk0 * dc.value / dh.value)
[ドキュメント] 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``.
Weinberg, 1972, pp 421-424; Weedman, 1986, pp 65-67; Peebles,
1993, pp 325-327.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Angular diameter distance in Mpc at each input redshift.
"""
if isiterable(z):
z = np.asarray(z)
return self.comoving_transverse_distance(z) / (1. + z)
[ドキュメント] 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.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
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
----------
Weinberg, 1972, pp 420-424; Weedman, 1986, pp 60-62.
"""
if isiterable(z):
z = np.asarray(z)
return (1. + z) * self.comoving_transverse_distance(z)
[ドキュメント] 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 : (N,) 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 : (N,) or scalar `~astropy.units.Quantity`
The angular diameter distance between each input redshift
pair.
Returns scalar if input is scalar, array else-wise.
"""
z1 = np.asanyarray(z1)
z2 = np.asanyarray(z2)
if np.any(np.less(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) / (1. + z2)
[ドキュメント] 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.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
d : float 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
"""
from scipy.integrate import quad
f = lambda red: quad(self._abs_distance_integrand_scalar, 0, red)[0]
return vectorize_if_needed(f, z)
[ドキュメント] 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 : array-like
Input redshifts. Must be 1D or scalar.
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. * np.log10(abs(self.luminosity_distance(z).value)) + 25.0
return u.Quantity(val, u.mag)
[ドキュメント] def comoving_volume(self, 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 0.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
V : `~astropy.units.Quantity`
Comoving volume in :math:`Mpc^3` at each input redshift.
"""
Ok0 = self._Ok0
if Ok0 == 0:
return 4. / 3. * pi * self.comoving_distance(z) ** 3
dh = self._hubble_distance.value # .value for speed
dm = self.comoving_transverse_distance(z).value
term1 = 4. * pi * dh ** 3 / (2. * 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. / sqrt(abs(Ok0)) * np.arcsinh(term3))
else:
return term1 * (term2 - 1. / sqrt(abs(Ok0)) * np.arcsin(term3))
[ドキュメント] 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 : array-like
Input redshifts.
Returns
-------
dV : `~astropy.units.Quantity`
Differential comoving volume per redshift per steradian at
each input redshift."""
dh = self._hubble_distance
dm = self.comoving_transverse_distance(z)
return dh * (dm ** 2.0) / u.Quantity(self.efunc(z), u.steradian)
[ドキュメント] def kpc_comoving_per_arcmin(self, z):
""" Separation in transverse comoving kpc corresponding to an
arcminute at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
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) *
arcmin_in_radians / u.arcmin)
[ドキュメント] def kpc_proper_per_arcmin(self, z):
""" Separation in transverse proper kpc corresponding to an
arcminute at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
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) *
arcmin_in_radians / u.arcmin)
[ドキュメント] def arcsec_per_kpc_comoving(self, z):
""" Angular separation in arcsec corresponding to a comoving kpc
at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
theta : `~astropy.units.Quantity` ['angle']
The angular separation in arcsec corresponding to a comoving kpc
at each input redshift.
"""
return u.arcsec / (self.comoving_transverse_distance(z).to(u.kpc) *
arcsec_in_radians)
[ドキュメント] def arcsec_per_kpc_proper(self, z):
""" Angular separation in arcsec corresponding to a proper kpc at
redshift ``z``.
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
theta : `~astropy.units.Quantity` ['angle']
The angular separation in arcsec corresponding to a proper kpc
at each input redshift.
"""
return u.arcsec / (self.angular_diameter_distance(z).to(u.kpc) *
arcsec_in_radians)
[ドキュメント]class LambdaCDM(FLRW):
"""FLRW cosmology with a cosmological constant and curvature.
This has no additional attributes beyond those of FLRW.
Parameters
----------
H0 : float or `~astropy.units.Quantity` ['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.
Ode0 : float
Omega dark energy: density of the cosmological constant in units of
the critical density at z=0.
Tcmb0 : float or scalar `~astropy.units.Quantity` ['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
Name for this cosmological object.
Examples
--------
>>> from astropy.cosmology import LambdaCDM
>>> cosmo = LambdaCDM(H0=70, Om0=0.3, Ode0=0.7)
The comoving distance in Mpc at redshift z:
>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)
"""
def __init__(self, H0, Om0, Ode0, Tcmb0=0, Neff=3.04,
m_nu=u.Quantity(0.0, u.eV), Ob0=None, name=None):
super().__init__(H0, Om0, Ode0, Tcmb0, Neff, m_nu, Ob0=Ob0, name=name)
# Please see "Notes about speeding up integrals" for discussion
# about what is being done here.
if self._Tcmb0.value == 0:
self._inv_efunc_scalar = scalar_inv_efuncs.lcdm_inv_efunc_norel
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0)
if self._Ok0 == 0:
self._optimize_flat_norad()
else:
self._comoving_distance_z1z2 = \
self._elliptic_comoving_distance_z1z2
elif not self._massivenu:
self._inv_efunc_scalar = scalar_inv_efuncs.lcdm_inv_efunc_nomnu
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0 + self._Onu0)
else:
self._inv_efunc_scalar = scalar_inv_efuncs.lcdm_inv_efunc
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0, self._neff_per_nu,
self._nmasslessnu,
self._nu_y_list)
def _optimize_flat_norad(self):
"""Set optimizations for flat LCDM cosmologies with no radiation.
"""
# Call out the Om0=0 (de Sitter) and Om0=1 (Einstein-de Sitter)
# The dS case is required because the hypergeometric case
# for Omega_M=0 would lead to an infinity in its argument.
# The EdS case is three times faster than the hypergeometric.
if self._Om0 == 0:
self._comoving_distance_z1z2 = \
self._dS_comoving_distance_z1z2
self._age = self._dS_age
self._lookback_time = self._dS_lookback_time
elif self._Om0 == 1:
self._comoving_distance_z1z2 = \
self._EdS_comoving_distance_z1z2
self._age = self._EdS_age
self._lookback_time = self._EdS_lookback_time
else:
self._comoving_distance_z1z2 = \
self._hypergeometric_comoving_distance_z1z2
self._age = self._flat_age
self._lookback_time = self._flat_lookback_time
[ドキュメント] def w(self, z):
"""Returns dark energy equation of state at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
w : ndarray or float
The dark energy equation of state.
Returns float if input scalar.
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. Here this is
:math:`w(z) = -1`.
"""
if np.isscalar(z):
return -1.0
else:
return -1.0 * np.ones(np.asanyarray(z).shape)
[ドキュメント] def de_density_scale(self, z):
""" Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns float if input scalar.
Notes
-----
The scaling factor, I, is defined by :math:`\\rho(z) = \\rho_0 I`,
and in this case is given by :math:`I = 1`.
"""
if np.isscalar(z):
return 1.
else:
return np.ones(np.asanyarray(z).shape)
def _elliptic_comoving_distance_z1z2(self, z1, z2):
""" Comoving transverse distance in Mpc between two redshifts.
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.
For Omega_rad = 0 the comoving distance can be directly calculated
as an elliptic integral.
Equation here taken from Kantowski, Kao, and Thomas, arXiv:0002334
Not valid or appropriate for flat cosmologies (Ok0=0).
Parameters
----------
z1, z2 : array-like
Input redshifts.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc between each input redshift.
"""
from scipy.special import ellipkinc
try:
z1, z2 = np.broadcast_arrays(z1, z2)
except ValueError as e:
raise ValueError("z1 and z2 have different shapes") from e
# The analytic solution is not valid for any of Om0, Ode0, Ok0 == 0.
# Use the explicit integral solution for these cases.
if self._Om0 == 0 or self._Ode0 == 0 or self._Ok0 == 0:
return self._integral_comoving_distance_z1z2(z1, z2)
b = -(27. / 2) * self._Om0**2 * self._Ode0 / self._Ok0**3
kappa = b / abs(b)
if (b < 0) or (2 < b):
def phi_z(Om0, Ok0, kappa, y1, A, z):
return np.arccos(((1 + z) * Om0 / abs(Ok0) + kappa * y1 - A) /
((1 + z) * Om0 / abs(Ok0) + kappa * y1 + A))
v_k = pow(kappa * (b - 1) + sqrt(b * (b - 2)), 1. / 3)
y1 = (-1 + kappa * (v_k + 1 / v_k)) / 3
A = sqrt(y1 * (3 * y1 + 2))
g = 1 / sqrt(A)
k2 = (2 * A + kappa * (1 + 3 * y1)) / (4 * A)
phi_z1 = phi_z(self._Om0, self._Ok0, kappa, y1, A, z1)
phi_z2 = phi_z(self._Om0, self._Ok0, kappa, y1, A, z2)
# Get lower-right 0<b<2 solution in Om0, Ode0 plane.
# Fot the upper-left 0<b<2 solution the Big Bang didn't happen.
elif (0 < b) and (b < 2) and self._Om0 > self._Ode0:
def phi_z(Om0, Ok0, y1, y2, z):
return np.arcsin(np.sqrt((y1 - y2) /
((1 + z) * Om0 / abs(Ok0) + y1)))
yb = cos(acos(1 - b) / 3)
yc = sqrt(3) * sin(acos(1 - b) / 3)
y1 = (1. / 3) * (-1 + yb + yc)
y2 = (1. / 3) * (-1 - 2 * yb)
y3 = (1. / 3) * (-1 + yb - yc)
g = 2 / sqrt(y1 - y2)
k2 = (y1 - y3) / (y1 - y2)
phi_z1 = phi_z(self._Om0, self._Ok0, y1, y2, z1)
phi_z2 = phi_z(self._Om0, self._Ok0, y1, y2, z2)
else:
return self._integral_comoving_distance_z1z2(z1, z2)
prefactor = self._hubble_distance / sqrt(abs(self._Ok0))
return prefactor * g * (ellipkinc(phi_z1, k2) - ellipkinc(phi_z2, k2))
def _dS_comoving_distance_z1z2(self, z1, z2):
""" Comoving line-of-sight distance in Mpc between objects at redshifts
z1 and z2 in a flat, Omega_Lambda=1 cosmology (de Sitter).
The comoving distance along the line-of-sight between two
objects remains constant with time for objects in the Hubble
flow.
The de Sitter case has an analytic solution.
Parameters
----------
z1, z2 : (N,) array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc between each input redshift.
"""
try:
z1, z2 = np.broadcast_arrays(z1, z2)
except ValueError as e:
raise ValueError("z1 and z2 have different shapes") from e
return self._hubble_distance * (z2 - z1)
def _EdS_comoving_distance_z1z2(self, z1, z2):
""" Comoving line-of-sight distance in Mpc between objects at redshifts
z1 and z2 in a flat, Omega_M=1 cosmology (Einstein - de Sitter).
The comoving distance along the line-of-sight between two
objects remains constant with time for objects in the Hubble
flow.
For OM=1, Omega_rad=0 the comoving distance has an analytic solution.
Parameters
----------
z1, z2 : (N,) array-like
Input redshifts. Must be 1D or scalar.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc between each input redshift.
"""
try:
z1, z2 = np.broadcast_arrays(z1, z2)
except ValueError as e:
raise ValueError("z1 and z2 have different shapes") from e
prefactor = 2 * self._hubble_distance
return prefactor * ((1+z1)**(-1./2) - (1+z2)**(-1./2))
def _hypergeometric_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.
For Omega_radiation = 0 the comoving distance can be directly calculated
as a hypergeometric function.
Equation here taken from Baes, Camps, Van De Putte, 2017, MNRAS, 468, 927.
Parameters
----------
z1, z2 : array-like
Input redshifts.
Returns
-------
d : `~astropy.units.Quantity` ['length']
Comoving distance in Mpc between each input redshift.
"""
try:
z1, z2 = np.broadcast_arrays(z1, z2)
except ValueError as e:
raise ValueError("z1 and z2 have different shapes") from e
s = ((1 - self._Om0) / self._Om0) ** (1./3)
# Use np.sqrt here to handle negative s (Om0>1).
prefactor = self._hubble_distance / np.sqrt(s * self._Om0)
return prefactor * (self._T_hypergeometric(s / (1 + z1)) -
self._T_hypergeometric(s / (1 + z2)))
def _T_hypergeometric(self, x):
""" Compute T_hypergeometric(x) using Gauss Hypergeometric function 2F1
T(x) = 2 \\sqrt(x) _{2}F_{1} \\left(\\frac{1}{6}, \\frac{1}{2}; \\frac{7}{6}; -x^3)
Note:
The scipy.special.hyp2f1 code already implements the hypergeometric
transformation suggested by Baes, Camps, Van De Putte, 2017, MNRAS, 468, 927.
for use in actual numerical evaulations.
"""
from scipy.special import hyp2f1
return 2 * np.sqrt(x) * hyp2f1(1./6, 1./2, 7./6, -x**3)
def _dS_age(self, z):
""" Age of the universe in Gyr at redshift ``z``.
The age of a de Sitter Universe is infinite.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
t : `~astropy.units.Quantity` ['time']
The age of the universe in Gyr at each input redshift.
"""
return self._hubble_time * inf_like(z)
def _EdS_age(self, z):
""" Age of the universe in Gyr at redshift ``z``.
For Omega_radiation = 0 (T_CMB = 0; massless neutrinos)
the age can be directly calculated as an elliptic integral.
See, e.g., Thomas and Kantowski, arXiv:0003463
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
t : `~astropy.units.Quantity` ['time']
The age of the universe in Gyr at each input redshift.
"""
if isiterable(z):
z = np.asarray(z)
return (2./3) * self._hubble_time * (1+z)**(-3./2)
def _flat_age(self, z):
""" Age of the universe in Gyr at redshift ``z``.
For Omega_radiation = 0 (T_CMB = 0; massless neutrinos)
the age can be directly calculated as an elliptic integral.
See, e.g., Thomas and Kantowski, arXiv:0003463
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
t : `~astropy.units.Quantity` ['time']
The age of the universe in Gyr at each input redshift.
"""
if isiterable(z):
z = np.asarray(z)
# Use np.sqrt, np.arcsinh instead of math.sqrt, math.asinh
# to handle properly the complex numbers for 1 - Om0 < 0
prefactor = (2./3) * self._hubble_time / \
np.lib.scimath.sqrt(1 - self._Om0)
arg = np.arcsinh(np.lib.scimath.sqrt((1 / self._Om0 - 1 + 0j) /
(1 + z)**3))
return (prefactor * arg).real
def _EdS_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``.
For Omega_radiation = 0 (T_CMB = 0; massless neutrinos)
the age can be directly calculated as an elliptic integral.
The lookback time is here calculated based on the age(0) - age(z)
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar
Returns
-------
t : `~astropy.units.Quantity` ['time']
Lookback time in Gyr to each input redshift.
"""
return self._EdS_age(0) - self._EdS_age(z)
def _dS_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``.
For Omega_radiation = 0 (T_CMB = 0; massless neutrinos)
the age can be directly calculated.
a = exp(H * t) where t=0 at z=0
t = (1/H) (ln 1 - ln a) = (1/H) (0 - ln (1/(1+z))) = (1/H) ln(1+z)
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
t : `~astropy.units.Quantity` ['time']
Lookback time in Gyr to each input redshift.
"""
if isiterable(z):
z = np.asarray(z)
return self._hubble_time * np.log(1+z)
def _flat_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``.
For Omega_radiation = 0 (T_CMB = 0; massless neutrinos)
the age can be directly calculated.
The lookback time is here calculated based on the age(0) - age(z)
Parameters
----------
z : array-like
Input redshifts. Must be 1D or scalar
Returns
-------
t : `~astropy.units.Quantity` ['time']
Lookback time in Gyr to each input redshift.
"""
return self._flat_age(0) - self._flat_age(z)
[ドキュメント] def efunc(self, z):
""" Function used to calculate H(z), the Hubble parameter.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`H(z) = H_0 E`.
"""
if isiterable(z):
z = np.asarray(z)
# We override this because it takes a particularly simple
# form for a cosmological constant
Om0, Ode0, Ok0 = self._Om0, self._Ode0, self._Ok0
if self._massivenu:
Or = self._Ogamma0 * (1. + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1.0 + z
return np.sqrt(zp1 ** 2 * ((Or * zp1 + Om0) * zp1 + Ok0) + Ode0)
[ドキュメント] def inv_efunc(self, z):
r""" Function used to calculate :math:`\frac{1}{H_z}`.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The inverse redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`H_z = H_0 /
E`.
"""
if isiterable(z):
z = np.asarray(z)
Om0, Ode0, Ok0 = self._Om0, self._Ode0, self._Ok0
if self._massivenu:
Or = self._Ogamma0 * (1 + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1.0 + z
return (zp1 ** 2 * ((Or * zp1 + Om0) * zp1 + Ok0) + Ode0)**(-0.5)
[ドキュメント]class FlatLambdaCDM(LambdaCDM):
"""FLRW cosmology with a cosmological constant and no curvature.
This has no additional attributes beyond those of FLRW.
Parameters
----------
H0 : float or `~astropy.units.Quantity` ['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.
Tcmb0 : float or scalar `~astropy.units.Quantity` ['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
Name for this cosmological object.
Examples
--------
>>> from astropy.cosmology import FlatLambdaCDM
>>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3)
The comoving distance in Mpc at redshift z:
>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)
"""
def __init__(self, H0, Om0, Tcmb0=0, Neff=3.04,
m_nu=u.Quantity(0.0, u.eV), Ob0=None, name=None):
super().__init__(H0, Om0, 0.0, Tcmb0, Neff, m_nu, Ob0=Ob0, name=name)
# Do some twiddling after the fact to get flatness
self._Ode0 = 1.0 - self._Om0 - self._Ogamma0 - self._Onu0
self._Ok0 = 0.0
# Please see "Notes about speeding up integrals" for discussion
# about what is being done here.
if self._Tcmb0.value == 0:
self._inv_efunc_scalar = scalar_inv_efuncs.flcdm_inv_efunc_norel
self._inv_efunc_scalar_args = (self._Om0, self._Ode0)
# Repeat the optimization reassignments here because the init
# of the LambaCDM above didn't actually create a flat cosmology.
# That was done through the explicit tweak setting self._Ok0.
self._optimize_flat_norad()
elif not self._massivenu:
self._inv_efunc_scalar = scalar_inv_efuncs.flcdm_inv_efunc_nomnu
self._inv_efunc_scalar_args = (self._Om0, self._Ode0,
self._Ogamma0 + self._Onu0)
else:
self._inv_efunc_scalar = scalar_inv_efuncs.flcdm_inv_efunc
self._inv_efunc_scalar_args = (self._Om0, self._Ode0,
self._Ogamma0, self._neff_per_nu,
self._nmasslessnu,
self._nu_y_list)
[ドキュメント] def efunc(self, z):
""" Function used to calculate H(z), the Hubble parameter.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`H(z) = H_0 E`.
"""
if isiterable(z):
z = np.asarray(z)
# We override this because it takes a particularly simple
# form for a cosmological constant
Om0, Ode0 = self._Om0, self._Ode0
if self._massivenu:
Or = self._Ogamma0 * (1 + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1.0 + z
return np.sqrt(zp1 ** 3 * (Or * zp1 + Om0) + Ode0)
[ドキュメント] def inv_efunc(self, z):
r"""Function used to calculate :math:`\frac{1}{H_z}`.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The inverse redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`H_z = H_0 / E`.
"""
if isiterable(z):
z = np.asarray(z)
Om0, Ode0 = self._Om0, self._Ode0
if self._massivenu:
Or = self._Ogamma0 * (1. + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1.0 + z
return (zp1 ** 3 * (Or * zp1 + Om0) + Ode0)**(-0.5)
def __repr__(self):
retstr = "{0}H0={1:.3g}, Om0={2:.3g}, Tcmb0={3:.4g}, "\
"Neff={4:.3g}, m_nu={5}, Ob0={6:s})"
return retstr.format(self._namelead(), self._H0, self._Om0,
self._Tcmb0, self._Neff, self.m_nu,
_float_or_none(self._Ob0))
[ドキュメント]class wCDM(FLRW):
"""FLRW cosmology with a constant dark energy equation of state
and curvature.
This has one additional attribute beyond those of FLRW.
Parameters
----------
H0 : float or `~astropy.units.Quantity` ['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.
Ode0 : float
Omega dark energy: density of dark energy in units of the critical
density at z=0.
w0 : float, optional
Dark energy equation of state at all redshifts. This is
pressure/density for dark energy in units where c=1. A cosmological
constant has w0=-1.0.
Tcmb0 : float or scalar `~astropy.units.Quantity` ['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
Name for this cosmological object.
Examples
--------
>>> from astropy.cosmology import wCDM
>>> cosmo = wCDM(H0=70, Om0=0.3, Ode0=0.7, w0=-0.9)
The comoving distance in Mpc at redshift z:
>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)
"""
def __init__(self, H0, Om0, Ode0, w0=-1., Tcmb0=0,
Neff=3.04, m_nu=u.Quantity(0.0, u.eV), Ob0=None, name=None):
super().__init__(H0, Om0, Ode0, Tcmb0, Neff, m_nu, Ob0=Ob0, name=name)
self._w0 = float(w0)
# Please see "Notes about speeding up integrals" for discussion
# about what is being done here.
if self._Tcmb0.value == 0:
self._inv_efunc_scalar = scalar_inv_efuncs.wcdm_inv_efunc_norel
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._w0)
elif not self._massivenu:
self._inv_efunc_scalar = scalar_inv_efuncs.wcdm_inv_efunc_nomnu
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0 + self._Onu0,
self._w0)
else:
self._inv_efunc_scalar = scalar_inv_efuncs.wcdm_inv_efunc
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0, self._neff_per_nu,
self._nmasslessnu,
self._nu_y_list, self._w0)
@property
def w0(self):
""" Dark energy equation of state"""
return self._w0
[ドキュメント] def w(self, z):
"""Returns dark energy equation of state at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
w : ndarray or float
The dark energy equation of state
Returns float if input scalar.
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. Here this is
:math:`w(z) = w_0`.
"""
if np.isscalar(z):
return self._w0
else:
return self._w0 * np.ones(np.asanyarray(z).shape)
[ドキュメント] def de_density_scale(self, z):
""" Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns float if input scalar.
Notes
-----
The scaling factor, I, is defined by :math:`\\rho(z) = \\rho_0 I`,
and in this case is given by
:math:`I = \\left(1 + z\\right)^{3\\left(1 + w_0\\right)}`
"""
if isiterable(z):
z = np.asarray(z)
return (1. + z) ** (3. * (1. + self._w0))
[ドキュメント] def efunc(self, z):
""" Function used to calculate H(z), the Hubble parameter.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`H(z) = H_0 E`.
"""
if isiterable(z):
z = np.asarray(z)
Om0, Ode0, Ok0, w0 = self._Om0, self._Ode0, self._Ok0, self._w0
if self._massivenu:
Or = self._Ogamma0 * (1. + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1.0 + z
return np.sqrt(zp1 ** 2 * ((Or * zp1 + Om0) * zp1 + Ok0) +
Ode0 * zp1 ** (3. * (1. + w0)))
[ドキュメント] def inv_efunc(self, z):
r""" Function used to calculate :math:`\frac{1}{H_z}`.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The inverse redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`H_z = H_0 / E`.
"""
if isiterable(z):
z = np.asarray(z)
Om0, Ode0, Ok0, w0 = self._Om0, self._Ode0, self._Ok0, self._w0
if self._massivenu:
Or = self._Ogamma0 * (1. + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1.0 + z
return (zp1 ** 2 * ((Or * zp1 + Om0) * zp1 + Ok0) +
Ode0 * zp1 ** (3. * (1. + w0)))**(-0.5)
def __repr__(self):
retstr = "{0}H0={1:.3g}, Om0={2:.3g}, Ode0={3:.3g}, w0={4:.3g}, "\
"Tcmb0={5:.4g}, Neff={6:.3g}, m_nu={7}, Ob0={8:s})"
return retstr.format(self._namelead(), self._H0, self._Om0,
self._Ode0, self._w0, self._Tcmb0, self._Neff,
self.m_nu, _float_or_none(self._Ob0))
[ドキュメント]class FlatwCDM(wCDM):
"""FLRW cosmology with a constant dark energy equation of state
and no spatial curvature.
This has one additional attribute beyond those of FLRW.
Parameters
----------
H0 : float or `~astropy.units.Quantity` ['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.
w0 : float, optional
Dark energy equation of state at all redshifts. This is
pressure/density for dark energy in units where c=1. A cosmological
constant has w0=-1.0.
Tcmb0 : float or scalar `~astropy.units.Quantity` ['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
Name for this cosmological object.
Examples
--------
>>> from astropy.cosmology import FlatwCDM
>>> cosmo = FlatwCDM(H0=70, Om0=0.3, w0=-0.9)
The comoving distance in Mpc at redshift z:
>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)
"""
def __init__(self, H0, Om0, w0=-1., Tcmb0=0,
Neff=3.04, m_nu=u.Quantity(0.0, u.eV), Ob0=None, name=None):
super().__init__(H0, Om0, 0., w0, Tcmb0, Neff, m_nu, Ob0=Ob0, name=name)
# Do some twiddling after the fact to get flatness
self._Ode0 = 1.0 - self._Om0 - self._Ogamma0 - self._Onu0
self._Ok0 = 0.0
# Please see "Notes about speeding up integrals" for discussion
# about what is being done here.
if self._Tcmb0.value == 0:
self._inv_efunc_scalar = scalar_inv_efuncs.fwcdm_inv_efunc_norel
self._inv_efunc_scalar_args = (self._Om0, self._Ode0,
self._w0)
elif not self._massivenu:
self._inv_efunc_scalar = scalar_inv_efuncs.fwcdm_inv_efunc_nomnu
self._inv_efunc_scalar_args = (self._Om0, self._Ode0,
self._Ogamma0 + self._Onu0,
self._w0)
else:
self._inv_efunc_scalar = scalar_inv_efuncs.fwcdm_inv_efunc
self._inv_efunc_scalar_args = (self._Om0, self._Ode0,
self._Ogamma0, self._neff_per_nu,
self._nmasslessnu,
self._nu_y_list, self._w0)
[ドキュメント] def efunc(self, z):
""" Function used to calculate H(z), the Hubble parameter.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`H(z) = H_0 E`.
"""
if isiterable(z):
z = np.asarray(z)
Om0, Ode0, w0 = self._Om0, self._Ode0, self._w0
if self._massivenu:
Or = self._Ogamma0 * (1. + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1. + z
return np.sqrt(zp1 ** 3 * (Or * zp1 + Om0) +
Ode0 * zp1 ** (3. * (1 + w0)))
[ドキュメント] def inv_efunc(self, z):
r""" Function used to calculate :math:`\frac{1}{H_z}`.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
E : ndarray or float
The inverse redshift scaling of the Hubble constant.
Returns float if input scalar.
Notes
-----
The return value, E, is defined such that :math:`H_z = H_0 / E`.
"""
if isiterable(z):
z = np.asarray(z)
Om0, Ode0, w0 = self._Om0, self._Ode0, self._w0
if self._massivenu:
Or = self._Ogamma0 * (1. + self.nu_relative_density(z))
else:
Or = self._Ogamma0 + self._Onu0
zp1 = 1. + z
return (zp1 ** 3 * (Or * zp1 + Om0) +
Ode0 * zp1 ** (3. * (1. + w0)))**(-0.5)
def __repr__(self):
retstr = "{0}H0={1:.3g}, Om0={2:.3g}, w0={3:.3g}, Tcmb0={4:.4g}, "\
"Neff={5:.3g}, m_nu={6}, Ob0={7:s})"
return retstr.format(self._namelead(), self._H0, self._Om0, self._w0,
self._Tcmb0, self._Neff, self.m_nu,
_float_or_none(self._Ob0))
[ドキュメント]class w0waCDM(FLRW):
"""FLRW cosmology with a CPL dark energy equation of state and curvature.
The equation for the dark energy equation of state uses the
CPL form as described in Chevallier & Polarski Int. J. Mod. Phys.
D10, 213 (2001) and Linder PRL 90, 91301 (2003):
:math:`w(z) = w_0 + w_a (1-a) = w_0 + w_a z / (1+z)`.
Parameters
----------
H0 : float or `~astropy.units.Quantity` ['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.
Ode0 : float
Omega dark energy: density of dark energy in units of the critical
density at z=0.
w0 : float, optional
Dark energy equation of state at z=0 (a=1). This is pressure/density
for dark energy in units where c=1.
wa : float, optional
Negative derivative of the dark energy equation of state with respect
to the scale factor. A cosmological constant has w0=-1.0 and wa=0.0.
Tcmb0 : float or scalar `~astropy.units.Quantity` ['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
Name for this cosmological object.
Examples
--------
>>> from astropy.cosmology import w0waCDM
>>> cosmo = w0waCDM(H0=70, Om0=0.3, Ode0=0.7, w0=-0.9, wa=0.2)
The comoving distance in Mpc at redshift z:
>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)
"""
def __init__(self, H0, Om0, Ode0, w0=-1., wa=0., Tcmb0=0,
Neff=3.04, m_nu=u.Quantity(0.0, u.eV), Ob0=None, name=None):
super().__init__(H0, Om0, Ode0, Tcmb0, Neff, m_nu, Ob0=Ob0, name=name)
self._w0 = float(w0)
self._wa = float(wa)
# Please see "Notes about speeding up integrals" for discussion
# about what is being done here.
if self._Tcmb0.value == 0:
self._inv_efunc_scalar = scalar_inv_efuncs.w0wacdm_inv_efunc_norel
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._w0, self._wa)
elif not self._massivenu:
self._inv_efunc_scalar = scalar_inv_efuncs.w0wacdm_inv_efunc_nomnu
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0 + self._Onu0,
self._w0, self._wa)
else:
self._inv_efunc_scalar = scalar_inv_efuncs.w0wacdm_inv_efunc
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0, self._neff_per_nu,
self._nmasslessnu,
self._nu_y_list, self._w0,
self._wa)
@property
def w0(self):
""" Dark energy equation of state at z=0"""
return self._w0
@property
def wa(self):
""" Negative derivative of dark energy equation of state w.r.t. a"""
return self._wa
[ドキュメント] def w(self, z):
"""Returns dark energy equation of state at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
w : ndarray or float
The dark energy equation of state
Returns float if input scalar.
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. Here this is
:math:`w(z) = w_0 + w_a (1 - a) = w_0 + w_a \\frac{z}{1+z}`.
"""
if isiterable(z):
z = np.asarray(z)
return self._w0 + self._wa * z / (1.0 + z)
[ドキュメント] def de_density_scale(self, z):
r""" Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns float if input scalar.
Notes
-----
The scaling factor, I, is defined by :math:`\\rho(z) = \\rho_0 I`,
and in this case is given by
.. math::
I = \left(1 + z\right)^{3 \left(1 + w_0 + w_a\right)}
\exp \left(-3 w_a \frac{z}{1+z}\right)
"""
if isiterable(z):
z = np.asarray(z)
zp1 = 1.0 + z
return zp1 ** (3 * (1 + self._w0 + self._wa)) * \
np.exp(-3 * self._wa * z / zp1)
def __repr__(self):
retstr = "{0}H0={1:.3g}, Om0={2:.3g}, "\
"Ode0={3:.3g}, w0={4:.3g}, wa={5:.3g}, Tcmb0={6:.4g}, "\
"Neff={7:.3g}, m_nu={8}, Ob0={9:s})"
return retstr.format(self._namelead(), self._H0, self._Om0,
self._Ode0, self._w0, self._wa,
self._Tcmb0, self._Neff, self.m_nu,
_float_or_none(self._Ob0))
[ドキュメント]class Flatw0waCDM(w0waCDM):
"""FLRW cosmology with a CPL dark energy equation of state and no
curvature.
The equation for the dark energy equation of state uses the
CPL form as described in Chevallier & Polarski Int. J. Mod. Phys.
D10, 213 (2001) and Linder PRL 90, 91301 (2003):
:math:`w(z) = w_0 + w_a (1-a) = w_0 + w_a z / (1+z)`.
Parameters
----------
H0 : float or `~astropy.units.Quantity` ['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.
w0 : float, optional
Dark energy equation of state at z=0 (a=1). This is pressure/density
for dark energy in units where c=1.
wa : float, optional
Negative derivative of the dark energy equation of state with respect
to the scale factor. A cosmological constant has w0=-1.0 and wa=0.0.
Tcmb0 : float or scalar `~astropy.units.Quantity` ['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
Name for this cosmological object.
Examples
--------
>>> from astropy.cosmology import Flatw0waCDM
>>> cosmo = Flatw0waCDM(H0=70, Om0=0.3, w0=-0.9, wa=0.2)
The comoving distance in Mpc at redshift z:
>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)
"""
def __init__(self, H0, Om0, w0=-1., wa=0., Tcmb0=0,
Neff=3.04, m_nu=u.Quantity(0.0, u.eV), Ob0=None, name=None):
super().__init__(H0, Om0, 0.0, w0=w0, wa=wa, Tcmb0=Tcmb0,
Neff=Neff, m_nu=m_nu, Ob0=Ob0, name=name)
# Do some twiddling after the fact to get flatness
self._Ode0 = 1.0 - self._Om0 - self._Ogamma0 - self._Onu0
self._Ok0 = 0.0
# Please see "Notes about speeding up integrals" for discussion
# about what is being done here.
if self._Tcmb0.value == 0:
self._inv_efunc_scalar = scalar_inv_efuncs.fw0wacdm_inv_efunc_norel
self._inv_efunc_scalar_args = (self._Om0, self._Ode0,
self._w0, self._wa)
elif not self._massivenu:
self._inv_efunc_scalar = scalar_inv_efuncs.fw0wacdm_inv_efunc_nomnu
self._inv_efunc_scalar_args = (self._Om0, self._Ode0,
self._Ogamma0 + self._Onu0,
self._w0, self._wa)
else:
self._inv_efunc_scalar = scalar_inv_efuncs.fw0wacdm_inv_efunc
self._inv_efunc_scalar_args = (self._Om0, self._Ode0,
self._Ogamma0, self._neff_per_nu,
self._nmasslessnu,
self._nu_y_list, self._w0,
self._wa)
def __repr__(self):
retstr = "{0}H0={1:.3g}, Om0={2:.3g}, "\
"w0={3:.3g}, Tcmb0={4:.4g}, Neff={5:.3g}, m_nu={6}, "\
"Ob0={7:s})"
return retstr.format(self._namelead(), self._H0, self._Om0, self._w0,
self._Tcmb0, self._Neff, self.m_nu,
_float_or_none(self._Ob0))
[ドキュメント]class wpwaCDM(FLRW):
"""FLRW cosmology with a CPL dark energy equation of state, a pivot
redshift, and curvature.
The equation for the dark energy equation of state uses the
CPL form as described in Chevallier & Polarski Int. J. Mod. Phys.
D10, 213 (2001) and Linder PRL 90, 91301 (2003), but modified
to have a pivot redshift as in the findings of the Dark Energy
Task Force (Albrecht et al. arXiv:0901.0721 (2009)):
:math:`w(a) = w_p + w_a (a_p - a) = w_p + w_a( 1/(1+zp) - 1/(1+z) )`.
Parameters
----------
H0 : float or `~astropy.units.Quantity` ['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.
Ode0 : float
Omega dark energy: density of dark energy in units of the critical
density at z=0.
wp : float, optional
Dark energy equation of state at the pivot redshift zp. This is
pressure/density for dark energy in units where c=1.
wa : float, optional
Negative derivative of the dark energy equation of state with respect
to the scale factor. A cosmological constant has wp=-1.0 and wa=0.0.
zp : float, optional
Pivot redshift -- the redshift where w(z) = wp
Tcmb0 : float or scalar `~astropy.units.Quantity` ['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
Name for this cosmological object.
Examples
--------
>>> from astropy.cosmology import wpwaCDM
>>> cosmo = wpwaCDM(H0=70, Om0=0.3, Ode0=0.7, wp=-0.9, wa=0.2, zp=0.4)
The comoving distance in Mpc at redshift z:
>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)
"""
def __init__(self, H0, Om0, Ode0, wp=-1., wa=0., zp=0,
Tcmb0=0, Neff=3.04, m_nu=u.Quantity(0.0, u.eV),
Ob0=None, name=None):
super().__init__(H0, Om0, Ode0, Tcmb0, Neff, m_nu, Ob0=Ob0, name=name)
self._wp = float(wp)
self._wa = float(wa)
self._zp = float(zp)
# Please see "Notes about speeding up integrals" for discussion
# about what is being done here.
apiv = 1.0 / (1.0 + self._zp)
if self._Tcmb0.value == 0:
self._inv_efunc_scalar = scalar_inv_efuncs.wpwacdm_inv_efunc_norel
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._wp, apiv, self._wa)
elif not self._massivenu:
self._inv_efunc_scalar = scalar_inv_efuncs.wpwacdm_inv_efunc_nomnu
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0 + self._Onu0,
self._wp, apiv, self._wa)
else:
self._inv_efunc_scalar = scalar_inv_efuncs.wpwacdm_inv_efunc
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0, self._neff_per_nu,
self._nmasslessnu,
self._nu_y_list, self._wp,
apiv, self._wa)
@property
def wp(self):
""" Dark energy equation of state at the pivot redshift zp"""
return self._wp
@property
def wa(self):
""" Negative derivative of dark energy equation of state w.r.t. a"""
return self._wa
@property
def zp(self):
""" The pivot redshift, where w(z) = wp"""
return self._zp
[ドキュメント] def w(self, z):
"""Returns dark energy equation of state at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
w : ndarray or float
The dark energy equation of state
Returns float if input scalar.
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. Here this is
:math:`w(z) = w_p + w_a (a_p - a)` where :math:`a = 1/1+z`
and :math:`a_p = 1 / 1 + z_p`.
"""
if isiterable(z):
z = np.asarray(z)
apiv = 1.0 / (1.0 + self._zp)
return self._wp + self._wa * (apiv - 1.0 / (1. + z))
[ドキュメント] def de_density_scale(self, z):
r""" Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns float if input scalar.
Notes
-----
The scaling factor, I, is defined by :math:`\\rho(z) = \\rho_0 I`,
and in this case is given by
.. math::
a_p = \frac{1}{1 + z_p}
I = \left(1 + z\right)^{3 \left(1 + w_p + a_p w_a\right)}
\exp \left(-3 w_a \frac{z}{1+z}\right)
"""
if isiterable(z):
z = np.asarray(z)
zp1 = 1. + z
apiv = 1. / (1. + self._zp)
return zp1 ** (3. * (1. + self._wp + apiv * self._wa)) * \
np.exp(-3. * self._wa * z / zp1)
def __repr__(self):
retstr = "{0}H0={1:.3g}, Om0={2:.3g}, Ode0={3:.3g}, wp={4:.3g}, "\
"wa={5:.3g}, zp={6:.3g}, Tcmb0={7:.4g}, Neff={8:.3g}, "\
"m_nu={9}, Ob0={10:s})"
return retstr.format(self._namelead(), self._H0, self._Om0,
self._Ode0, self._wp, self._wa, self._zp,
self._Tcmb0, self._Neff, self.m_nu,
_float_or_none(self._Ob0))
[ドキュメント]class w0wzCDM(FLRW):
"""FLRW cosmology with a variable dark energy equation of state
and curvature.
The equation for the dark energy equation of state uses the
simple form: :math:`w(z) = w_0 + w_z z`.
This form is not recommended for z > 1.
Parameters
----------
H0 : float or `~astropy.units.Quantity` ['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.
Ode0 : float
Omega dark energy: density of dark energy in units of the critical
density at z=0.
w0 : float, optional
Dark energy equation of state at z=0. This is pressure/density for
dark energy in units where c=1.
wz : float, optional
Derivative of the dark energy equation of state with respect to z.
A cosmological constant has w0=-1.0 and wz=0.0.
Tcmb0 : float or scalar `~astropy.units.Quantity` ['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
Name for this cosmological object.
Examples
--------
>>> from astropy.cosmology import w0wzCDM
>>> cosmo = w0wzCDM(H0=70, Om0=0.3, Ode0=0.7, w0=-0.9, wz=0.2)
The comoving distance in Mpc at redshift z:
>>> z = 0.5
>>> dc = cosmo.comoving_distance(z)
"""
def __init__(self, H0, Om0, Ode0, w0=-1., wz=0., Tcmb0=0,
Neff=3.04, m_nu=u.Quantity(0.0, u.eV), Ob0=None,
name=None):
super().__init__(H0, Om0, Ode0, Tcmb0, Neff, m_nu, Ob0=Ob0, name=name)
self._w0 = float(w0)
self._wz = float(wz)
# Please see "Notes about speeding up integrals" for discussion
# about what is being done here.
if self._Tcmb0.value == 0:
self._inv_efunc_scalar = scalar_inv_efuncs.w0wzcdm_inv_efunc_norel
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._w0, self._wz)
elif not self._massivenu:
self._inv_efunc_scalar = scalar_inv_efuncs.w0wzcdm_inv_efunc_nomnu
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0 + self._Onu0,
self._w0, self._wz)
else:
self._inv_efunc_scalar = scalar_inv_efuncs.w0wzcdm_inv_efunc
self._inv_efunc_scalar_args = (self._Om0, self._Ode0, self._Ok0,
self._Ogamma0, self._neff_per_nu,
self._nmasslessnu,
self._nu_y_list, self._w0,
self._wz)
@property
def w0(self):
""" Dark energy equation of state at z=0"""
return self._w0
@property
def wz(self):
""" Derivative of the dark energy equation of state w.r.t. z"""
return self._wz
[ドキュメント] def w(self, z):
"""Returns dark energy equation of state at redshift ``z``.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
w : ndarray or float
The dark energy equation of state
Returns float if input scalar.
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. Here this is given by
:math:`w(z) = w_0 + w_z z`.
"""
if isiterable(z):
z = np.asarray(z)
return self._w0 + self._wz * z
[ドキュメント] def de_density_scale(self, z):
r""" Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : array-like
Input redshifts.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns float if input scalar.
Notes
-----
The scaling factor, I, is defined by :math:`\\rho(z) = \\rho_0 I`,
and in this case is given by
.. math::
I = \left(1 + z\right)^{3 \left(1 + w_0 - w_z\right)}
\exp \left(-3 w_z z\right)
"""
if isiterable(z):
z = np.asarray(z)
zp1 = 1. + z
return zp1 ** (3. * (1. + self._w0 - self._wz)) *\
np.exp(-3. * self._wz * z)
def __repr__(self):
retstr = "{0}H0={1:.3g}, Om0={2:.3g}, "\
"Ode0={3:.3g}, w0={4:.3g}, wz={5:.3g} Tcmb0={6:.4g}, "\
"Neff={7:.3g}, m_nu={8}, Ob0={9:s})"
return retstr.format(self._namelead(), self._H0, self._Om0,
self._Ode0, self._w0, self._wz, self._Tcmb0,
self._Neff, self.m_nu, _float_or_none(self._Ob0))
def _float_or_none(x, digits=3):
""" Helper function to format a variable that can be a float or None"""
if x is None:
return str(x)
fmtstr = "{0:.{digits}g}".format(x, digits=digits)
return fmtstr.format(x)
def vectorize_if_needed(func, *x):
""" Helper function to vectorize functions on array inputs"""
if any(map(isiterable, x)):
return np.vectorize(func)(*x)
else:
return func(*x)
def inf_like(x):
"""Return the shape of x with value infinity and dtype='float'.
Preserves 'shape' for both array and scalar inputs.
But always returns a float array, even if x is of integer type.
>>> inf_like(0.) # float scalar
inf
>>> inf_like(1) # integer scalar should give float output
inf
>>> inf_like([0., 1., 2., 3.]) # float list
array([inf, inf, inf, inf])
>>> inf_like([0, 1, 2, 3]) # integer list should give float output
array([inf, inf, inf, inf])
"""
if np.isscalar(x):
return np.inf
else:
return np.full_like(x, np.inf, dtype='float')