# Copyright (c) 2016-2017 Weitian LI <liweitianux@live.com>
# MIT license
"""
Flat ΛCDM cosmological model.
References
----------
.. [unibonn-wiki]
https://astro.uni-bonn.de/~pavel/WIKIPEDIA/Lambda-CDM_model.html
.. [wikipedia]
https://en.wikipedia.org/wiki/Lambda-CDM_model
.. [randall2002]
Randall, Sarazin & Ricker 2002, ApJ, 577, 579
http://adsabs.harvard.edu/abs/2002ApJ...577..579R
Sec.(2)
.. [hogg1999]
Hogg 1999, arXiv:astro-ph/9905116
http://adsabs.harvard.edu/abs/1999astro.ph..5116H
.. [thomas2000]
Thomas & Kantowski 2000, Physical Review D, 62, 103507
http://adsabs.harvard.edu/abs/2000PhRvD..62j3507T
.. [ellis2007]
Ellis 2007, General Relativity and Gravitation, 39, 1047
http://adsabs.harvard.edu/abs/2007GReGr..39.1047E
.. [cassano2005]
Cassano & Brunetti 2005, MNRAS, 357, 1313
http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
"""
import logging
import numpy as np
from scipy import integrate
from scipy import interpolate
from astropy.cosmology import FlatLambdaCDM
from .units import (UnitConversions as AUC, Constants as AC)
logger = logging.getLogger(__name__)
class Cosmology:
"""
Flat ΛCDM cosmological model.
Attributes
----------
H0 : float
Hubble parameter at present day (z=0)
Unit: [km/s/Mpc]
Om0 : float
Density parameter of (dark and baryon) matter at present day
Ob0 : float
Density parameter of baryon at present day
Ode0 : float
Density parameter of dark energy at present day
Tcmb0 : float
Present-day CMB temperature
Unit: [K]
sigma8 : float
Present-day rms density fluctuation on a scale of 8 h^-1 [Mpc]
ns : float
Scalar spectral index
Internal attributes
-------------------
_cosmo : `~astropy.cosmology.Cosmology`
Astropy cosmology instance to help calculations.
_growth_factor0 : float
Present day (z=0) growth factor
"""
# Present day (z=0) growth factor
_growth_factor0 = None
def __init__(self, H0=71.0, Om0=0.27, Ob0=0.046,
Tcmb0=2.725, sigma8=0.81, ns=0.96):
self.setup(H0=H0, Om0=Om0, Ob0=Ob0, Tcmb0=Tcmb0, sigma8=sigma8, ns=ns)
def setup(self, **kwargs):
"""
Setup/update the parameters of the cosmology model.
"""
for key, value in kwargs.items():
if key in ["H0", "Om0", "Ob0", "Tcmb0", "sigma8", "ns"]:
setattr(self, key, value)
else:
raise ValueError("unknown parameter: %s" % key)
self.Ode0 = 1.0 - self.Om0
self._cosmo = FlatLambdaCDM(H0=self.H0, Om0=self.Om0, Ob0=self.Ob0,
Tcmb0=self.Tcmb0)
self._growth_factor0 = None
logger.info("Setup cosmology with: {0}".format(kwargs))
@property
def h(self):
"""
Dimensionless/reduced Hubble parameter
"""
return self.H0 / 100.0
@property
def M8(self):
"""
Mass contained in a sphere of radius of 8 h^-1 [Mpc].
Unit: [Msun]
"""
r = 8 * AUC.Mpc2cm / self.h # [cm]
M8 = (4*np.pi/3) * r**3 * self.rho_crit(0) # [g]
M8 *= AUC.g2Msun # [Msun]
return M8
def E(self, z):
"""
Redshift evolution factor.
"""
return np.sqrt(self.Om0 * (1+z)**3 + self.Ode0)
def H(self, z):
"""
Hubble parameter at redshift z.
Unit: [km/s/Mpc]
"""
return self.H0 * self.E(z)
def Dc(self, z):
"""
Comoving distance at redshift z.
Unit: [Mpc]
"""
return self._cosmo.comoving_distance(z).value
def Dc_to_redshift(self, Dc, zmin=0, zmax=3, zstep=0.01):
"""
Calculate the redshifts corresponding to the given comoving
distances by interpolation.
Parameters
----------
Dc : float, or `~numpy.ndarray`
Comoving distances
Unit: [Mpc]
zmin, zmax : float, optional
The minimum and maximum redshift within which the input
comoving distances are enclosed; otherwise, a error will be
raised during the calculation.
zstep : float, optional
The redshift step size adopted to do the interpolation.
Returns
-------
redshift : float, or `~numpy.ndarray`
Calculated redshifts w.r.t. the input comoving distances.
Raises
------
ValueError :
The ``zmin`` or ``zmax`` is not enough to enclose the input
comoving distance range.
"""
Dc_min, Dc_max = self.Dc([zmin, zmax]) # [Mpc]
if np.sum(Dc < Dc_min) > 0:
raise ValueError("zmin=%s is too big for input Dc" % zmin)
if np.sum(Dc > Dc_max) > 0:
raise ValueError("zmax=%s is too small for input Dc" % zmax)
z_ = np.arange(zmin, zmax+zstep/2, zstep)
Dc_ = self.Dc(z_)
Dc_interp = interpolate.interp1d(Dc_, z_, kind="linear")
return Dc_interp(Dc)
def DA(self, z):
"""
Angular diameter distance at redshift z.
Unit: [Mpc]
Defined as the ratio of an object's physical transverse size
to its (observed) angular size (in radians). It is used to
convert the observed angular separations between sources into
their proper separations.
NOTE
----
This distance is NOT increasing indefinitely as z -> ∞.
Reference: Ref.[hogg1999]
"""
return self._cosmo.angular_diameter_distance(z).value
def DL(self, z):
"""
Luminosity distance at redshift z.
Unit: [Mpc]
Defined by the relationship between the measured bolometric
(i.e., integrated over all frequencies) flux S_bolo and the
object's intrinsic bolometric luminosity L_bolo.
NOTE
----
DL = DA * (1+z)^2
This is the general reciprocity theorem in General Relativity.
Reference
---------
* Ref.[hogg1999],Eq.(20,21)
* Ref.[ellis2007]
"""
return self._cosmo.luminosity_distance(z).value
@property
def hubble_time(self):
"""
Hubble time.
Unit: [Gyr]
"""
uconv = AUC.Mpc2km * AUC.s2Gyr
t_H = (1.0/self.H0) * uconv # [Gyr]
return t_H
def age(self, z):
"""
Cosmic time (age) at redshift z.
Parameters
----------
z : `~numpy.ndarray`
Redshift
Returns
-------
age : `~numpy.ndarray`
Age of the universe (cosmic time) at the given redshift.
Unit: [Gyr]
References: Ref.[thomas2000],Eq.(18)
"""
z = np.asarray(z)
t_H = self.hubble_time
t = ((2*t_H / 3 / np.sqrt(1-self.Om0)) *
np.arcsinh(np.sqrt((1/self.Om0 - 1) / (1+z)**3)))
return t
@property
def age0(self):
"""
Present age of the universe.
"""
return self.age(0)
def redshift(self, age):
"""
Invert the above ``self.age(z)`` formula analytically, to calculate
the redshift corresponding to the given cosmic time (age).
Parameters
----------
age : `~numpy.ndarray`
Age of the universe (i.e., cosmic time)
Unit: [Gyr]
Returns
-------
z : `~numpy.ndarray`
Redshift corresponding to the specified age.
"""
age = np.asarray(age)
t_H = self.hubble_time
term1 = (1/self.Om0) - 1
term2 = np.sinh(3*age * np.sqrt(1-self.Om0) / (2*t_H)) ** 2
z = (term1 / term2) ** (1/3) - 1
return z
def rho_crit(self, z):
"""
Critical density at redshift z.
Unit: [g/cm^3]
"""
rho = 3 * self.H(z)**2 / (8*np.pi * AC.G)
rho *= AUC.km2Mpc**2
return rho
def Om(self, z):
"""
Density parameter of matter at redshift z.
"""
return self.Om0 * (1+z)**3 / self.E(z)**2
@property
def baryon_fraction(self):
"""
The cosmological mean baryon fraction (w.r.t. matter).
XXX: assumed to be *constant* regardless of redshifts!
"""
return self.Ob0 / self.Om0
@property
def darkmatter_fraction(self):
"""
The cosmological mean dark matter fraction (w.r.t. matter),
assumed to be *constant* regardless of redshifts!
See also: ``self.baryon_fraction``
"""
return 1 - self.baryon_fraction
def overdensity_virial(self, z):
"""
Calculate the virial overdensity, which generally used to
determine the virial radius of a cluster.
References: Ref.[cassano2005],Eqs.(10,A4)
"""
omega_z = (1 / self.Om(z)) - 1
Delta_c = 18*np.pi**2 * (1 + 0.4093 * omega_z**0.9052)
return Delta_c
def overdensity_crit(self, z):
"""
Critical (linear) overdensity for a region to collapse
at a redshift z.
References: Ref.[randall2002],Eq.(A1)
"""
coef = 3 * (12*np.pi) ** (2/3) / 20
D0 = self.growth_factor0
D_z = self.growth_factor(z)
Om_z = self.Om(z)
delta_c = coef * (D0 / D_z) * (1 + 0.0123*np.log10(Om_z))
return delta_c
def growth_factor(self, z):
"""
Growth factor at redshift z.
References: Ref.[randall2002],Eq.(A7)
"""
x0 = (2 * self.Ode0 / self.Om0) ** (1/3)
x = x0 / (1 + z)
coef = np.sqrt(x**3 + 2) / (x**1.5)
integral = integrate.quad(lambda y: y**1.5 * (y**3+2)**(-1.5),
a=0, b=x, epsabs=1e-5, epsrel=1e-5)[0]
D = coef * integral
return D
@property
def growth_factor0(self):
"""
Present-day (z=0) growth factor.
"""
if self._growth_factor0 is None:
self._growth_factor0 = self.growth_factor(0)
return self._growth_factor0
def dVc(self, z):
"""
Calculate the differential comoving volume.
The dimensions is [Mpc^3]/[sr]/[unit redshift].
"""
return self._cosmo.differential_comoving_volume(z).value