# Copyright (c) 2016-2017 Weitian LI <liweitianux@live.com>
# MIT license
"""
Flat ΛCDM cosmological model.
"""
import logging
import numpy as np
from scipy import integrate
from astropy.cosmology import FlatLambdaCDM
from ..configs import configs
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
sigma8 : float
Present-day rms density fluctuation on a scale of 8 h^-1 Mpc.
Internal attributes
-------------------
_cosmo : `~astropy.cosmology.Cosmology`
Astropy cosmology instance to help calculations.
_growth_factor0 : float
Present day (z=0) growth factor
References
----------
[1] https://astro.uni-bonn.de/~pavel/WIKIPEDIA/Lambda-CDM_model.html
[2] https://en.wikipedia.org/wiki/Lambda-CDM_model
[3] Randall, Sarazin & Ricker 2002, ApJ, 577, 579
http://adsabs.harvard.edu/abs/2002ApJ...577..579R
Sec.(2)
"""
# Present day (z=0) growth factor
_growth_factor0 = None
def __init__(self,
H0=configs.cosmology["H0"],
Om0=configs.cosmology["Om0"],
Ob0=configs.cosmology["Ob0"],
sigma8=configs.cosmology["sigma8"]):
self.setup(H0=H0, Om0=Om0, Ob0=Ob0, sigma8=sigma8)
def setup(self, **kwargs):
"""
Setup/update the parameters of the cosmology model.
"""
for key, value in kwargs.items():
if key in ["H0", "Om0", "Ob0", "sigma8"]:
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)
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.
"""
return self.H0 * self.E(z)
def DL(self, z):
"""
Luminosity distance at redshift z.
Unit: [Mpc]
"""
return self._cosmo.luminosity_distance(z).value
def DA(self, z):
"""
Angular diameter distance at redshift z.
Unit: [Mpc]
"""
return self._cosmo.angular_diameter_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 : float
Redshift
Returns
-------
age : float
Age of the universe (cosmic time) at the given redshift.
Unit: [Gyr]
References
----------
[1] Thomas & Kantowski 2000, Physical Review D, 62, 103507
http://adsabs.harvard.edu/abs/2000PhRvD..62j3507T
Eq.(18)
"""
t_H = self.hubble_time
t = (t_H * (2/3/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 : float
Age of the universe (cosmic time), unit [Gyr]
Returns
-------
z : float
Redshift corresponding to the specified 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
def overdensity_virial(self, z):
"""
Calculate the virial overdensity, which generally used to
determine the virial radius of a cluster.
References
----------
[1] Cassano & Brunetti 2005, MNRAS, 357, 1313
http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
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
----------
[1] Randall, Sarazin & Ricker 2002, ApJ, 577, 579
http://adsabs.harvard.edu/abs/2002ApJ...577..579R
Appendix.A, 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
----------
[1] Randall, Sarazin & Ricker 2002, ApJ, 577, 579
http://adsabs.harvard.edu/abs/2002ApJ...577..579R
Appendix.A, 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