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# Copyright (c) 2016-2017 Weitian LI <liweitianux@live.com>
# MIT license
"""
Flat ΛCDM cosmological model.
"""
import numpy as np
from scipy import integrate
from astropy.cosmology import FlatLambdaCDM
from .units import (UnitConversions as AUC, Constants as AC)
class Cosmology:
"""
Flat ΛCDM cosmological model.
Attributes
----------
H0 : float
Hubble parameter at present day (z=0)
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.
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)
"""
def __init__(self, H0=71.0, Om0=0.27, Ob0=0.046, sigma8=0.834):
self.H0 = H0 # [km/s/Mpc]
self.Om0 = Om0
self.Ob0 = Ob0
self.Ode0 = 1.0 - Om0
self.sigma8 = sigma8
self._cosmo = FlatLambdaCDM(H0=H0, Om0=Om0, Ob0=Ob0)
@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)
@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
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 not hasattr(self, "_growth_factor0"):
self._growth_factor0 = self.growth_factor(0)
return self._growth_factor0
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