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# Copyright (c) 2016-2017 Weitian LI <liweitianux@live.com>
# MIT license
"""
Cosmological models
"""
import logging
import numpy as np
from scipy import integrate
import astropy.units as au
from astropy.cosmology import LambdaCDM, z_at_value
logger = logging.getLogger(__name__)
class Cosmology:
"""
Cosmological model.
Attributes
----------
H0 : float
Hubble parameter at present day (z=0)
Om0 : float
Density parameter of matter at present day
Ode0 : float
Density parameter of dark energy at present day
model : str
Type of the current cosmological model:
* open : Om0 < 1, Ode0 = 0
* closed : Om0 > 1, Ode0 = 0
* EdS (Einstein-de Sitter) : Om0 = 1, Ode0 = 0
* flatLambdaCDM : Om0 + Ode0 = 1, Ode0 > 0
References
----------
[1] https://astro.uni-bonn.de/~pavel/WIKIPEDIA/Lambda-CDM_model.html
[2] https://en.wikipedia.org/wiki/Lambda-CDM_model
"""
def __init__(self, H0=71.0, Om0=0.27, Ob0=0.046, Ode0=None, sigma8=None):
Ode0 = 1.0 - Om0 if Ode0 is None else Ode0
if (Ode0 > 0) and abs(Om0 + Ode0 - 1) > 1e-5:
raise ValueError("non-flat LambdaCDM model not supported!")
self.H0 = H0 # [km/s/Mpc]
self.Om0 = Om0
self.Ob0 = Ob0
self.Ode0 = Ode0
self._sigma8 = sigma8
self.cosmo = LambdaCDM(H0=H0, Om0=Om0, Ob0=Ob0, Ode0=Ode0)
logger.info("Cosmological model: {0}".format(self.model))
@property
def model(self):
if self.Ode0 < 1e-5:
if self.Om0 < 1:
model = "open"
elif self.Om0 == 1:
model = "EdS"
else:
model = "closed"
else:
model = "flatLambdaCDM"
return model
@property
def h(self):
"""
Dimensionless/reduced Hubble parameter
"""
return self.H0 / 100.0
@property
def sigma8(self):
"""
Present-day rms density fluctuation on a scale of 8 h^-1 Mpc.
References
----------
[1] Randall, Sarazin & Ricker 2002, ApJ, 577, 579
http://adsabs.harvard.edu/abs/2002ApJ...577..579R
Sec.2
"""
if hasattr(self, "_sigma8"):
return self._sigma8
#
if self.model == "open":
sigma8 = 0.827
elif self.model == "closed":
raise NotImplementedError
elif self.model == "EdS":
sigma8 = 0.514
elif self.model == "flatLambdaCDM":
sigma8 = 0.834
else:
raise ValueError("unknown model: {0}".format(self.model))
return sigma8
@property
def M8(self):
"""
Mass contained in a sphere of radius of 8 h^-1 Mpc.
Unit: [Msun]
"""
r = 8 * au.Mpc.to(au.cm) / self.h # [cm]
M8 = (4*np.pi/3) * r**3 * self.rho_crit(0)
return (M8 * au.g.to(au.solMass))
def age(self, z):
"""
Cosmic time at redshift z.
Parameters
----------
z : float
Redshift
Returns
-------
age : float
Age of the universe (cosmic time) at the given redshift.
Unit: [Gyr]
"""
return self.cosmo.age(z).value
@property
def age0(self):
"""
Present age of the universe.
"""
if not hasattr(self, "_age0"):
self._age0 = self.age(0)
return self._age0
def redshift(self, age):
"""
Invert the above age calculation, to return the redshift
corresponding to the given cosmic time.
Parameters
----------
age : float
Age of the universe (cosmic time), unit [Gyr]
Returns
-------
z : float
Redshift corresponding to the input age.
"""
return z_at_value(self.age, age)
def rho_crit(self, z):
"""
Critical density at redshift z.
Unit: [g/cm^3]
"""
return self.cosmo.critical_density(0).value
def OmegaM(self, z):
"""
Density parameter of matter at redshift z.
"""
return self.Om0 * (1+z)**3 / self.cosmo.efunc(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)
"""
if self.model == "open":
raise NotImplementedError
elif self.model == "closed":
raise NotImplementedError
elif self.model == "EdS":
Delta_c = 18 * np.pi**2
elif self.model == "flatLambdaCDM":
omega_z = (1 / self.OmegaM(z)) - 1
Delta_c = 18*np.pi**2 * (1 + 0.4093 * omega_z**0.9052)
else:
raise ValueError("unknown model: {0}".format(self.model))
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)
"""
if self.model == "open":
raise NotImplementedError
elif self.model == "closed":
raise NotImplementedError
elif self.model == "EdS":
coef = 3 * (12*np.pi) ** (2/3) / 20
delta_c = coef * (self.age(0) / self.age(z)) ** (2/3)
elif self.model == "flatLambdaCDM":
coef = 3 * (12*np.pi) ** (2/3) / 20
D0 = self.growth_factor(0)
D_z = self.growth_factor(z)
Om_z = self.OmegaM(z)
delta_c = coef * (D0 / D_z) * (1 + 0.0123*np.log10(Om_z))
else:
raise ValueError("unknown model: {0}".format(self.model))
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)
"""
if self.model == "open":
raise NotImplementedError
elif self.model == "closed":
raise NotImplementedError
elif self.model == "EdS":
raise NotImplementedError
elif self.model == "flatLambdaCDM":
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),
0, x)[0]
D = coef * integral
else:
raise ValueError("unknown model: {0}".format(self.model))
return D
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