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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