# Copyright (c) 2016-2017,2019 Weitian LI # MIT License """ Cosmology calculator in a flat ΛCDM universe. References ---------- .. [unibonn-wiki] https://astro.uni-bonn.de/~pavel/WIKIPEDIA/Lambda-CDM_model.html .. [wikipedia-lcdm] 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 .. [wikipedia-virial] https://en.wikipedia.org/wiki/Virial_mass .. [bryan1998] http://adsabs.harvard.edu/abs/1998ApJ...495...80B """ 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). """ 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.[bryan1998],Eqs.(5,6) * Ref.[wikipedia-virial] """ x = self.Om(z) - 1 return 18*np.pi**2 + 82*x - 39 * x**2 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