def omega_M_z(z, **cosmo):
"""Matter density omega_M as a function of redshift z.
Notes
-----
From Lahav et al. (1991, MNRAS 251, 128) equations 11b-c. This is
equivalent to equation 10 of Eisenstein & Hu (1999 ApJ 511 5).
"""
if get_omega_k_0(**cosmo) == 0:
return 1.0 / (1. + (1. - cosmo['omega_M_0'])/
(cosmo['omega_M_0'] * (1. + z)**3.))
else:
return (cosmo['omega_M_0'] * (1. + z)**3. /
cd.e_z(z, **cosmo)**2.)
def optical_depth_instant(z_r, x_ionH=1.0, x_ionHe=1.0, z_rHe = None,
return_tau_star=False, verbose=0, **cosmo):
"""Optical depth assuming instantaneous reionization and a flat
universe.
Calculates the optical depth due to Thompson scattering off free
electrons in the IGM.
Parameters
----------
z_r:
Redshift of instantaneos reionization.
x_ionH:
Ionized fraction of hydrogen after reionization.
x_ionHe:
Set to 2.0 for fully ionized helium. Set to 1.0 for singly
ionized helium. Set to 0.0 for neutral helium. This value
equals X_HeII + 2 * X_HeIII after z_r (where X_HeII is the
fraction of helium that is singly ionized, and X_HeII is the
fraction of helium that is doubly ionized).
z_rHe (optional):
Redshift of instantaneos Helium reionization, i.e. when helium
becomes doubly ionized. z_rHe should be less than z_r.
return_tau_star: Boolean
whether or not to return the value of tau_star, as defined by
Griffiths et al. (arxiv:astro-ph/9812125v3)
cosmo: cosmological parameters
Returns
-------
tau: array
optical depth to election
tau_star: array or scalar
Notes
-----
See, e.g. Griffiths et al. (arxiv:astro-ph/9812125v3, note that
the published version [ 1999MNRAS.308..854G] has typos)
"""
if numpy.any(cden.get_omega_k_0(**cosmo) != 0):
raise ValueError("Not valid for non-flat (omega_k_0 !=0) cosmology.")
if z_rHe is not None:
# Optical depth from z_rHe to 0 with He fully (twice) ionized.
tau_short_all = optical_depth_instant(z_rHe, x_ionH, x_ionHe=2.0,
**cosmo)
# Optical depth from z_rHe to 0 without He fully ionized.
tau_short_H = optical_depth_instant(z_rHe, x_ionH, x_ionHe, **cosmo)
# Difference due to fully ionized He (added to tau later):
tau_short_He = tau_short_all - tau_short_H
if(verbose > 0) :
print ("tau_short_He = ", tau_short_He)
rho_crit, rho_0, n_He_0, n_H_0 = cden.baryon_densities(**cosmo)
# comoving Mpc^-1
n_p = n_H_0 + 2. * n_He_0
# comoving Mpc^-1
n_e = (n_H_0 * x_ionH) + (n_He_0 * x_ionHe)
# fraction of electrons that are free
x = n_e / n_p
if(verbose > 0) :
print ("n_He/n_H = ", n_He_0 / n_H_0)
print ("x = ne/np = ", x)
print ("n_e/n_H_0 = ", n_e/n_H_0)
H_0 = cc.H100_s * cosmo['h']
# Mpc s^-1 * Mpc^2 * Mpc^-3 / s^-1 -> unitless
tau_star = cc.c_light_Mpc_s * cc.sigma_T_Mpc * n_p * x / H_0
### The tau_star expressions above and below are mathematically identical.
#tau_star = cc.c_light_Mpc_s * cc.sigma_T_Mpc * n_H_0 * (n_e/n_H_0) / H_0
e_z_reion = cd.e_z(z_r, **cosmo)
tau = 2. * tau_star * (e_z_reion - 1.0) / (3. * cosmo['omega_M_0'])
if z_rHe is not None:
# Add in the Helium reionization contribution:
tau += tau_short_He
if return_tau_star:
return tau, tau_star
else:
return tau
def optical_depth_instant(z_r, x_ionH=1.0, x_ionHe=1.0, z_rHe = None,
return_tau_star=False, verbose=0, **cosmo):
"""Optical depth assuming instantaneous reionization and a flat
universe.
Calculates the optical depth due to Thompson scattering off free
electrons in the IGM.
Parameters
----------
z_r:
Redshift of instantaneos reionization.
x_ionH:
Ionized fraction of hydrogen after reionization.
x_ionHe:
Set to 2.0 for fully ionized helium. Set to 1.0 for singly
ionized helium. Set to 0.0 for neutral helium. This value
equals X_HeII + 2 * X_HeIII after z_r (where X_HeII is the
fraction of helium that is singly ionized, and X_HeII is the
fraction of helium that is doubly ionized).
z_rHe (optional):
Redshift of instantaneos Helium reionization, i.e. when helium
becomes doubly ionized. z_rHe should be less than z_r.
return_tau_star: Boolean
whether or not to return the value of tau_star, as defined by
Griffiths et al. (arxiv:astro-ph/9812125v3)
cosmo: cosmological parameters
Returns
-------
tau: array
optical depth to election
tau_star: array or scalar
Notes
-----
See, e.g. Griffiths et al. (arxiv:astro-ph/9812125v3, note that
the published version [ 1999MNRAS.308..854G] has typos)
"""
if numpy.any(cden.get_omega_k_0(**cosmo) != 0):
raise ValueError, "Not valid for non-flat (omega_k_0 !=0) cosmology."
if z_rHe is not None:
# Optical depth from z_rHe to 0 with He fully (twice) ionized.
tau_short_all = optical_depth_instant(z_rHe, x_ionH, x_ionHe=2.0,
**cosmo)
# Optical depth from z_rHe to 0 without He fully ionized.
tau_short_H = optical_depth_instant(z_rHe, x_ionH, x_ionHe, **cosmo)
# Difference due to fully ionized He (added to tau later):
tau_short_He = tau_short_all - tau_short_H
if(verbose > 0) :
print "tau_short_He = ", tau_short_He
rho_crit, rho_0, n_He_0, n_H_0 = cden.baryon_densities(**cosmo)
# comoving Mpc^-1
n_p = n_H_0 + 2. * n_He_0
# comoving Mpc^-1
n_e = (n_H_0 * x_ionH) + (n_He_0 * x_ionHe)
# fraction of electrons that are free
x = n_e / n_p
if(verbose > 0) :
print "n_He/n_H = ", n_He_0 / n_H_0
print "x = ne/np = ", x
print "n_e/n_H_0 = ", n_e/n_H_0
H_0 = cc.H100_s * cosmo['h']
# Mpc s^-1 * Mpc^2 * Mpc^-3 / s^-1 -> unitless
tau_star = cc.c_light_Mpc_s * cc.sigma_T_Mpc * n_p * x / H_0
### The tau_star expressions above and below are mathematically identical.
#tau_star = cc.c_light_Mpc_s * cc.sigma_T_Mpc * n_H_0 * (n_e/n_H_0) / H_0
e_z_reion = cd.e_z(z_r, **cosmo)
tau = 2. * tau_star * (e_z_reion - 1.0) / (3. * cosmo['omega_M_0'])
if z_rHe is not None:
# Add in the Helium reionization contribution:
tau += tau_short_He
if return_tau_star:
return tau, tau_star
else:
return tau
