def ionization_from_luminosity(z, ratedensityfunc, xHe=1.0,
rate_is_tfunc = False,
ratedensityfunc_args = (),
method = 'romberg',
**cosmo):
"""Integrate the ionization history given an ionizing luminosity
function, ignoring recombinations.
Parameters
----------
ratedensityfunc: callable
function giving comoving ionizing photon emission rate
density, or ionizing emissivity (photons s^-1 Mpc^-3) as a
function of redshift (or time).
rate_is_tfunc: boolean
Set to true if ratedensityfunc is a function of time rather than z.
Notes
-----
Ignores recombinations.
The ionization rate is computed as ratedensity / nn, where nn = nH
+ xHe * nHe. So if xHe is 1.0, we are assuming that helium becomes
singly ionized at proportionally the same rate as hydrogen. If xHe
is 2.0, we are assuming helium becomes fully ionizing at
proportionally the same rate as hydrogen.
The returened x is therefore the ionized fraction of hydrogen, and
the ionized fraction of helium is xHe * x.
"""
cosmo = cd.set_omega_k_0(cosmo)
rhoc, rho0, nHe, nH = cden.baryon_densities(**cosmo)
nn = (nH + xHe * nHe)
if rate_is_tfunc:
t = cd.age(z, **cosmo)[0]
def dx_dt(t1):
return numpy.nan_to_num(ratedensityfunc(t1, *ratedensityfunc_args) /
nn)
sorti = numpy.argsort(t)
x = numpy.empty(t.shape)
x[sorti] = cu.integrate_piecewise(dx_dt, t[sorti], method = method)
return x
else:
dt_dz = lambda z1: cd.lookback_integrand(z1, **cosmo)
def dx_dz(z1):
z1 = numpy.abs(z1)
return numpy.nan_to_num(dt_dz(z1) *
ratedensityfunc(z1, *ratedensityfunc_args) /
nn)
sorti = numpy.argsort(-z)
x = numpy.empty(z.shape)
x[sorti] = cu.integrate_piecewise(dx_dz, -z[sorti], method = method)
return x
def ionization_from_luminosity(z, ratedensityfunc, xHe=1.0,
rate_is_tfunc = False,
ratedensityfunc_args = (),
method = 'romberg',
**cosmo):
"""Integrate the ionization history given an ionizing luminosity
function, ignoring recombinations.
Parameters
----------
ratedensityfunc: callable
function giving comoving ionizing photon emission rate
density, or ionizing emissivity (photons s^-1 Mpc^-3) as a
function of redshift (or time).
rate_is_tfunc: boolean
Set to true if ratedensityfunc is a function of time rather than z.
Notes
-----
Ignores recombinations.
The ionization rate is computed as ratedensity / nn, where nn = nH
+ xHe * nHe. So if xHe is 1.0, we are assuming that helium becomes
singly ionized at proportionally the same rate as hydrogen. If xHe
is 2.0, we are assuming helium becomes fully ionizing at
proportionally the same rate as hydrogen.
The returened x is therefore the ionized fraction of hydrogen, and
the ionized fraction of helium is xHe * x.
"""
cosmo = cd.set_omega_k_0(cosmo)
rhoc, rho0, nHe, nH = cden.baryon_densities(**cosmo)
nn = (nH + xHe * nHe)
if rate_is_tfunc:
t = cd.age(z, **cosmo)[0]
def dx_dt(t1):
return numpy.nan_to_num(ratedensityfunc(t1, *ratedensityfunc_args) /
nn)
sorti = numpy.argsort(t)
x = numpy.empty(t.shape)
x[sorti] = cu.integrate_piecewise(dx_dt, t[sorti], method = method)
return x
else:
dt_dz = lambda z1: cd.lookback_integrand(z1, **cosmo)
def dx_dz(z1):
z1 = numpy.abs(z1)
return numpy.nan_to_num(dt_dz(z1) *
ratedensityfunc(z1, *ratedensityfunc_args) /
nn)
sorti = numpy.argsort(-z)
x = numpy.empty(z.shape)
x[sorti] = cu.integrate_piecewise(dx_dz, -z[sorti], method = method)
return x
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 integrate_optical_depth(x_ionH, x_ionHe, z, **cosmo):
"""The electron scattering optical depth given ionized filling
factor vs. redshift.
Parameters
----------
x_ionH: array
Ionized fraction of hydrogen as a function of z. Should be [0,1].
x_ionHe: array
Set x_ionHE to X_HeII + 2 * X_HeIII, where X_HeII is the
fraction of helium that is singly ionized, and X_HeII is the
fraction of helium that is doubly ionized. See Notes below.
z: array
Redshift values at which the filling factor is specified.
cosmo: cosmological parameters
uses: 'X_H' and/or 'Y_He', plus parameters needed for hubble_z
Returns
-------
tau: array
The optical depth as a function of z.
Notes
-----
The precision of your result depends on the spacing of the input
arrays. When in doubt, try doubling your z resolution and see if
the optical depth values have converged.
100% singly ionized helium means x_ionHe = 1.0, 100% doubly
ionized helium means x_ionHe = 2.0
If you want helium to be singly ionized at the same rate as
hydrogen, set x_ionHe = x_ionH.
If you want helium to be doubly ionized at the same rate as
hydrogen is ionized, set x_ionHe = 2 * x_ionH.
"""
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
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
# s^-1
H_z = cd.hubble_z(z, **cosmo)
# Mpc^3 s^-1 * Mpc^-3 / s^-1 -> unitless
integrand = -1. * tau_star * x * ((1. + z)**2.) / H_z
integral = numpy.empty(integrand.shape)
integral[...,1:] = si.cumtrapz(integrand, z)
integral[...,0] = 0.0
return numpy.abs(integral)
def integrate_ion_recomb_collapse(z, coeff_ion,
temp_min = 1e4,
passed_min_mass = False,
temp_gas=1e4,
alpha_B=None,
clump_fact_func = clumping_factor_BKP,
**cosmo):
"""IGM ionization state with recombinations from halo collapse
fraction. Integrates an ODE describing IGM ionization and
recombination rates.
z: array
The redshift values at which to calculate the ionized
fraction. This array should be in reverse numerical order. The
first redshift specified should be early enough that the
universe is still completely neutral.
coeff_ion:
The coefficient converting the collapse fraction to ionized
fraction, neglecting recombinations. Equivalent to the product
(f_star * f_esc_gamma * N_gamma) in the BKP paper.
temp_min:
See docs for ionization_from_collapse. Either the minimum virial
temperature or minimum mass of halos contributing to
reionization.
passed_temp_min:
See documentation for ionization_from_collapse.
temp_gas:
Gas temperature used to calculate the recombination coefficient
if alpha_b is not specified.
alpha_B:
Optional recombination coefficient in units of cm^3
s^-1. In alpha_B=None, it is calculated from temp_gas.
clump_fact_func: function
Function returning the clumping factor when given a redshift.
cosmo: dict
Dictionary specifying the cosmological parameters.
We assume, as is fairly standard, that the ionized
fraction is contained in fully ionized bubbles surrounded by a
fully neutral IGM. The output is therefore the volume filling
factor of ionized regions, not the ionized fraction of a
uniformly-ionized IGM.
I have also made the standard assumption that all ionized photons
are immediately absorbed, which allows the two differential
equations (one for ionization-recombination and one for
emission-photoionizaion) to be combined into a single ODE.
"""
# Determine recombination coefficient.
if alpha_B is None:
alpha_B_cm = recomb_rate_coeff_HG(temp_gas, 'H', 'B')
else:
alpha_B_cm = alpha_B
alpha_B = alpha_B_cm / (cc.Mpc_cm**3.)
print ("Recombination rate alpha_B = %.4g (Mpc^3 s^-1) = %.4g (cm^3 s^-1)"
% (alpha_B, alpha_B_cm))
# Normalize power spectrum.
if 'deltaSqr' not in cosmo:
cosmo['deltaSqr'] = cp.norm_power(**cosmo)
# Calculate useful densities.
rho_crit, rho_0, n_He_0, n_H_0 = cden.baryon_densities(**cosmo)
# Function used in the integration.
# Units: (Mpc^3 s^-1) * Mpc^-3 = s^-1
coeff_rec_func = lambda z: (clump_fact_func(z)**2. *
alpha_B *
n_H_0 * (1.+z)**3.)
# Generate a function that converts redshift to age of the universe.
redfunc = cd.quick_redshift_age_function(zmax = 1.1 * numpy.max(z),
zmin = -0.0,
**cosmo)
# Function used in the integration.
ionfunc = quick_ion_col_function(coeff_ion,
temp_min,
passed_min_mass = passed_min_mass,
zmax = 1.1 * numpy.max(z),
zmin = -0.0,
zstep = 0.1, **cosmo)
# Convert specified redshifts to cosmic time (age of the universe).
t = cd.age(z, **cosmo)
# Integrate to find u(z) = x(z) - w(z), where w is the ionization fraction
u = si.odeint(_udot, y0=0.0, t=t,
args=(coeff_rec_func, redfunc, ionfunc))
u = u.flatten()
w = ionization_from_collapse(z, coeff_ion, temp_min,
passed_min_mass = passed_min_mass,
**cosmo)
x = u + w
x[x > 1.0] = 1.0
return x, w, t
def integrate_ion_recomb(z,
ion_func,
clump_fact_func,
xHe=1.0,
temp_gas=1e4,
alpha_B=None,
bubble=True,
**cosmo):
"""Integrate IGM ionization and recombination given an ionization function.
Parameters:
z: array
The redshift values at which to calculate the ionized
fraction. This array should be in reverse numerical order. The
first redshift specified should be early enough that the
universe is still completely neutral.
ion_func:
A function giving the ratio of the total density of emitted
ionizing photons to the density hydrogen atoms (or hydrogen
plus helium, if you prefer) as a function of redshift.
temp_gas:
Gas temperature used to calculate the recombination coefficient
if alpha_b is not specified.
alpha_B:
Optional recombination coefficient in units of cm^3
s^-1. In alpha_B=None, it is calculated from temp_gas.
clump_fact_func: function
Function returning the clumping factor when given a redshift,
defined as <n_HII^2>/<n_HII>^2.
cosmo: dict
Dictionary specifying the cosmological parameters.
Notes:
We only track recombination of hydrogen, but if xHe > 0, then the
density is boosted by the addition of xHe * nHe. This is
eqiuvalent to assuming the the ionized fraction of helium is
always proportional to the ionized fraction of hydrogen. If
xHe=1.0, then helium is singly ionized in the same proportion as
hydrogen. If xHe=2.0, then helium is fully ionized in the same
proportion as hydrogen.
We assume, as is fairly standard, that the ionized
fraction is contained in fully ionized bubbles surrounded by a
fully neutral IGM. The output is therefore the volume filling
factor of ionized regions, not the ionized fraction of a
uniformly-ionized IGM.
I have also made the standard assumption that all ionized photons
are immediately absorbed, which allows the two differential
equations (one for ionization-recombination and one for
emission-photoionizaion) to be combined into a single ODE.
"""
# Determine recombination coefficient.
if alpha_B is None:
alpha_B_cm = recomb_rate_coeff_HG(temp_gas, 'H', 'B')
else:
alpha_B_cm = alpha_B
alpha_B = alpha_B_cm * cc.Gyr_s / (cc.Mpc_cm**3.)
print ("Recombination rate alpha_B = %.4g (Mpc^3 Gyr^-1) = %.4g (cm^3 s^-1)"
% (alpha_B, alpha_B_cm))
# Normalize power spectrum.
if 'deltaSqr' not in cosmo:
cosmo['deltaSqr'] = cp.norm_power(**cosmo)
# Calculate useful densities.
rho_crit, rho_0, n_He_0, n_H_0 = cden.baryon_densities(**cosmo)
# Boost density to approximately account for helium.
nn = (n_H_0 + xHe * n_He_0)
# Function used in the integration.
# Units: (Mpc^3 Gyr^-1) * Mpc^-3 = Gyr^-1
coeff_rec_func = lambda z1: (clump_fact_func(z1) *
alpha_B *
nn * (1.+z1)**3.)
# Generate a function that converts age of the universe to z.
red_func = cd.quick_redshift_age_function(zmax = 1.1 * numpy.max(z),
zmin = -0.0,
dz = 0.01,
**cosmo)
ref_func_Gyr = lambda t1: red_func(t1 * cc.Gyr_s)
# Convert specified redshifts to cosmic time (age of the universe).
t = cd.age(z, **cosmo)[0]/cc.Gyr_s
# Integrate to find u(z) = x(z) - w(z), where w is the ionization fraction
u = si.odeint(_udot, y0=0.0, t=t,
args=(coeff_rec_func, ref_func_Gyr, ion_func, bubble))
u = u.flatten()
w = ion_func(z)
x = u + w
#x[x > 1.0] = 1.0
return x, w, t
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 integrate_optical_depth(x_ionH, x_ionHe, z, **cosmo):
"""The electron scattering optical depth given ionized filling
factor vs. redshift.
Parameters
----------
x_ionH: array
Ionized fraction of hydrogen as a function of z. Should be [0,1].
x_ionHe: array
Set x_ionHE to X_HeII + 2 * X_HeIII, where X_HeII is the
fraction of helium that is singly ionized, and X_HeII is the
fraction of helium that is doubly ionized. See Notes below.
z: array
Redshift values at which the filling factor is specified.
cosmo: cosmological parameters
uses: 'X_H' and/or 'Y_He', plus parameters needed for hubble_z
Returns
-------
tau: array
The optical depth as a function of z.
Notes
-----
The precision of your result depends on the spacing of the input
arrays. When in doubt, try doubling your z resolution and see if
the optical depth values have converged.
100% singly ionized helium means x_ionHe = 1.0, 100% doubly
ionized helium means x_ionHe = 2.0
If you want helium to be singly ionized at the same rate as
hydrogen, set x_ionHe = x_ionH.
If you want helium to be doubly ionized at the same rate as
hydrogen is ionized, set x_ionHe = 2 * x_ionH.
"""
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
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
# s^-1
H_z = cd.hubble_z(z, **cosmo)
# Mpc^3 s^-1 * Mpc^-3 / s^-1 -> unitless
integrand = -1. * tau_star * x * ((1. + z)**2.) / H_z
integral = numpy.empty(integrand.shape)
integral[...,1:] = si.cumtrapz(integrand, z)
integral[...,0] = 0.0
return numpy.abs(integral)
def integrate_ion_recomb_collapse(z, coeff_ion,
temp_min = 1e4,
passed_min_mass = False,
temp_gas=1e4,
alpha_B=None,
clump_fact_func = clumping_factor_BKP,
**cosmo):
"""IGM ionization state with recombinations from halo collapse
fraction. Integrates an ODE describing IGM ionization and
recombination rates.
z: array
The redshift values at which to calculate the ionized
fraction. This array should be in reverse numerical order. The
first redshift specified should be early enough that the
universe is still completely neutral.
coeff_ion:
The coefficient converting the collapse fraction to ionized
fraction, neglecting recombinations. Equivalent to the product
(f_star * f_esc_gamma * N_gamma) in the BKP paper.
temp_min:
See docs for ionization_from_collapse. Either the minimum virial
temperature or minimum mass of halos contributing to
reionization.
passed_temp_min:
See documentation for ionization_from_collapse.
temp_gas:
Gas temperature used to calculate the recombination coefficient
if alpha_b is not specified.
alpha_B:
Optional recombination coefficient in units of cm^3
s^-1. In alpha_B=None, it is calculated from temp_gas.
clump_fact_func: function
Function returning the clumping factor when given a redshift.
cosmo: dict
Dictionary specifying the cosmological parameters.
We assume, as is fairly standard, that the ionized
fraction is contained in fully ionized bubbles surrounded by a
fully neutral IGM. The output is therefore the volume filling
factor of ionized regions, not the ionized fraction of a
uniformly-ionized IGM.
I have also made the standard assumption that all ionized photons
are immediately absorbed, which allows the two differential
equations (one for ionization-recombination and one for
emission-photoionizaion) to be combined into a single ODE.
"""
# Determine recombination coefficient.
if alpha_B is None:
alpha_B_cm = recomb_rate_coeff_HG(temp_gas, 'H', 'B')
else:
alpha_B_cm = alpha_B
alpha_B = alpha_B_cm / (cc.Mpc_cm**3.)
print ("Recombination rate alpha_B = %.4g (Mpc^3 s^-1) = %.4g (cm^3 s^-1)"
% (alpha_B, alpha_B_cm))
# Normalize power spectrum.
if 'deltaSqr' not in cosmo:
cosmo['deltaSqr'] = cp.norm_power(**cosmo)
# Calculate useful densities.
rho_crit, rho_0, n_He_0, n_H_0 = cden.baryon_densities(**cosmo)
# Function used in the integration.
# Units: (Mpc^3 s^-1) * Mpc^-3 = s^-1
coeff_rec_func = lambda z: (clump_fact_func(z)**2. *
alpha_B *
n_H_0 * (1.+z)**3.)
# Generate a function that converts redshift to age of the universe.
redfunc = cd.quick_redshift_age_function(zmax = 1.1 * numpy.max(z),
zmin = -0.0,
**cosmo)
# Function used in the integration.
ionfunc = quick_ion_col_function(coeff_ion,
temp_min,
passed_min_mass = passed_min_mass,
zmax = 1.1 * numpy.max(z),
zmin = -0.0,
zstep = 0.1, **cosmo)
# Convert specified redshifts to cosmic time (age of the universe).
t = cd.age(z, **cosmo)
# Integrate to find u(z) = x(z) - w(z), where w is the ionization fraction
u = si.odeint(_udot, y0=0.0, t=t,
args=(coeff_rec_func, redfunc, ionfunc))
u = u.flatten()
w = ionization_from_collapse(z, coeff_ion, temp_min,
passed_min_mass = passed_min_mass,
**cosmo)
x = u + w
x[x > 1.0] = 1.0
return x, w, t
def integrate_ion_recomb(z,
ion_func,
clump_fact_func,
xHe=1.0,
temp_gas=1e4,
alpha_B=None,
bubble=True,
**cosmo):
"""Integrate IGM ionization and recombination given an ionization function.
Parameters:
z: array
The redshift values at which to calculate the ionized
fraction. This array should be in reverse numerical order. The
first redshift specified should be early enough that the
universe is still completely neutral.
ion_func:
A function giving the ratio of the total density of emitted
ionizing photons to the density hydrogen atoms (or hydrogen
plus helium, if you prefer) as a function of redshift.
temp_gas:
Gas temperature used to calculate the recombination coefficient
if alpha_b is not specified.
alpha_B:
Optional recombination coefficient in units of cm^3
s^-1. In alpha_B=None, it is calculated from temp_gas.
clump_fact_func: function
Function returning the clumping factor when given a redshift,
defined as <n_HII^2>/<n_HII>^2.
cosmo: dict
Dictionary specifying the cosmological parameters.
Notes:
We only track recombination of hydrogen, but if xHe > 0, then the
density is boosted by the addition of xHe * nHe. This is
eqiuvalent to assuming the the ionized fraction of helium is
always proportional to the ionized fraction of hydrogen. If
xHe=1.0, then helium is singly ionized in the same proportion as
hydrogen. If xHe=2.0, then helium is fully ionized in the same
proportion as hydrogen.
We assume, as is fairly standard, that the ionized
fraction is contained in fully ionized bubbles surrounded by a
fully neutral IGM. The output is therefore the volume filling
factor of ionized regions, not the ionized fraction of a
uniformly-ionized IGM.
I have also made the standard assumption that all ionized photons
are immediately absorbed, which allows the two differential
equations (one for ionization-recombination and one for
emission-photoionizaion) to be combined into a single ODE.
"""
# Determine recombination coefficient.
if alpha_B is None:
alpha_B_cm = recomb_rate_coeff_HG(temp_gas, 'H', 'B')
else:
alpha_B_cm = alpha_B
alpha_B = alpha_B_cm * cc.Gyr_s / (cc.Mpc_cm**3.)
print ("Recombination rate alpha_B = %.4g (Mpc^3 Gyr^-1) = %.4g (cm^3 s^-1)"
% (alpha_B, alpha_B_cm))
# Normalize power spectrum.
if 'deltaSqr' not in cosmo:
cosmo['deltaSqr'] = cp.norm_power(**cosmo)
# Calculate useful densities.
rho_crit, rho_0, n_He_0, n_H_0 = cden.baryon_densities(**cosmo)
# Boost density to approximately account for helium.
nn = (n_H_0 + xHe * n_He_0)
# Function used in the integration.
# Units: (Mpc^3 Gyr^-1) * Mpc^-3 = Gyr^-1
coeff_rec_func = lambda z1: (clump_fact_func(z1) *
alpha_B *
nn * (1.+z1)**3.)
# Generate a function that converts age of the universe to z.
red_func = cd.quick_redshift_age_function(zmax = 1.1 * numpy.max(z),
zmin = -0.0,
dz = 0.01,
**cosmo)
ref_func_Gyr = lambda t1: red_func(t1 * cc.Gyr_s)
# Convert specified redshifts to cosmic time (age of the universe).
t = cd.age(z, **cosmo)[0]/cc.Gyr_s
# Integrate to find u(z) = x(z) - w(z), where w is the ionization fraction
u = si.odeint(_udot, y0=0.0, t=t,
args=(coeff_rec_func, ref_func_Gyr, ion_func, bubble))
u = u.flatten()
w = ion_func(z)
x = u + w
#x[x > 1.0] = 1.0
return x, w, t