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_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 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