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