def optical_depth(self, use_recomb=True, store=True):
"""Calculate Thompson electron scattering optical depth."""
# First calculate the optical depth from the fully-ionized
# epoch up to our minimum redshift.
tau_z0 = cr.optical_depth_instant(self.z[-1], **self.cosmo)
# Then the contribution from the partially-ionized epoch.
if use_recomb:
xPhys = self.xr.copy()
else:
xPhys = self.xEsc.copy()
xPhys[xPhys > 1.0] = 1.0
tau_later = (tau_z0 +
cr.integrate_optical_depth(xPhys[...,::-1],
xPhys[...,::-1],
self.z[::-1],
**self.cosmo))
tau = tau_later[...,::-1]
if store:
if use_recomb:
self.tau = tau
else:
self.tauEsc = tau
return tau
def test_GBL_tau_inst():
"""Test match between analytical and numerical tau with instant
reionization.
Also makes a plot reproducing figure 1 of arXiv:astro-ph/9812125v3.
"""
dz = 0.05
z = numpy.arange(0., 80. + 1.5 * dz, dz)
# Fully ionized H and He
x_ionH = 1.0
x_ionHe = 2.0
cosmo = {}
cosmo['omega_M_0'] = numpy.array([[0.3], [0.6], [1.0]])
cosmo['omega_lambda_0'] = 1. - cosmo['omega_M_0']
cosmo['h'] = 0.65
cosmo['omega_b_0'] = 0.02 / cosmo['h']**2.
cosmo['Y_He'] = 0.24
cd.set_omega_k_0(cosmo)
tau_inst = cr.optical_depth_instant(z,
x_ionH=x_ionH,
x_ionHe=x_ionHe,
**cosmo)
tau_int = cr.integrate_optical_depth(x_ionH, x_ionHe, z, **cosmo)
linestyle = ['-', ':', '--']
pylab.figure()
pylab.subplot(2, 1, 1)
pylab.title("Compare to GB&L fig. 1 (astro-ph/9812125v3.)")
for i in range(len(linestyle)):
pylab.plot(z, tau_inst[i], ls=linestyle[i], color='b')
pylab.plot(z, tau_int[i], ls=linestyle[i], color='r')
pylab.xlim(0, 80)
pylab.ylim(0, 1)
pylab.xlabel(r"$\mathrm{z_{ion}}$")
pylab.ylabel(r"$\tau$")
pylab.subplot(2, 1, 2)
for i in range(len(linestyle)):
pylab.plot(z,
1.e4 * (tau_int[i] - tau_inst[i]) / tau_inst[i],
ls=linestyle[i],
color='k')
diff = (tau_int[i] - tau_inst[i]) / tau_inst[i]
diff[numpy.isnan(diff)] = 0.0
print("max fractional error in num. int. = %.3g" %
numpy.max(numpy.abs(diff)))
ntest.assert_array_less(numpy.abs(diff),
numpy.zeros(diff.shape) + 2.e-4)
pylab.xlim(0, 40)
pylab.xlabel(r"$\mathrm{z_{ion}}$")
pylab.ylabel(r"$\mathrm{10^4 \times (num.\tau - ana.\tau)/ana.\tau}$")
def test_GBL_tau_inst():
"""Test match between analytical and numerical tau with instant
reionization.
Also makes a plot reproducing figure 1 of arXiv:astro-ph/9812125v3.
"""
dz = 0.05
z = numpy.arange(0., 80. + 1.5*dz, dz)
# Fully ionized H and He
x_ionH = 1.0
x_ionHe = 2.0
cosmo = {}
cosmo['omega_M_0'] = numpy.array([[0.3],[0.6],[1.0]])
cosmo['omega_lambda_0'] = 1. - cosmo['omega_M_0']
cosmo['h'] = 0.65
cosmo['omega_b_0'] = 0.02 / cosmo['h']**2.
cosmo['Y_He'] = 0.24
cd.set_omega_k_0(cosmo)
tau_inst = cr.optical_depth_instant(z, x_ionH=x_ionH, x_ionHe=x_ionHe,
**cosmo)
tau_int = cr.integrate_optical_depth(x_ionH, x_ionHe, z, **cosmo)
linestyle = ['-', ':', '--']
pylab.figure()
pylab.subplot(2,1,1)
pylab.title("Compare to GB&L fig. 1 (astro-ph/9812125v3.)")
for i in range(len(linestyle)):
pylab.plot(z, tau_inst[i], ls=linestyle[i], color='b')
pylab.plot(z, tau_int[i], ls=linestyle[i], color='r')
pylab.xlim(0,80)
pylab.ylim(0,1)
pylab.xlabel(r"$\mathrm{z_{ion}}$")
pylab.ylabel(r"$\tau$")
pylab.subplot(2,1,2)
for i in range(len(linestyle)):
pylab.plot(z,
1.e4 * (tau_int[i] - tau_inst[i])/tau_inst[i],
ls=linestyle[i], color='k')
diff = (tau_int[i] - tau_inst[i]) / tau_inst[i]
diff[numpy.isnan(diff)] = 0.0
print ("max fractional error in num. int. = %.3g" %
numpy.max(numpy.abs(diff))
)
ntest.assert_array_less(numpy.abs(diff),
numpy.zeros(diff.shape) + 2.e-4)
pylab.xlim(0,40)
pylab.xlabel(r"$\mathrm{z_{ion}}$")
pylab.ylabel(r"$\mathrm{10^4 \times (num.\tau - ana.\tau)/ana.\tau}$")
def test_tau_instant():
"""Check the optical depth we get using the WMAP z_reion.
"""
dz = 0.1
z = numpy.arange(80., 0. - 1.5 * dz, -1. * dz)
# Can't match WMAP7 optical depths exactly. Need to look into new
# treatment in CAMB as mentioned in the WMAP7 paper (see
# parameters.py).
cosmos = [
cparam.WMAP7_BAO_H0_mean(flat=True),
cparam.WMAP7_ML(flat=True),
cparam.WMAP5_BAO_SN_mean(flat=True),
cparam.WMAP5_ML(flat=True),
cparam.WMAP5_mean(flat=True)
]
# The WMAP5 numbers apparently assume He is neutral, while WMAP7
# includes simultaneous singly ionized He plus Helium (double)
# reionization at z=3.5.
x_ionHe_list = [1.0, 1.0, 0, 0, 0]
z_rHe_list = [3.5, 3.5, None, None, None] #[3.5, None, None, None]
for cosmo in cosmos:
# Fully ionized H
x_ionH = 1.0
# He ionization with H?
x_ionHe = x_ionHe_list.pop(0)
# Redshift for Helium double reionization?
z_rHe = z_rHe_list.pop(0)
zr = cosmo['z_reion']
tau_zr = cosmo['tau']
tau_calc = cr.optical_depth_instant(zr,
x_ionH=x_ionH,
x_ionHe=x_ionHe,
z_rHe=z_rHe,
verbose=1,
**cosmo)
print "z_r = %f, testing tau:" % zr,
print tu.fractional_diff_string(tau_zr, tau_calc, 3)
ntest.assert_approx_equal(tau_calc,
tau_zr,
significant=2,
err_msg="Optical depth doesn't match WMAP")
def test_GBL_tau_star():
"""Test tau_* against GB&L astro-ph/9812125v3.
tau_* is a quantity used in optical_depth_instant.
"""
z = 1.0
# Fully ionized H and He
x_ionH = 1.0
x_ionHe = 2.0
cosmo = {}
cosmo['omega_M_0'] = numpy.array([[0.3],[0.6],[1.0]])
cosmo['omega_lambda_0'] = 1. - cosmo['omega_M_0']
cosmo['h'] = 0.65
cosmo['omega_b_0'] = 0.02 / cosmo['h']**2.
cosmo['Y_He'] = 0.24
cd.set_omega_k_0(cosmo)
tau_inst, tau_star = cr.optical_depth_instant(z,
x_ionH=x_ionH,
x_ionHe=x_ionHe,
return_tau_star=True,
**cosmo)
print "tau_star = %.7f" % tau_star
print ("tau_star/(h Omega_b) = %.7f =? 0.061" %
(tau_star / (cosmo['h'] * cosmo['omega_b_0'])))
ntest.assert_approx_equal(tau_star / (cosmo['h'] * cosmo['omega_b_0']),
0.061,
2)
print "(1 - Y_He/2) = %.3f =? 0.88" % (1. - (cosmo['Y_He']/2.))
ntest.assert_approx_equal((1. - (cosmo['Y_He']/2.)),
0.88,
7)
H_0 = cc.H100_s * cosmo['h']
# s^-1 * Mpc s^-1 * Mpc^2 / Mpc^3 msun^-1 s^-2 / Msun ->
tau_star_explicit = ((1. - (cosmo['Y_He']/2.)) *
((3. * H_0 * cosmo['omega_b_0'] * cc.c_light_Mpc_s *
cc.sigma_T_Mpc) /
(8. * math.pi * cc.G_const_Mpc_Msun_s *
(cc.m_p_g/cc.M_sun_g))))
print "tau_star_explicit = %.7f =? tau_star" % tau_star_explicit
ntest.assert_approx_equal(tau_star, tau_star_explicit, 3)
def test_tau_instant():
"""Check the optical depth we get using the WMAP z_reion.
"""
dz = 0.1
z = numpy.arange(80., 0. - 1.5 * dz, -1. * dz)
# Can't match WMAP7 optical depths exactly. Need to look into new
# treatment in CAMB as mentioned in the WMAP7 paper (see
# parameters.py).
cosmos = [cparam.WMAP7_BAO_H0_mean(flat=True),
cparam.WMAP7_ML(flat=True),
cparam.WMAP5_BAO_SN_mean(flat=True),
cparam.WMAP5_ML(flat=True),
cparam.WMAP5_mean(flat=True)]
# The WMAP5 numbers apparently assume He is neutral, while WMAP7
# includes simultaneous singly ionized He plus Helium (double)
# reionization at z=3.5.
x_ionHe_list = [1.0, 1.0, 0, 0, 0]
z_rHe_list = [3.5, 3.5, None, None, None] #[3.5, None, None, None]
for cosmo in cosmos:
# Fully ionized H
x_ionH = 1.0
# He ionization with H?
x_ionHe = x_ionHe_list.pop(0)
# Redshift for Helium double reionization?
z_rHe = z_rHe_list.pop(0)
zr = cosmo['z_reion']
tau_zr = cosmo['tau']
tau_calc = cr.optical_depth_instant(zr,
x_ionH=x_ionH,
x_ionHe=x_ionHe,
z_rHe=z_rHe,
verbose = 1,
**cosmo)
print "z_r = %f, testing tau:" % zr,
print tu.fractional_diff_string(tau_zr, tau_calc, 3)
ntest.assert_approx_equal(tau_calc, tau_zr, significant=2,
err_msg="Optical depth doesn't match WMAP")
def test_GBL_tau_star():
"""Test tau_* against GB&L astro-ph/9812125v3.
tau_* is a quantity used in optical_depth_instant.
"""
z = 1.0
# Fully ionized H and He
x_ionH = 1.0
x_ionHe = 2.0
cosmo = {}
cosmo['omega_M_0'] = numpy.array([[0.3], [0.6], [1.0]])
cosmo['omega_lambda_0'] = 1. - cosmo['omega_M_0']
cosmo['h'] = 0.65
cosmo['omega_b_0'] = 0.02 / cosmo['h']**2.
cosmo['Y_He'] = 0.24
cd.set_omega_k_0(cosmo)
tau_inst, tau_star = cr.optical_depth_instant(z,
x_ionH=x_ionH,
x_ionHe=x_ionHe,
return_tau_star=True,
**cosmo)
print("tau_star = %.7f" % (tau_star))
print("tau_star/(h Omega_b) = %.7f =? 0.061" %
(tau_star / (cosmo['h'] * cosmo['omega_b_0'])))
ntest.assert_approx_equal(tau_star / (cosmo['h'] * cosmo['omega_b_0']),
0.061, 2)
print("(1 - Y_He/2) = %.3f =? 0.88" % (1. - (cosmo['Y_He'] / 2.)))
ntest.assert_approx_equal((1. - (cosmo['Y_He'] / 2.)), 0.88, 7)
H_0 = cc.H100_s * cosmo['h']
# s^-1 * Mpc s^-1 * Mpc^2 / Mpc^3 msun^-1 s^-2 / Msun ->
tau_star_explicit = ((1. - (cosmo['Y_He'] / 2.)) * (
(3. * H_0 * cosmo['omega_b_0'] * cc.c_light_Mpc_s * cc.sigma_T_Mpc) /
(8. * math.pi * cc.G_const_Mpc_Msun_s * (cc.m_p_g / cc.M_sun_g))))
print("tau_star_explicit = %.7f =? tau_star" % (tau_star_explicit))
ntest.assert_approx_equal(tau_star, tau_star_explicit, 3)
def test_tau_BKP():
"""Reionization consistency check with BK&P 2009MNRAS.397..971B
Bagla J.~S., Kulkarni G., Padmanabhan T., 2009 (2009MNRAS.397..971B).
Take their cannonical set of cosmological parameters and ionization
coefficients and make sure I get the same optical depth value.
The first figure demonstrates the calculation of the ionization
fraction and the optical depth (not shown in the paper).
The second figure shows dependence of the f_* f_esc,gamma values on
tau (actually calculated the other way around, f_* f_esc,gamma ->
tau). The plotted point (x) marks the value from the paper. The
agreement between the calculated optical depth and the WMAP tau value
is printed in the text output.
"""
N_gamma = 6804.
# f_* f_esc N_gamma
f_ion = numpy.transpose(numpy.atleast_2d(numpy.arange(10., 200., 10.)))
alpha_B = 1e-13 # cm^3 s^-1
m_min = 1e8 # M_sun
x_ionHe = 2.0
dz = 0.1
z = numpy.arange(20., 6. - 1.5 * dz, -1. * dz)
#z = numpy.arange(6., 20. + 1.5 * dz, dz)
cosmos = [cparam.WMAP5_ML(flat=True),
cparam.WMAP5_mean(flat=True)]
pylab.figure(figsize=(8,8))
colors=['r', 'k']
names = ["WMAP5 ML", "WMAP5 mean"]
i = -1
for cosmo in cosmos:
i += 1
print("\n%s" % (names[i]))
# calculate ionized fraction, including recombinations
x_rec, w_rec, t = cr.integrate_ion_recomb_collapse(z, f_ion,
m_min,
passed_min_mass=True,
alpha_B=alpha_B,
**cosmo)
# calculate the optical depth in this scenario
tau_z0 = cr.optical_depth_instant(z[-1],
x_ionH=1.0,
x_ionHe=2.0,
**cosmo)
tau_later = cr.integrate_optical_depth(x_rec, x_ionHe * x_rec,
z, **cosmo)
tau_0 = tau_later[:, -1] + tau_z0
tau = tau_later
print("tau(WMAP) = %.3f" % (cosmo['tau']))
for j in range(len(f_ion.flat)):
if round(f_ion[j],1) != 50.0:
continue
print("with f_* f_esc_gamma N_gamma = %.1f:" % (f_ion[j]))
pylab.plot(z, x_rec[j], ls='-', color=colors[i])
pylab.plot(z, w_rec[j], ls=':', color=colors[i])
#pylab.plot(z, 10. * tau[j], ls='--', color=colors[i])
pylab.plot(z, 10. * (tau_0[j] - tau[j]), ls='--', color=colors[i])
pylab.axhline(y=10. * tau_0[j], ls='--', color=colors[i])
print("tau(z=0) = %.4f" % (tau_0[j]))
print("fractional diff. = %.3g" % ((tau_0[j] - cosmo['tau']) /
cosmo['tau']))
if i==1:
# Make sure we recover the WMAP value.
ntest.assert_approx_equal(tau_0[j], cosmo['tau'], 2)
pylab.ylim(0,1.01)
pylab.xlabel("redshift z")
pylab.ylabel(r"ionized fraction or optical depth $\tau \times 10$")
pylab.figure(figsize=(8,8))
pylab.plot(tau_0, f_ion / N_gamma, '-', color='k')
pylab.plot([cosmo['tau']], [50.0 / N_gamma], 'x', color='b')
pylab.xlim(0.06, 0.12)
pylab.ylim(0., 0.022)
pylab.xlabel(r'$\tau$')
pylab.ylabel(r'$f_* f_{esc,\gamma}$')
def test_tau_BKP():
"""Reionization consistency check with BK&P 2009MNRAS.397..971B
Bagla J.~S., Kulkarni G., Padmanabhan T., 2009 (2009MNRAS.397..971B).
Take their cannonical set of cosmological parameters and ionization
coefficients and make sure I get the same optical depth value.
The first figure demonstrates the calculation of the ionization
fraction and the optical depth (not shown in the paper).
The second figure shows dependence of the f_* f_esc,gamma values on
tau (actually calculated the other way around, f_* f_esc,gamma ->
tau). The plotted point (x) marks the value from the paper. The
agreement between the calculated optical depth and the WMAP tau value
is printed in the text output.
"""
N_gamma = 6804.
# f_* f_esc N_gamma
f_ion = numpy.transpose(numpy.atleast_2d(numpy.arange(10., 200., 10.)))
alpha_B = 1e-13 # cm^3 s^-1
m_min = 1e8 # M_sun
x_ionHe = 2.0
dz = 0.1
z = numpy.arange(20., 6. - 1.5 * dz, -1. * dz)
#z = numpy.arange(6., 20. + 1.5 * dz, dz)
cosmos = [cparam.WMAP5_ML(flat=True),
cparam.WMAP5_mean(flat=True)]
pylab.figure(figsize=(8,8))
colors=['r', 'k']
names = ["WMAP5 ML", "WMAP5 mean"]
i = -1
for cosmo in cosmos:
i += 1
print
print names[i]
# calculate ionized fraction, including recombinations
x_rec, w_rec, t = cr.integrate_ion_recomb_collapse(z, f_ion,
m_min,
passed_min_mass=True,
alpha_B=alpha_B,
**cosmo)
# calculate the optical depth in this scenario
tau_z0 = cr.optical_depth_instant(z[-1],
x_ionH=1.0,
x_ionHe=2.0,
**cosmo)
tau_later = cr.integrate_optical_depth(x_rec, x_ionHe * x_rec,
z, **cosmo)
tau_0 = tau_later[:, -1] + tau_z0
tau = tau_later
print "tau(WMAP) = %.3f" % cosmo['tau']
for j in range(len(f_ion.flat)):
if round(f_ion[j],1) != 50.0:
continue
print "with f_* f_esc_gamma N_gamma = %.1f:" % f_ion[j]
pylab.plot(z, x_rec[j], ls='-', color=colors[i])
pylab.plot(z, w_rec[j], ls=':', color=colors[i])
#pylab.plot(z, 10. * tau[j], ls='--', color=colors[i])
pylab.plot(z, 10. * (tau_0[j] - tau[j]), ls='--', color=colors[i])
pylab.axhline(y=10. * tau_0[j], ls='--', color=colors[i])
print "tau(z=0) = %.4f" % tau_0[j]
print " fractional diff. = %.3g" % ((tau_0[j] - cosmo['tau']) /
cosmo['tau'])
if i==1:
# Make sure we recover the WMAP value.
ntest.assert_approx_equal(tau_0[j], cosmo['tau'], 2)
pylab.ylim(0,1.01)
pylab.xlabel("redshift z")
pylab.ylabel(r"ionized fraction or optical depth $\tau \times 10$")
pylab.figure(figsize=(8,8))
pylab.plot(tau_0, f_ion / N_gamma, '-', color='k')
pylab.plot([cosmo['tau']], [50.0 / N_gamma], 'x', color='b')
pylab.xlim(0.06, 0.12)
pylab.ylim(0., 0.022)
pylab.xlabel(r'$\tau$')
pylab.ylabel(r'$f_* f_{esc,\gamma}$')