Cosmic Recombination

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Cosmic Recombination

This page describes the recombination physics implemented by the abstract NcRecomb class and its concrete Seager solver NcRecombSeager. Further background is given in Weinberg (2008), Cosmology.

Abundances and Notation

Let \(n_\Hy\) be the total number density of hydrogen nuclei (ionized or not), split into neutral \(n_\HyI\) and ionized \(n_\HyII\) with \(n_\HyI + n_\HyII = n_\Hy\). Similarly \(n_\He\) is the number density of helium nuclei, with neutral, single, and double ionized fractions \(n_\HeI\), \(n_\HeII\), and \(n_\HeIII\).

The helium primordial abundance is the ratio of the helium mass to the total baryonic mass, \[ Y_p = \frac{n_\He m_\He}{n_\He m_\He + n_\Hy m_\Hy}, \] where \(m_\Hy\) and \(m_\He\) are the hydrogen and helium masses. Element abundances are defined relative to the number of free protons \(n_p \equiv n_\Hy\), \[ X_f = \frac{n_f}{n_p}, \] for any element \(f\), with \(\e\) denoting free electrons. These fractions satisfy \[ \begin{aligned} X_\HyI + X_\HyII &= 1, \\ X_\HeI + X_\HeII + X_\HeIII &= X_\He, \\ X_\He &\equiv \frac{m_p}{m_\He}\frac{Y_p}{1 - Y_p}. \end{aligned} \] Assuming a neutral universe, the free-electron fraction is \[ X_\e = X_\HyII + X_\HeII + 2 X_\HeIII. \]

Equilibrium Fractions

In equilibrium, the ionized-to-neutral ratios follow the Saha equation. For hydrogen, \[ \frac{X_\HyII X_\e}{X_\HyI} = \frac{e^{-\HyI_{1s}/(k_B T)}}{n_\Hy \lambda_\e^3}, \] where \(\HyI_{1s}\) is the hydrogen \(1s\) binding energy and \(\lambda_\e\) is the electron thermal wavelength, \[ \lambda_\e = \sqrt{\frac{2\pi\hbar^2}{m_\e k_B T}}, \] with \(k_B\) the Boltzmann constant, \(m_\e\) the electron mass, and \(\hbar\) the reduced Planck constant. For singly ionized helium, \[ \frac{X_\HeII X_\e}{X_\HeI} = \frac{e^{-\HeI_{1s}/(k_B T)}}{4 n_\Hy \lambda_\e^3}, \] and for doubly ionized helium, \[ \frac{X_\HeIII X_\e}{X_\HeII} = \frac{e^{-\HeII_{1s}/(k_B T)}}{4 n_\Hy \lambda_\e^3}, \] where \(\HeI_{1s}\) and \(\HeII_{1s}\) are the corresponding \(1s\) binding energies.

Optical Depth and Visibility Function

Using the redshift time \(\lambda \equiv -\ln(x) = -\ln(1+z)\), the derivative of the optical depth \(\tau\) is \[ \frac{\mathrm{d}\tau}{\mathrm{d}\lambda} = -\frac{c\,\sigma_T n_B X_\e}{H}, \] where \(c\) is the speed of light, \(\sigma_T\) the Thomson cross section, \(n_B\) the baryon number density, and \(H\) the Hubble function. The optical depth is integrated from the present time, \[ \tau = \int_0^\lambda \frac{\mathrm{d}\tau}{\mathrm{d}\lambda}, \] and the visibility function is \[ v_\tau = \frac{\mathrm{d}\tau}{\mathrm{d}\lambda}\,e^{-\tau}. \]

Seager Solver

NcRecombSeager implements cosmic recombination following Seager (1999) and Seager (2000), with the modifications of recfast 1.5.2 — including those discussed in Wong (2008). The matter-temperature modification of Scott (2009) is not included, as the more robust integration method used here makes it unnecessary.

The code solves the coupled system for the singly ionized hydrogen \(X_\HyII\), singly ionized helium \(X_\HeII\), and the baryon temperature \(T_m\): \[ \begin{aligned} \frac{\mathrm{d}X_\HyII}{\mathrm{d}x} &= \frac{X_\HyII X_\e n_\Hy - X_\HyI B_{\HyI, 1s\,{}^2\!S_{1/2}}(T_m)}{H x}\left[\alpha_\Hy(T_m)\frac{n_\Hy K_\HyI X_\HyI \Lambda_\Hy + 1}{n_\Hy K_\HyI X_\HyI \left[\Lambda_\Hy + B_{\HyI, 2s\,{}^2\!S_{1/2}}(T_m)\,\alpha_\Hy(T_m)\right] + 1}\right], \\ \frac{\mathrm{d}T_m}{\mathrm{d}x} &= \frac{c}{Hx}\frac{8\sigma_\mathrm{T} a_\mathrm{R} T_r^4}{3 m_\mathrm{e} c^2}\frac{X_\e(T_m - T_r)}{1 + X_\He + X_\e} + \frac{2T_m}{x}, \\ \frac{\mathrm{d}X_\HeII}{\mathrm{d}x} &= \frac{X_\HeII X_\e n_\Hy - X_\HeI B_{\HeI, 1s\,{}^1\!S_0}(T_m)}{H x}\Bigg\{\left[\alpha_\He(T_m)\frac{n_\Hy K_\HeI X_\HeI \Lambda_\He + B^{\HeI, 2s\,{}^1\!S_0}_{\HeI, 2p\,{}^1\!P_1}(T_m)}{n_\Hy K_\HeI X_\HeI \left[\Lambda_\He + B_{\HeI, 2s\,{}^1\!S_0}(T_m)\,\alpha_\He(T_m)\right] + B^{\HeI, 2s\,{}^1\!S_0}_{\HeI, 2p\,{}^1\!P_1}(T_m)}\right] \\ &\quad + \alpha_\He^\mathrm{t}(T_m)\frac{1}{n_\Hy K_\HeI^\mathrm{t} X_\HeI B_{\HeI, 2p\,{}^3\!P_\mathrm{mean}}(T_m)\,\alpha_\He^\mathrm{t}(T_m) + 1}\Bigg\}. \end{aligned} \]

Boltzmann Factors

The Boltzmann factors for the hydrogen levels are \[ B_{\HyI, l}(T_m) = k_\mathrm{e}^3(T_m)\,\exp\left[-E_{\HyI, l}/(k_B T_m)\right], \] for \(l = 1s\,{}^2\!S_{1/2},\, 2s\,{}^2\!S_{1/2}\), where \(k_\mathrm{e}\) is the electron thermal wavenumber. For the helium-I levels, \[ B_{\HeI, l}(T_m) = 4 k_\mathrm{e}^3(T_m)\,\exp\left[-E_{\HeI, l}/(k_B T_m)\right], \] with levels \(l = 1s\,{}^1\!S_0,\, 2s\,{}^1\!S_0,\, 2p\,{}^1\!P_1\). The symbol \(B^{\HeI, 2s\,{}^1\!S_0}_{\HeI, 2p\,{}^1\!P_1}(T_m)\) denotes the ratio of two Boltzmann factors, \[ B^{\HeI, 2s\,{}^1\!S_0}_{\HeI, 2p\,{}^1\!P_1}(T_m) = \exp\left[-\left(E_{\HeI, 2s\,{}^1\!S_0} - E_{\HeI, 2p\,{}^1\!P_1}\right)/(k_B T_m)\right]. \]

Rates and Coefficients

The two-photon decay rates are \(\Lambda_\Hy\) for hydrogen and \(\Lambda_\He\) for helium-I. The Case B recombination coefficient for hydrogen, \(\alpha_\Hy\), uses the Pequignot (1991) fit; the helium-I coefficients \(\alpha_\He\) and \(\alpha_\He^\mathrm{t}\) (triplet) use the Hummer (1998) fits. The NcRecombSeagerOpt flags select which \(K\) factors are used and whether the triplet contribution is included in the helium rate.

The integration starting point in \(\lambda\) is obtained from the helium Saha equilibrium, and the system is integrated over all components without switching or approximation.

API Reference

See NcRecomb (abstract base) and NcRecombSeager (Seager solver). The most relevant methods are:

Physical constants and binding energies are provided by NcmC.