A WENO algorithm for radiative transfer with resonant scattering and the Wouthuysen-Field coupling

We develop a numerical solver for the integral-differential equations, which describe the radiative transfer of photon distribution in the frequency space with resonant scattering of Lyα photons by hydrogen gas in the early universe. The time-dependent solutions of this equation is crucial to the estimation of the effect of the Wouthuysen-Field (WF) coupling in relation to the 21 cm emission and absorption at the epoch of reionization. However, the time-dependent solutions of this equation have not yet been well performed. The resonant scattering leads to the photon distribution in the frequency space to be piecewise smooth containing sharp changes. The weighted essentially nonoscillatory (WENO) scheme is suitable to handle this problem, as this algorithm has been found to be highly stable and robust for solving Boltzmann equation. We test this numerical solver by (1) the analytic solutions of the evolution of the photon distribution in rest background; (2) the analytic solution in expanding background, but without resonant scattering; (3) the formation of local Boltzmann distribution around the resonant frequency with the temperature to be the same as that of atom for recoil. We find that the evolution of the photon distribution due to resonant scattering with and without recoil generally undergoes three phases. First, the profile of the photon distribution is similar to the initial one. Second, an extremely flat plateau (without recoil) or local Boltzmann distribution (with recoil) form around the resonant frequency, and the width and height of the flat plateau or local Boltzmann distribution increase with time. Finally, the distribution around the resonant frequency is saturated when the photons from the source is balanced by the redshift of the expansion. This result indicates that the onset of the W-F coupling should not be determined by the third phase, but by the time scale of the second phase. We found that the time scale of the W-F coupling is equal to about a few hundreds of the mean free flight time of photons with resonant frequency, and it basically is independent of the Sobolev parameter if this parameter is much less than 1.