A WENO algorithm for radiative transfer with resonant scattering: the time scale of the Wouthuysen-Field Coupling

We develop a numerical solver for the integral-differential equations, which describes the radiative transfer of photon distribution in the frequency space with resonant scattering of Lyalpha 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. 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 against analytic solutions of the evolution of the photon distribution in rest background, analytic solution in expanding background without resonant scattering and formation of local Boltzmann distribution around the resonant frequency with the temperature same as that of atom for recoil. We find that evolution of photon distribution undergoes three phases; profile is similar to the initial one, a flat plateau (without recoil) or local Boltzmann distribution (with recoil) forms around the resonant frequency, and 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 is independent of the Sobolev parameter if this parameter is much less than 1.
Comment: 20 pages, 13 figures