Semi-analytical approach to Ly$\alpha$ multiple-scattering in 21-cm signal simulations

A crucial physical quantity in determining the 21-cm signal during cosmic dawn is the inhomogeneous background of Ly$\alpha$ photons originating from the first galaxies. As these photons travel through the intergalactic medium, their scattering cross-section is often approximated as a delta function at resonance due to computational cost. That is, photons with emitted wavelengths between Ly$\alpha$ and Ly$\beta$ are assumed to travel in straight lines until they redshift into the Ly$\alpha$ resonance. However, due to the damping wing in the Ly$\alpha$ cross-section, this approximation fails as the frequency of the photon approaches the resonant frequency, resulting in multiple scatterings events that could be separated by non-negligible distances. Some previous works studied this effect of Ly$\alpha$ multiple scattering by running computationally heavy radiative-transfer simulations. However, robustly interpreting the cosmic 21cm signal requires exploring a large parameter space of astrophysical uncertainties, motivating more computationally-efficient approaches. Here we incorporate Ly$\alpha$ multiple scatterings in the public, semi-numerical simulation 21cmFAST. We employ Monte Carlo simulations to study the trajectories of Ly$\alpha$ photons on different scales. We find that the distance distributions of Ly$\alpha$ photons with respect to the absorption point can be modeled as analytical functions that are governed by a single parameter. Upon implementing the distance distributions in 21cmFAST, we find that the multiple scattering effect is important (about 50% difference in the 21-cm power spectrum) only at high redshifts before the spin temperature is fully coupled to the kinetic temperature. Furthermore, we find that Ly$\alpha$ multiple scattering does not enhance Ly$\alpha$ heating, and that the combined effect is negligible, especially under realistic X-ray heating scenarios.