The heating rate of the Intergalactic Medium by Lyman-$\alpha$ photon scattering

The strength of the 21-cm Cosmic Dawn radio absorption signature against the Cosmic Microwave Background or against a bright background radio source depends on the temperature of the neutral hydrogen in the intergalactic medium. While x-rays from forming galaxies will likely dominate the heating rate, recent models suggest scenarios in which heating by the scattering of Lyman-$\alpha$ photons sourced by the galaxies contributes non-negligibly as well, and may even dominate. The efficiency of Lyman-$\alpha$ photon heating for a point source at high redshift using a numerical solution to the exact radiative transfer equation is provided here. It is found to be a factor of several smaller than the rate computed in the commonly used diffusion approximation to the radiative transfer equation.