Soft Photon Heating: A Semi-Analytic Framework and Applications to $21$cm Cosmology

The presence of an abundant population of low frequency photons at high redshifts (such as a radio background) can source leading order effects on the evolution of the matter and spin temperatures through rapid free-free absorptions. This effect, known as soft photon heating, can have a dramatic impact on the differential brightness temperature, $\Delta T_{\rm b}$, a central observable in $21$cm cosmology. Here, we introduce a semi-analytic framework to describe the dynamics of soft photon heating, providing a simplified set of evolution equations and a useful numerical scheme which can be used to study this generic effect. We also perform quasi-instantaneous and continuous soft photon injections to elucidate the different regimes in which soft photon heating is expected to impart a significant contribution to the global $21$cm signal and its fluctuations. We find that soft photon backgrounds produced after recombination with spectral index $\gamma > 3.0$ undergo significant free-free absorption, and therefore this heating effect cannot be neglected. The effect becomes stronger with steeper spectral index, and in some cases the injection of a synchrotron-like spectrum ($\gamma = 3.6$) can suppress the amplitude of $\Delta T_{\rm b}$ relative to the standard model prediction, making the global $21$cm signal even more difficult to detect in these scenarios.