Radiative transfer of Lyman-$\alpha$ photons at cosmic dawn with realistic gas physics

Lyman-$\alpha$ photons enable the cosmic dawn 21-cm signal through a process called the Wouthuysen-Field effect. An accurate model of the signal in this epoch hinges on the accuracy of the computation of the Ly$\alpha$ coupling, which requires one to calculate the specific intensity of Ly$\alpha$ photons emitted from the first stars. Most traditional calculations of the Ly$\alpha$ coupling assume a delta-function scattering cross-section, as the resonant nature of the Ly$\alpha$ scattering makes an accurate radiative transfer (RT) solution computationally expensive. Attempts to improve upon this traditional approach using numerical RT have recently emerged. However, some of these treatments suffer from assumptions such as a uniform gas distribution, coherent scattering in the gas frame and isotropic scattering. While others which do not account for these only do so through certain schemes along with core-skipping algorithms. We present results from a self-consistent Monte Carlo RT simulations devoid of any of the assumptions in the previous work for the first time. We find that gas bulk motion is the most important effect to account for in RT resulting in an RMS difference of 38% in the 21-cm signal and anisotropic scattering being the least important effect contributing to less than 3% RMS difference in 21-cm signal. We also evaluate the 21-cm power spectrum and compare that with the traditional results at cosmic dawn. This work points the way towards higher-accuracy models to enable better inferences from future measurements.