Lyman-{\alpha} polarization from cosmological ionization fronts: I. Radiative transfer simulations

In this paper, we present the formalism of simulating Lyman-$\alpha$ emission and polarization around reionization ($z$ = 8) from a plane-parallel ionization front. We accomplish this by using a Monte Carlo method to simulate the production of a Lyman-$\alpha$ photon, its propagation through an ionization front, and the eventual escape of this photon. This paper focuses on the relation of the input parameters of ionization front speed $U$, blackbody temperature $T_{\rm bb}$, and neutral hydrogen density $n_{\rm HI}$, on intensity $I$ and polarized intensity $P$ as seen by a distant observer. The resulting values of intensity range from $3.18\times 10^{-14}$ erg/cm$^{2}$/s/sr to $1.96 \times 10^{-9}$ erg/cm$^{2}$/s/sr , and the polarized intensity ranges from $5.73\times 10^{-17}$ erg/cm$^{2}$/s/sr to $5.31 \times 10^{-12}$ erg/cm$^{2}$/s/sr. We found that higher $T_{\rm bb}$, higher $U$, and higher $n_{\rm HI}$ contribute to higher intensity, as well as polarized intensity, though the strongest dependence was on the hydrogen density. The dependence of viewing angle of the front is also explored. We present tests to support the validity model, which makes the model suitable for further use in a following paper where we will calculate the intensity and polarized intensity power spectrum on a full reionization simulation.