Lyman α resonant-line radiative transfer in expanding media

ABSTRACT The Lyman $\alpha$ (Ly$\alpha$) line of atomic hydrogen encodes crucial information about both the intrinsic sources and surrounding environments of star-forming regions throughout the Universe. Due to the complexity of resonant scattering, analytical solutions remain scarce, with most studies focusing on idealized, static configurations. However, observations of Ly$\alpha$ emitting galaxies consistently reveal signatures of outflows, imprinted through red-peak dominance in spectral line profiles. We derive novel analytical solutions for resonant-line radiative transfer in moving media, specifically homologous-like cloud expansion and unbounded cosmological flows, which capture the main physics of velocity gradients. Our framework accounts for spatial diffusion, partial frequency redistribution, and kinematic effects. To validate these analytical solutions and identify regimes where diffusion-based assumptions hold, we introduce a robust Gridless Monte Carlo Radiative Transfer (GMCRT) method. By integrating optical depths exactly in the comoving frame, GMCRT updates photon frequencies continuously to account for Doppler shifts induced by velocity gradients. We demonstrate excellent consistency between GMCRT and our analytical solutions in regimes where diffusion approximations apply. At higher velocities or lower optical depths, discrepancies highlight the limitations of simplified formalisms. We also provide scaling relations for a point source in a cloud with a maximum-to-thermal velocity ratio $\beta = V_\text{max} / v_\text{th}$, as modifying the standard dependence on line centre optical depth of $\propto (a\tau _0)^{1/3}$ by additional factors, e.g. characteristic escape frequency scale as $x_\text{esc} \propto \beta ^{1/3}$, force multipliers as $M_\text{F} \propto \beta ^{-1/3}$, and trapping time as $t_\text{trap} \propto \beta ^{-2/3}$. Our work complements numerical simulations by improving physical intuition about non-static environments when interpreting Ly$\alpha$ observations and guiding subgrid prescriptions in galaxy formation models.