Lyman-alpha resonant-line radiative transfer in expanding media

The Lyman-alpha (LyA) 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, analytic solutions remain scarce, with most studies focusing on idealized, static configurations. However, observations of LyA emitting galaxies consistently reveal signatures of outflows, imprinted through red-peak dominance in spectral line profiles. In this paper, we derive novel analytic 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. To validate these analytic 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 analytic 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_max / v_th, as modifying the standard dependence on line centre optical depth of (atau_0)^1/3 by additional factors, e.g. characteristic escape frequency scale as x_esc ~ beta^1/3, force multipliers as M_F ~ beta^-1/3, and trapping time as t_trap ~ beta^-2/3. Our work complements numerical simulations by improving physical intuition about nonstatic environments when interpreting LyA observations and guiding future subgrid prescriptions in galaxy formation models.