Astraeus VIII: A new framework for Lyman-$\alpha$ emitters applied to different reionisation scenarios

We use the {\sc astraeus} framework to investigate how the visibility and spatial distribution of Lyman-$\alpha$ (Ly$\alpha$) emitters (LAEs) during reionisation is sensitive to a halo mass-dependent fraction of ionising radiation escaping from the galactic environment ($f_\mathrm{esc}$) and the ionisation topology. To this end, we consider the two physically plausible bracketing scenarios of $f_\mathrm{esc}$ increasing and decreasing with rising halo mass. We derive the corresponding observed Ly$\alpha$ luminosities of galaxies for three different analytic Ly$\alpha$ line profiles and associated Ly$\alpha$ escape fraction ($f_\mathrm{esc}^\mathrm{Ly\alpha}$) models: importantly, we introduce two novel analytic Ly$\alpha$ line profile models that describe the surrounding interstellar medium (ISM) as dusty gas clumps. They are based on parameterising results from radiative transfer simulations, with one of them relating $f_\mathrm{esc}^\mathrm{Ly\alpha}$ to $f_\mathrm{esc}$ by assuming the ISM of being interspersed with low-density tunnels. Our key findings are: (i) for dusty gas clumps, the Ly$\alpha$ line profile develops from a central to double peak profile as a galaxy's halo mass increases; (ii) LAEs are galaxies with $M_h\gtrsim10^{10}M_\odot$ located in overdense and highly ionised regions; (iii) for this reason, the spatial distribution of LAEs is primarily sensitive to the global ionisation fraction and only weakly in second-order to the ionisation topology or a halo mass-dependent $f_\mathrm{esc}$; (iv) furthermore, as the observed Ly$\alpha$ luminosity functions reflect the Ly$\alpha$ emission from more massive galaxies, there is a degeneracy between the $f_\mathrm{esc}$-dependent intrinsic Ly$\alpha$ luminosity and the Ly$\alpha$ attenuation by dust in the ISM if $f_\mathrm{esc}$ does not exceed $\sim50\%$.