The Intrinsic Distribution of Lyman-$α$ Halos
The emission and escape of Lyman-$α$ photons from star-forming galaxies is determined through complex interactions between the emitted photons and a galaxy's interstellar and circumgalactic gas, causing Lyman-$α$ emitters (LAEs) to commonly appear not as point sources but in spatially extended halos with complex spectral profiles. We develop a 3D spatial-spectral model of Lyman-$α$ halos (LAHs) to replicate LAH observations in integral field spectroscopic studies, such as those made with VLT/MUSE. The profile of this model is a function of 6 key halo properties: the halo- and compact-source exponential scale lengths ($r_{sH}$ and $r_{sC}$), the halo flux fraction ($f_H$), the compact component ellipticity ($q$), the spectral line width ($σ$), and the spectral line skewness parameter ($γ$). Placing a series of model LAHs into datacubes reflecting observing conditions in the MUSE UDF-Mosaic survey, we test their detection recoverability and determine that $σ$, $r_{sH}$, and $f_H$ are expected to have the most significant effect on the detectability of the overall LAH at a given central wavelength and intrinsic line luminosity. We develop a general selection function model spanning a grid of these halo parameters, and with a sample of 145 UDF-Mosaic LAHs with measured halo properties, we derive completeness-corrected, intrinsic distributions of the values of $σ$, $r_{sH}$, and $f_H$ for $3<z<5$ LAHs. We present best-fit functional forms of the distributions, and a $σ$ distribution corrected for instrumental line-spread function (LSF) broadening, and thereby show the physical line spread distribution of the intrinsic population. Finally, we discuss implications of these distributions for Ly$α$ emission through the circumgalactic medium, finding that observations undercount LAHs with extended halo scale lengths compared to the intrinsic population.