Mapping the intergalactic medium (IGM) in Lyman-$alpha$ emission would yield unprecedented tomographic information on the large-scale distribution of baryons and potentially provide new constraints on the UV background and various feedback processes relevant to galaxy formation. Here, we use a cosmological hydrodynamical simulation to examine the Lyman-$alpha$ emission of the IGM due to collisional excitations and recombinations in the presence of a UV background. We focus on gas in large-scale-structure filaments in which Lyman-$alpha$ radiative transfer effects are expected to be moderate. At low density the emission is primarily due to fluorescent re-emission of the ionising UV background due to recombinations, while collisional excitations dominate at higher densities. We discuss prospects of current and future observational facilities to detect this emission and find that the emission of filaments of the cosmic web will typically be dominated by the halos and galaxies embedded in them, rather than by the lower density filament gas outside halos. Detecting filament gas directly would require a very long exposure with a MUSE-like instrument on the ELT. Our most robust predictions that act as lower limits indicate this would be slightly less challenging at lower redshifts ($z lesssim 4$). We also find that there is a large amount of variance between fields in our mock observations. High-redshift protoclusters appear to be the most promising environment to observe the filamentary IGM in Lyman-$alpha$ emission.
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