Prospects for detecting cosmic filaments in Lyman-alpha emission across redshifts $z=2-5$

The standard $\rm \Lambda$CDM cosmological model predicts that a large amount of diffuse neutral hydrogen distributes in cosmic filaments, which could be mapped through Lyman-alpha (Ly$\alpha$) emission observations. We use the hydrodynamical simulation Illustris-TNG50 to investigate the evolution of surface brightness and detectability of neutral hydrogen in cosmic filaments across redshifts $z=2-5$. While the HI column density of cosmic filaments decreases with redshift, due to the rising temperature with cosmic time in filaments, the surface brightness of Ly$\alpha$ emission in filaments is brighter at lower redshifts, suggesting that the detection of cosmic filaments is more feasible at lower redshifts. However, most of the Ly$\alpha$ emission from cosmic filaments is around $10^{-21}$ $\rm erg\ s^{-1}cm^{-2}arsec^{-2}$, making it extremely challenging to detect with current observational instruments. We further generate mock images using the Multi-Unit Spectroscopic Explorer (MUSE) spectrograph installed on both the Very Large Telescope (VLT) and the upcoming Extremely Large Telescope (ELT). Our finding indicates that while the VLT can only detect filamentary structures made of dense gas in galactic centers, the ELT is expected to reveal much finer filamentary structures from diffuse neutral hydrogen outside of galaxies. Compared to the VLT, both the number density and the longest length of filaments are greatly boosted with the ELT. Hence the forthcoming ELT is highly promising to provide a clearer view of cosmic filaments in Ly$\alpha$ emission.