Constraints on the evolution of the relationship between HI mass and halo mass in the last 12 Gyr

The neutral hydrogen (HI) content of dark matter haloes forms an intermediate state in the baryon cycle that connects the hot shock-heated gas and cold star-forming gas in haloes. Measurement of the relationship between HI mass and halo mass therefore puts important constraints on galaxy formation models. We combine radio observations of HI in emission at low redshift ($z\sim 0$) with optical/UV observations of HI in absorption at high redshift ($1<z<4$) to derive constraints on the evolution of the HI-mass halo-mass (HIHM) relation from redshift $z=4$ to $z=0$. We find that one can model the HIHM relation similar to the stellar-mass halo-mass (SHM) relation at $z \sim 0$. At $z=0$, haloes with mass $10^{11.7}$ M$_\odot$ have the highest HI mass fraction ($\sim 1\%$), which is about four times smaller than their stellar mass fraction. We model the evolution of the HIHM relation in a manner similar to that of the SHM relation. Combining this parameterisation with a redshift- and mass-dependent modified Navarro-Frenk-White (NFW) profile for the HI density within a halo, we draw constraints on the evolution of the HIHM relation from the observed HI column density, incidence rate, and clustering bias at high redshift. We compare these findings with results from hydrodynamical simulations and other approaches in the literature, and find the models to be consistent with each other at the $68\%$ confidence level.