MIGHTEE-HI: The HI mass-stellar mass relation of massive galaxies and the HI mass function at 0.25<z<0.5

The relationship between the already formed stellar mass in a galaxy and the gas reservoir of neutral atomic hydrogen, is a key element in our understanding of how gas is turned into stars in galaxy haloes. In this paper, we measure the $M_{\mathrm{H}\mathrm{I}}-M_{\star }$ relation based on a stellar-mass selected sample at $0.25<z<0.5$ and the MIGHTEE-HI DR1 spectral data. Using a powerful Bayesian stacking technique, for the first time we are also able to measure the underlying bivariate distribution of HI mass and stellar mass of galaxies with $M_{\star }>{10}^{9.5}$ M${}_{\odot }$, finding that an asymmetric underlying HI distribution is strongly preferred by our complete samples. We define the concepts of the average of the logarithmic HI mass, $⟨{\log }_{10}(M_{\mathrm{H}\mathrm{I}})⟩$, and the logarithmic average of the HI mass, ${\log }_{10}(⟨M_{\mathrm{H}\mathrm{I}}⟩)$, and find that the difference between $⟨{\log }_{10}(M_{\mathrm{H}\mathrm{I}})⟩$ and ${\log }_{10}(⟨M_{\mathrm{H}\mathrm{I}}⟩)$ can be as large as $\sim$0.5 dex for the preferred asymmetric HI distribution. We observe shallow slopes in the underlying $M_{\mathrm{H}\mathrm{I}}-M_{\star }$ scaling relations, suggesting the presence of an upper HI mass limit beyond which a galaxy can no longer retain further HI gas. From our bivariate distribution we also infer the HI mass function at this redshift and find tentative evidence for a decrease of 2-10 times in the co-moving space density of the most HI massive galaxies up to $z\sim 0.5$.