MIGHTEE-H  <scp>i</scp> : the <i>M</i> H  <scp>i</scp> - <i>M</i>  * relation of massive galaxies and the H  <scp>i</scp> mass function at 0.25 &amp;lt; <i>z</i> &amp;lt; 0.5

ABSTRACT 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 $M_{\rm H\, {\small I}}-M_{\star }$ relation based on a stellar-mass selected sample at $0.25 &amp;lt; z &amp;lt; 0.5$ and the MeerKAT International GHz Tiered Extragalactic Exploration-H i Data Release 1 spectral data. Using a powerful Bayesian stacking technique, for the first time we are also able to measure the underlying bivariate distribution of H i mass and stellar mass of galaxies with $M_\star &amp;gt; 10^{9.5}$ M$_{\odot }$, finding that an asymmetric underlying H i distribution is strongly preferred by our complete samples. We define the concepts of the average of the logarithmic H i mass, $\langle \log _{10}(M_{\rm H\, {\small I}})\rangle$, and the logarithmic average of the H i mass, $\log _{10}(\langle M_{\rm H\, {\small I}}\rangle)$, and find that the difference between $\langle \log _{10}(M_{\rm H\, {\small I}})\rangle$ and $\log _{10}(\langle M_{\rm H\, {\small I}}\rangle)$ can be as large as $\sim$0.5 dex for the preferred asymmetric H i distribution. We observe shallow slopes in the underlying $M_{\rm H\, {\small I}}-M_{\star }$ scaling relations, suggesting the presence of an upper H i mass limit beyond which a galaxy can no longer retain further H i gas. From our bivariate distribution we also infer the H i mass function at this redshift and find tentative evidence for a decrease of 2–10 times in the comoving space density of the most H i massive galaxies up to $z\sim 0.5$.