Measuring the HI mass function below the detection threshold

We present a Bayesian Stacking technique to directly measure the HI mass function (HIMF) and its evolution with redshift using galaxies formally below the nominal detection threshold. We generate galaxy samples over several sky areas given an assumed HIMF described by a Schechter function and simulate the HI emission lines with different levels of background noise to test the technique. We use Multinest to constrain the parameters of the HIMF in a broad redshift bin, demonstrating that the HIMF can be accurately reconstructed, using the simulated spectral cube far below the HI mass limit determined by the $5\sigma$ flux-density limit, i.e. down to $M_{\rm HI} = 10^{7.5}$ M$_{\odot}$ over the redshift range $0 < z < 0.55$ for this particular simulation, with a noise level similar to that expected for the MIGHTEE survey. We also find that the constraints on the parameters of the Schechter function, $\phi_{\star}$, $M_\star$ and $\alpha$ can be reliably fit, becoming tighter as the background noise decreases as expected, although the constraints on the redshift evolution are not significantly affected. All the parameters become better constrained as the survey area increases. In summary, we provide an optimal method for estimating the HI mass at cosmological distances that allows us to constrain the HI mass function below the detection threshold in forthcoming HI surveys. This study is a first step towards the measurement of the HIMF at high ($z>0.1$) redshifts.