  We explore the galaxy formation physics governing the low mass end of the HI
mass function in the local Universe. Specifically, we predict the effects on
the HI mass function of varying i) the strength of photoionisation feedback and
the redshift of the end of the epoch of reionization, ii) the cosmology, iii)
the supernovae feedback prescription, and iv) the efficiency of star formation.
We find that the shape of the low-mass end of the HI mass function is most
affected by the critical halo mass below which galaxy formation is suppressed
by photoionisation heating of the intergalactic medium. We model the redshift
dependence of this critical dark matter halo mass by requiring a match to the
low-mass end of the HI mass function. The best fitting critical dark matter
halo mass decreases as redshift increases in this model, corresponding to a
circular velocity of $\sim 50 \, {\rm km \,s}^{-1}$ at $z=0$, $\sim 30 \, {\rm
km\, s}^{-1}$ at $z \sim 1$ and $\sim 12 \, {\rm km \, s}^{-1}$ at $z=6$. We
find that an evolving critical halo mass is required to explain both the shape
and abundance of galaxies in the HI mass function below $M_{\rm HI} \sim 10^{8}
h^{-2} {\rm M_{\odot}}$. The model makes specific predictions for the
clustering strength of HI-selected galaxies with HI masses > $10^{6} h^{-2}
{\rm M_{\odot}}$ and $> 10^{7} h^{-2} {\rm M_{\odot}}$ and for the relation
between the HI and stellar mass contents of galaxies which will be testable
with upcoming surveys with the Square Kilometre Array and its pathfinders. We
conclude that measurements of the HI mass function at $z \ge 0$ will lead to an
improvement in our understanding of the net effect of photoionisation feedback
on galaxy formation and evolution.
