The HI-halo mass relation at redshift $z \sim 1$ from the Minkowski functionals of 21 cm intensity maps

The mean and the scatter of the HI content of a dark-matter halo as a function of the halo mass are useful statistics that can be used to test models of structure and galaxy formation. We investigate the possibility of constraining this HI-halo mass relation (HIHMR) from intensity maps of the redshifted 21 cm line. In particular, we use the geometry and topology of the brightness-temperature isocontours in a single frequency channel as quantified by the Minkowski functionals. First, we generate mock maps from a large N-body simulation considering the impact of thermal noise and foreground removal. We then use the Fisher information formalism to forecast constraints on a parametric model for the HIHMR. We consider a 20,000 deg$^2$ survey (originally proposed for dark-energy science) conducted with the Square Kilometre Array Phase 1 (SKA-1) MID observatory operating in single-dish mode. For a channel bandwidth of 2 MHz, we show that an integration time of a few$\,\times\,10^4$ s per pointing is sufficient to image the smoothed HI distribution at redshift $z \simeq 1$ and to measure the HIHMR in a nearly optimal way from the Minkowski functionals. Tighter constraints on some of the parameters can be obtained by using also an independent measurement of the mean HI density. Combining the results from different frequency channels provides exquisite constraints on the evolution of the HIHMR, especially in the central frequency range of the data cube.