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We estimate the cosmic number density of the recently identified class ofHI-bearing ultra-diffuse sources (HUDs) based on the completeness limits of theALFALFA survey. These objects fall in the range $8.5 < \logM_{\rm{HI}}/\rm{M_{\odot}} < 9.5$, have average $r$-band surface brightnessesfainter than 24 mag arsec$^{-2}$, half-light radii greater than 1.5 kpc, andare separated from neighbours by at least 350 kpc. We find that HUDs contributeat most 6% of the population of HI-bearing dwarfs, have a total cosmic numberdensity of $(1.5 \pm 0.6) \times 10^{-3}$ $\rm{Mpc^{-3}}$, and an HI massdensity of $(6.0 \pm 0.8) \times 10^{5}$ $\rm{M_{\odot}\,Mpc^{-3}}$. Weestimate that this is similar to the total cosmic number density of UDGs ingroups and clusters, and conclude that the relation between the number of UDGshosted in a halo and the halo mass, must have a break below $M_{200} \sim10^{12}$ $\rm{M_{\odot}}$ in order to account for the abundance of HUDs. Thedistribution of the velocity widths of HUDs rises steeply towards low values,indicating a preference for slow rotation rates. These objects have been absentfrom measurements of the galaxy stellar mass function owing to their lowsurface brightness. However, we estimate that due to their low number density,their inclusion would constitute a correction of less than 1%. Comparison withthe Santa Cruz SAM shows that it produces HI-rich central UDGs that havesimilar colours to HUDs, but these are currently produced in much great anumber. While previous results from this sample have favoured formationscenarios where HUDs form in high spin parameter halos, comparisons with theresults of Rong et al. 2017, which invokes that formation mechanism, revealthat this model produces an order of magnitude more field UDGs than we observein the HUD population.(Abridged)

The contribution of HI-bearing ultra-diffuse galaxies to the cosmic number density of galaxies