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