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 have HI masses approximately in the range 8.5 < \log M _ { \mathrm { HI } } / \mathrm { M _ { \odot } } < 9.5 , average r -band surface brightnesses fainter than 24 \mathrm { mag } \mathrm { arcsec } ^ { -2 } , half-light radii greater than 1.5 kpc , and are separated from neighbours by at least 350 kpc .
In this work we demonstrate that they contribute at most \sim 6 % of the population of HI-bearing dwarfs detected by ALFALFA ( with similar HI masses ) , have a total cosmic number density of ( 1.5 \pm 0.6 ) \times 10 ^ { -3 } \mathrm { Mpc ^ { -3 } } , and an HI mass density of ( 6.0 \pm 0.8 ) \times 10 ^ { 5 } \mathrm { M _ { \odot } Mpc ^ { -3 } } .
We estimate that this is similar to the total cosmic number density of ultra-diffuse galaxies ( 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 } \mathrm { M _ { \odot } } in order to account for the abundance of HUDs in the field .
The distribution of the velocity widths of HUDs rises steeply towards low values , indicating a preference for slow rotation rates compared to the global HI-rich dwarf population .
These objects were already included in previous measurements of the HI mass function , but 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 semi-analytic model 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 , which invokes that formation mechanism , reveal that this model produces an order of magnitude more field UDGs than we observe in the HUD population , and these have an occurrence rate ( relative to other dwarfs ) that is approximately double what we observe .
In addition , the colours of HUDs are bluer than those predicted by , although we suspect this is due to a systematic problem in reproducing the star formation histories of low-mass galaxies rather than being specific to the ultra-diffuse nature of these sources .