  The ongoing Arecibo Legacy Fast ALFA (ALFALFA) survey is a wide-area,
extragalactic HI-line survey conducted at the Arecibo Observatory. Sources have
so far been extracted over ~3,000 sq.deg of sky (40% of its final area),
resulting in the largest HI-selected sample to date. We measure the space
density of HI-bearing galaxies as a function of their observed velocity width
(uncorrected for inclination) down to w = 20 km/s, a factor of 2 lower than the
previous generation HI Parkes All-Sky Survey. We confirm previous results that
indicate a substantial discrepancy between the observational distribution and
the theoretical one expected in a cold dark matter (CDM) universe, at low
widths. In particular, a comparison with synthetic galaxy samples populating
state-of-the-art CDM simulations imply a factor of ~8 difference in the
abundance of galaxies with w = 50 km/s (increasing to a factor of ~100 when
extrapolated to the ALFALFA limit of w = 20 km/s). We furthermore identify
possible solutions, including a keV warm dark matter scenario and the fact that
HI disks in low mass galaxies are usually not extended enough to probe the full
amplitude of the galactic rotation curve. In this latter case, we can
statistically infer the relationship between the measured HI rotational
velocity of a galaxy and the mass of its host CDM halo. Observational
verification of the presented relationship at low velocities would provide an
important test of the validity of the established dark matter model.
