  We present a new approach for quantifying the abundance of galaxy clusters
and constraining cosmological parameters using dynamical measurements. In the
standard method, galaxy line-of-sight (LOS) velocities, $v$, or velocity
dispersions are used to infer cluster masses, $M$, in order to quantify the
halo mass function (HMF), $dn(M)/d\log(M)$, which is strongly affected by mass
measurement errors. In our new method, the probability distribution of
velocities for each cluster in the sample are summed to create a new statistic
called the velocity distribution function (VDF), $dn(v)/dv$. The VDF can be
measured more directly and precisely than the HMF and it can also be robustly
predicted with cosmological simulations which capture the dynamics of subhalos
or galaxies. We apply these two methods to mock cluster catalogs and forecast
the bias and constraints on the matter density parameter $\Omega_m$ and the
amplitude of matter fluctuations $\sigma_8$ in flat $\Lambda$CDM cosmologies.
For an example observation of 200 massive clusters, the VDF with (without)
velocity errors constrains the parameter combination $\sigma_8\Omega_m^{0.29\
(0.29)} = 0.587 \pm 0.011\ (0.583 \pm 0.011)$ and shows only minor bias.
However, the HMF with dynamical mass errors is biased to low $\Omega_m$ and
high $\sigma_8$ and the fiducial model lies well outside of the forecast
constraints, prior to accounting for Eddington bias. When the VDF is combined
with constraints from the cosmic microwave background (CMB), the degeneracy
between cosmological parameters can be significantly reduced. Upcoming
spectroscopic surveys that probe larger volumes and fainter magnitudes will
provide a larger number of clusters for applying the VDF as a cosmological
probe.
