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), $dn(M)/dlog(M)$, 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 $sigma_8Omega_m^{0.29 (0.29)} = 0.587 pm 0.011 (0.583 pm 0.011)$ 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.
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