Large Scale Power Spectrum from Peculiar Velocities Via Likelihood Analysis

The power spectrum (PS) of {\it mass} density fluctuations, independent of ``biasing", is estimated from the Mark3 Catalog of Peculiar Velocities of galaxies using Bayesian statistics. A parametric model is assumed for the PS, and the free parameters are determined by maximizing the posterior probability of the model given the data. The method has been tested using detailed mock catalogs. We use generalized CDM models with and without COBE normalization. The robust result for all the models tested is $P(k) \Omega^{1.2} = (4.1 \pm 0.7)x10^3 (Mpc h^{-1})^3$ at $k=0.1 h Mpc^{-1}$, with the peak constrained to the range $0.03 < k < 0.06 h Mpc^{-1}$. It is consistent with a direct computation of the PS (Kolatt \& Dekel 1996). When compared to galaxy-density surveys, the implied values for $\beta$ ($\equiv \Omega^{0.6}/b$) are of order unity to within 25\%. A $\Gamma$-shape model, free of COBE normalization, is constrained by the velocity data to $\Gamma=0.5\pm 0.15$ and $\sigma_8 \Omega^{0.6}=0.85\pm 0.1$. Within the family of COBE-normalized CDM models, the best tilted model ($\Omega=1$, $n \approx 0.84 h_{50}^{-0.65}$) and the best open model ($n=1$, $\Omega \approx 0.75 h_{50}^{-1.0}$) are more likely than the best $\Lambda$ model ($n=1$, $\Lambda=1-\Omega$, $\Omega\approx 0.70 h_{50}^{-1.2}$). The most likely CDM model with $\Omega\leq 1$ is found to be of $\Omega=1$, $h\approx 0.75$, and a {\it tilted} spectrum of $n= 0.8\pm 0.02$ with tensor fluctuations.The tightest constraint for the tilted-$\Lambda$ models is of the sort $\Omega h_{50}^{1.2} n^\nu = 0.7 \pm 0.08$, with $\nu=3.8$ and $1.85$ with and without tensor fluctuations respectively.