The Finite‐Size Effect of Galaxies on the Cosmic Virial Theorem and the Pairwise Peculiar Velocity Dispersions

We discuss the effect of the finite size of galaxies on estimatingsmall-scale relative pairwise peculiar velocity dispersions from the cosmicvirial theorem (CVT). Specifically we evaluate the effect by incorporating thefinite core radius $r_c$ in the two-point correlation function of mass, i.e.$\xi_\rho(r) \propto (r+r_c)^{-\gamma}$ and the effective gravitational forcesoftening $r_s$ on small scales. We analytically obtain the lowest-ordercorrection term for $\gamma <2$ which is in quantitative agreement with thefull numerical evaluation. With a nonzero $r_s$ and/or $r_c$ the cosmic virialtheorem is no longer limited to the case of $\gamma<2$. We present accuratefitting formulae for the CVT predicted pairwise velocity dispersion for thecase of $\gamma>2$. Compared with the idealistic point-mass approximation($r_s=r_c=0$), the finite size effect can significantly reduce the small-scalevelocity dispersions of galaxies at scales much larger than $r_s$ and $r_c$.Even without considering the finite size of galaxies, nonzero values for $r_c$are generally expected, for instance, for cold dark matter (CDM) models with ascale-invariant primordial spectrum. For these CDM models, a reasonable forcesoftening $r_s\le 100 \hikpc$ would have rather tiny effect. We present the CVTpredictions for the small-scale pairwise velocity dispersion in the CDM modelsnormalized by the COBE observation. The implication of our results forconfrontation of observations of galaxy pair-wise velocity dispersions andtheoretical predictions of the CVT is also discussed.Comment: 18 pages. LaTeX text and 8 postcript figures. submitted to Ap