The finite size effect of galaxies on the cosmic virial theorem and the pairwise peculiar velocity dispersions

Abstract

We discuss the effect of the finite size of galaxies on estimating small-scale relative pairwise peculiar velocity dispersions from the cosmic virial theorem (CVT). Specifically we evaluate the effect by incorporating the finite core radius $r_c$ in the two-point correlation function of mass, i.e. $ξ_ρ(r) \propto (r+r_c)^{-γ}$ and the effective gravitational force softening $r_s$ on small scales. We analytically obtain the lowest-order correction term for $γ<2$ which is in quantitative agreement with the full numerical evaluation. With a nonzero $r_s$ and/or $r_c$ the cosmic virial theorem is no longer limited to the case of $γ<2$. We present accurate fitting formulae for the CVT predicted pairwise velocity dispersion for the case of $γ>2$. Compared with the idealistic point-mass approximation ($r_s=r_c=0$), the finite size effect can significantly reduce the small-scale velocity 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 a scale-invariant primordial spectrum. For these CDM models, a reasonable force softening $r_s\le 100 \hikpc$ would have rather tiny effect. We present the CVT predictions for the small-scale pairwise velocity dispersion in the CDM models normalized by the COBE observation. The implication of our results for confrontation of observations of galaxy pair-wise velocity dispersions and theoretical predictions of the CVT is also discussed.