3.3. Cosmic Virial Theorem

In the cosmic virial theorem (CVT), clustering is assumed to be statistically stable on scales 1 h^-1Mpc, such that the ensemble-average contribution of a third mass particle to the relative acceleration of a pair of galaxies is balanced by the relative motions of the ensemble of pairs. The kinetic energy is represented by the pairwise relative velocity dispersion of galaxies [₁₂(r)], and the potential energy involves a combination of integrals of the 2- and 3-point galaxy correlation functions [26]. A very crude approximation to the actual relation is

(10)

where (r) is the two-point correlation function. In some ways this is like stacking many groups of galaxies to get a mean M/L, so it can be regarded as a statistical version of the mass-to-light ratio method.

On scales 1 - 10 h^-1Mpc the more useful statistics for dynamical mass estimates is the mean (first moment) pairwise velocity in comparison with an integral of the galaxy two-point correlation function.

In the approach of the Layzer-Irvine (LI) equation, the kinetic energy is associated with the absolute rms velocity (in the CMB frame) of individual galaxies and the potential energy is an integral over the 2-point correlation function.

New developments: It has been realized that the pair velocity dispersion is an unstable statistic that is dominated by galaxies in clusters. Attempts are being made to apply the method while excluding clusters. Filtered versions of the LI approach help truncate the (weak) divergence in estimating the potential energy and make the kinetic energy term easier to estimate from data.

Pro: There is no need to associate galaxies with separate groups and clusters; it is all statistical. In the LI method the energies are dominated by contributions from the large-scale mass distribution, and it is therefore less contaminated by clusters and less biased by the assumption of point masses. The two-point correlation for the LI method can be measured with reasonable accuracy.

Con: The relevant correlation integrals, and in particular the 3-point correlation function that enters the CVT, are very difficult to measure. The pair velocity dispersion in the CVT is an unstable statistic, that is dominated by pairs of galaxies in clusters. A robust, self-consistent way to avoid the clusters is yet to be found. The absolute velocity dispersion which enters the LI equation is hard to measure. The CVT assumes that galaxies are point masses (or of finite size), which overestimates the force that they exert, and biases _m low. The overlap of extended galactic halos makes a significant difference. The methods depend on the assumption that the statistical distribution of galaxies and mass are similar (as in the M/L method). Otherwise, they must refer to a specific model for galaxy biasing.

Current Results: The line-of-sight pair velocity dispersion outside of clusters is in the range _f 300 ± 100 km s^-1 [27] [28]. With Peebles' old estimate of the correlation functions, assuming point masses and no biasing, he obtains from the CVT _m 0.15 [6]. From the mean pairwise velocity in IRAS 1.2Jy [27]: _m ~ 0.25. However, it has been demonstrated that with an extended mass distribution in galactic halos the above CVT estimates are lower bounds, and that the observations may be consistent with _m 1 [29].