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3.1. Dynamical Mass Estimates

Here is an example that illustrates features common to many dynamical estimates of the mean mass density.

We are near the edge of a concentration of galaxies that de Vaucouleurs (1956) called the Local Supercluster. It is centered near the Virgo cluster, at distance

Equation 3 (3)

where Hubble's constant is written as

Equation 4 (4)

and the dimensionless parameter is thought to be in the range 0.5 ltapprox h ltapprox 0.8. In the Friedmann-Lemaître model the gravitational attraction of the mass excess in this region produces a peculiar motion of inflow (relative to the general expansion of the universe). In linear perturbation theory the mass conservation law relates the peculiar velocity barv(barr, t) and the mass density contrast delta (barr, t) = deltarho / rho by the equation

Equation 5 (5)

The divergence is with respect to comoving coordinates x, where a physical length interval is delta r = a (t)delta x (and the expansion parameter a (t) appears in eq. [1]). The density fluctuations are assumed to have grown by gravity out of small primeval irregularities, so the mass density contrast varies as delta propto D (t), where D (t) is the growing solution to the time evolution of delta in linear perturbation theory. The result of integrating equation (5) over a sphere of radius R and applying Gauss's theorem is the wanted relation,

Equation 6 (6)

Here n is the unit normal of the sphere, barv is the radial inward peculiar velocity averaged over the surface, and bardelta = delta M/M is the mass contrast averaged within the sphere. The dimensionless factor f is

Equation 7 (7)

The power law is a good approximation if Lambda = 0 or space curvature vanishes.

For a sphere centered on the Virgo Cluster, with us on the surface, estimates of the mean radial flow through the sphere and the contrast deltag = delta M/M in galaxy counts within the sphere are

Equation 8a Equation 8b (8)

The velocity is from a survey in progress by Tonry et al. (1998); it is consistent with earlier measurements (eg. Faber & Burstein 1988). The density contrast in counts of IRAS galaxies within our distance from the Virgo cluster is bardelta = 1.4 (Strauss et al. 1992). IRAS galaxies are detected because they are rich in gas and have high star formation rates, making them prominent in the 60 to 100 micron range of the IRAS satellite survey. IRAS galaxies avoid dense regions, likely because collisions and the ram pressure of intracluster gas have stripped the galaxies of the gas that fuels bursts of star formation and high infrared luminosity; a commonly used correction factor of 1.4 would bring the contrast for optical galaxy counts to deltag = 2. A preliminary analysis of the Optical Redshift Survey (Santiago et al. 1996) by Strauss (1998) gives deltag ~ 3. With bardelta = bardeltag, equations (6) to (8) give

Equation 9 (9)

This is plotted as the right-hand point in Figure 1.

Figure 1

Figure 1. Dynamical estimates of the mean mass density from galaxy velocities in and near the Local Group (left point), in rich clusters of galaxies, and from the flow of galaxies toward the Local Supercluster (right-hand point).

There are three key assumptions. First, the analysis uses conventional gravity physics. An alternative, Milgrom's (1995) modified Newtonian dynamics (MOND), has been quite durable in applications to individual galaxies (de Blok & McGaugh 1998). An extension to the analysis of large-scale flows would be interesting, but the focus here is the test of cosmology based on the conventional gravity physics of general relativity theory. Second, the relation between peculiar velocities and the mass distribution follows from the assumption that structure grew by gravity out of small primeval departures from homogeneity. (Thus the boundary condition for eq. [7] is D (t) -> 0 at z >> 1.) Most dynamical measures of Omegam use this assumption; the exception is relaxed systems that have forgotten their initial conditions (as in the velocity dispersion measure used by Marzke et al. 1995.) We have no viable alternative to the gravitational instability picture for structure formation on large scales, but it will be checked by consistency with all the other cosmological tests, when they are better established. Third, and most contentious, equation (9) assumes the mass clusters with the galaxies. If the mass contrast were reduced to delta ~ 0.2 deltag then the other numbers would be consistent with Omegam = 1. The concept that the galaxy distribution may be a biased measure of the mass distribution has been influential, and rightly so; this important issue had to be explored. But as discussed next I think it is also fair to say that there never was any evidence for what I would expect to be the distinctive signature of biasing: void galaxies.

If Omegam = 1 we must decide where most of the mass is. It can't be in groups and clusters of galaxies: Figure 1 shows that analyses similar to the above yield similar values of Omegam in the Local Group (Peebles 1996) and in clusters of galaxies (Carlberg et al. 1996). That leaves the voids, spaces between the concentrations of giant galaxies. We know the voids are not undisturbed: absorption lines in quasar spectra show that at redshift z = 3 space was filled with clouds of hydrogen. Thus voids would have to be regions where star or galaxy formation was suppressed. I find it hard to believe the suppression was so complete as to leave nothing observable; surely there would be irregulars or dwarfs from almost failed seeds. Searches in relatively nearby voids, where galaxies are observable well into the faint end of the luminosity function, reveal no such population. Perhaps the gravitational growth of clustering swept the void galaxies into the concentrations of normal ones, but in that case gravity would have pulled the mass with the galaxies, suppressing biasing (Tegmark & Peebles 1998). Perhaps our picture for structure formation needs tuning; that will be checked as the cosmological tests improve. The straightforward reading is that biasing is not a strong factor; Omegam is substantially less than unity. This is the basis for the grades in line 1a in Table 2. The grades are subject to negotiation, of course; the discovery of a population of void galaxies would make a big difference to me.

The theory of the origin of the light elements requires baryon density parameter Omegabaryon = 0.02 / h2 ~ 0.04 for h = 0.7. It is not easy to reconcile the dynamical analyses with such a small value for Omegam. As I noted in the last section, the common assumption is that the mass of the universe is dominated by nonbaryonic dark matter. There certainly is nothing unreasonable about the idea---Nature need not have put most of the matter in a readily observable form---but the cosmology certainly would be cleaner if we had a laboratory detection of this hypothetical mass component.

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