3.3 Matter density

Many cosmological tests, such as the two just discussed, will constrain some combination of _M and . It is therefore useful to consider tests of _M alone, even if our primary goal is to determine . (In truth, it is also hard to constrain _M alone, as almost all methods actually constrain some combination of _M and the Hubble constant h = H₀ / (100 km/sec/Mpc); the HST Key Project on the extragalactic distance scale finds h = 0.71 ± 0.06 [86], which is consistent with other methods [87], and what I will assume below.)

For years, determinations of _M based on dynamics of galaxies and clusters have yielded values between approximately 0.1 and 0.4 - noticeably larger than the density parameter in baryons as inferred from primordial nucleosynthesis, _B = (0.019 ± 0.001) h^-2 0.04 [88, 89], but noticeably smaller than the critical density. The last several years have witnessed a number of new methods being brought to bear on the question; the quantitative results have remained unchanged, but our confidence in them has increased greatly.

A thorough discussion of determinations of _M requires a review all its own, and good ones are available [90, 91, 92, 87, 93]. Here I will just sketch some of the important methods.

The traditional method to estimate the mass density of the universe is to ``weigh'' a cluster of galaxies, divide by its luminosity, and extrapolate the result to the universe as a whole. Although clusters are not representative samples of the universe, they are sufficiently large that such a procedure has a chance of working. Studies applying the virial theorem to cluster dynamics have typically obtained values _M = 0.2 ± 0.1 [94, 90, 91]. Although it is possible that the global value of M/L differs appreciably from its value in clusters, extrapolations from small scales do not seem to reach the critical densit [95]. New techniques to weigh the clusters, including gravitational lensing of background galaxies [96] and temperature profiles of the X-ray gas [97], while not yet in perfect agreement with each other, reach essentially similar conclusions.

Rather than measuring the mass relative to the luminosity density, which may be different inside and outside clusters, we can also measure it with respect to the baryon density [98], which is very likely to have the same value in clusters as elsewhere in the universe, simply because there is no way to segregate the baryons from the dark matter on such large scales. Most of the baryonic mass is in the hot intracluster gas [99], and the fraction f_gas of total mass in this form can be measured either by direct observation of X-rays from the gas [100] or by distortions of the microwave background by scattering off hot electrons (the Sunyaev-Zeldovich effect) [101], typically yielding 0.1 f_gas 0.2. Since primordial nucleosynthesis provides a determination of _B ~ 0.04, these measurements imply

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consistent with the value determined from mass to light ratios.

Another handle on the density parameter in matter comes from properties of clusters at high redshift. The very existence of massive clusters has been used to argue in favor of _M ~ 0.2 [102], and the lack of appreciable evolution of clusters from high redshifts to the present [103, 104] provides additional evidence that _M < 1.0.

The story of large-scale motions is more ambiguous. The peculiar velocities of galaxies are sensitive to the underlying mass density, and thus to _M, but also to the ``bias'' describing the relative amplitude of fluctuations in galaxies and mass [90, 105]. Difficulties both in measuring the flows and in disentangling the mass density from other effects make it difficult to draw conclusions at this point, and at present it is hard to say much more than 0.2 _M 1.0.

Finally, the matter density parameter can be extracted from measurements of the power spectrum of density fluctuations (see for example [106]). As with the CMB, predicting the power spectrum requires both an assumption of the correct theory and a specification of a number of cosmological parameters. In simple models (e.g., with only cold dark matter and baryons, no massive neutrinos), the spectrum can be fit (once the amplitude is normalized) by a single ``shape parameter'', which is found to be equal to = _M h. (For more complicated models see [107].) Observations then yield ~ 0.25, or _M ~ 0.36. For a more careful comparison between models and observations, see [108, 109, 110, 111]. Thus, we have a remarkable convergence on values for the density parameter in matter:

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Even without the supernova results, this determination in concert with the CMB measurements favoring a flat universe provide a strong case for a nonzero cosmological constant.