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In the previous lecture we set up the tools required to analyze the kinematics and dynamics of homogeneous and isotropic cosmologies in general relativity. In this lecture we turn to the actual universe in which we live, and discuss the remarkable properties cosmologists have discovered in the last ten years. Most remarkable among them is the fact that the universe is dominated by a uniformly-distributed and slowly-varying source of "dark energy," which may be a vacuum energy (cosmological constant), a dynamical field, or something even more dramatic.

3.1. Matter: Ordinary and Dark

In the years before we knew that dark energy was an important constituent of the universe, and before observations of galaxy distributions and CMB anisotropies had revolutionized the study of structure in the universe, observational cosmology sought to measure two numbers: the Hubble constant H0 and the matter density parameter OmegaM. Both of these quantities remain undeniably important, even though we have greatly broadened the scope of what we hope to measure. The Hubble constant is often parameterized in terms of a dimensionless quantity h as

Equation 52 (52)

After years of effort, determinations of this number seem to have zeroed in on a largely agreed-upon value; the Hubble Space Telescope Key Project on the extragalactic distance scale [21] finds

Equation 53 (53)

which is consistent with other methods [22], and what we will assume henceforth.

For years, determinations of OmegaM based on dynamics of galaxies and clusters have yielded values between approximately 0.1 and 0.4, noticeably smaller than the critical density. The last several years have witnessed a number of new methods being brought to bear on the question; here we sketch some of the most important ones.

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 OmegaM = 0.2 ± 0.1 [23, 24, 25]. 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 density [26]. New techniques to weigh the clusters, including gravitational lensing of background galaxies [27] and temperature profiles of the X-ray gas [28], 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 [29], 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 [30], and the fraction fgas of total mass in this form can be measured either by direct observation of X-rays from the gas [31] or by distortions of the microwave background by scattering off hot electrons (the Sunyaev-Zeldovich effect) [32], typically yielding 0.1 leq fgas leq 0.2. Since primordial nucleosynthesis provides a determination of OmegaB ~ 0.04, these measurements imply

Equation 54 (54)

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 OmegaM ~ 0.2 [33], and the lack of appreciable evolution of clusters from high redshifts to the present [34, 35] provides additional evidence that OmegaM < 1.0. On the other hand, a recent measurement of the relationship between the temperature and luminosity of X-ray clusters measured with the XMM-Newton satellite [36] has been interpreted as evidence for OmegaM near unity. This last result seems at odds with a variety of other determinations, so we should keep a careful watch for further developments in this kind of study.

The story of large-scale motions is more ambiguous. The peculiar velocities of galaxies are sensitive to the underlying mass density, and thus to OmegaM, but also to the "bias" describing the relative amplitude of fluctuations in galaxies and mass [24, 37]. Nevertheless, recent advances in very large redshift surveys have led to relatively firm determinations of the mass density; the 2df survey, for example, finds 0.1 leq OmegaM leq 0.4 [38].

Finally, the matter density parameter can be extracted from measurements of the power spectrum of density fluctuations (see for example [39]). 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 Gamma = OmegaM h. (For more complicated models see [40].) Observations then yield Gamma ~ 0.25, or OmegaM ~ 0.36. For a more careful comparison between models and observations, see [41, 42, 43, 44].

Thus, we have a remarkable convergence on values for the density parameter in matter:

Equation 55 (55)

As we will see below, this value is in excellent agreement with that which we would determine indirectly from combinations of other measurements.

As you are undoubtedly aware, however, matter comes in different forms; the matter we infer from its gravitational influence need not be the same kind of ordinary matter we are familiar with from our experience on Earth. By "ordinary matter" we mean anything made from atoms and their constituents (protons, neutrons, and electrons); this would include all of the stars, planets, gas and dust in the universe, immediately visible or otherwise. Occasionally such matter is referred to as "baryonic matter", where "baryons" include protons, neutrons, and related particles (strongly interacting particles carrying a conserved quantum number known as "baryon number"). Of course electrons are conceptually an important part of ordinary matter, but by mass they are negligible compared to protons and neutrons; the mass of ordinary matter comes overwhelmingly from baryons.

Ordinary baryonic matter, it turns out, is not nearly enough to account for the observed matter density. Our current best estimates for the baryon density [45, 46] yield

Equation 56 (56)

where these error bars are conservative by most standards. This determination comes from a variety of methods: direct counting of baryons (the least precise method), consistency with the CMB power spectrum (discussed later in this lecture), and agreement with the predictions of the abundances of light elements for Big-Bang nucleosynthesis (discussed in the next lecture). Most of the matter density must therefore be in the form of non-baryonic dark matter, which we will abbreviate to simply "dark matter". (Baryons can be dark, but it is increasingly common to reserve the terminology for the non-baryonic component.) Essentially every known particle in the Standard Model of particle physics has been ruled out as a candidate for this dark matter. One of the few things we know about the dark matter is that is must be "cold" - not only is it non-relativistic today, but it must have been that way for a very long time. If the dark matter were "hot", it would have free-streamed out of overdense regions, suppressing the formation of galaxies. The other thing we know about cold dark matter (CDM) is that it should interact very weakly with ordinary matter, so as to have escaped detection thus far. In the next lecture we will discuss some currently popular candidates for cold dark matter.

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