Next Contents Previous

D. The cold dark matter model

Some of the present cosmological tests were understood in the 1930s; others are based on new ideas about structure formation. A decade ago a half dozen models for structure formation were under discussion 38; now the known viable models have been winnowed to one class: cold dark matter (CDM) and variants. We comment on the present state of tests of the CDM model in Sec. IV.A.2, and in connection with the cosmological tests in Sec. IV.B.

The CDM model assumes the mass of the universe now is dominated by dark matter that is nonbaryonic and acts like a gas of massive, weakly interacting particles with negligibly small primeval velocity dispersion. Structure is supposed to have formed by the gravitational growth of primeval departures from homogeneity that are adiabatic, scale-invariant, and Gaussian. The early discussions also assume an Einstein-de Sitter universe. These features all are naturally implemented in simple models for inflation, and the CDM model may have been inspired in part by the developing ideas of inflation. But the motivation in writing down this model was to find a simple way to show that the observed present-day mass fluctuations can agree with the growing evidence that the anisotropy of the 3 K thermal cosmic microwave background radiation is very small (Peebles, 1982). The first steps toward turning this picture into a model for structure formation were taken by Blumenthal et al. (1984).

In the decade commencing about 1985 the standard cosmology for many active in research in this subject was the Einstein-de Sitter model, and for good reason: it eliminates the coincidences problem, it avoids the curiosity of nonzero dark energy, and it fits the condition from conventional inflation that space sections have zero curvature. But unease about the astronomical problems with the high mass density of the Einstein-de Sitter model led to occasional discussions of a low density universe with or without a cosmological constant, and the CDM model played an important role in these considerations, as we now discuss.

When the CDM model was introduced it was known that the observations disfavor the high mass density of the Einstein-de Sitter model, unless the mass is more smoothly distributed than the visible matter (Sec. III.C). The key papers showing that this wanted biased distribution of visible galaxies relative to the distribution of all of the mass can follow in a natural way in the CDM theory are Kaiser (1984) and Davis et al. (1985). In short, where the mass density is high enough to lead to the gravitational assembly of a large galaxy the mass density tends to be high nearby, favoring the formation of neighboring large galaxies

The biasing concept is important and certainly had to be explored. But in 1985 there was little empirical evidence for the effect and there were significant arguments against it, mainly the empty state of the voids between the concentrations of large galaxies. 39 In the biasing picture the voids contain most of the mass of an Einstein-de Sitter universe, but few of the galaxies, because galaxy formation there is supposed to have been suppressed. But it is hard to see how galaxy formation could be entirely extinguished: the CDM model would be expected to predict a void population of irregular galaxies, that show signs of a difficult youth. Many irregular galaxies are observed, but they avoid the voids. The straightforward reading of the observations thus is that the voids are empty, and that the dynamics of the motions of the visible galaxies therefore say OmegaM0 is well below unity, and that the mass is not more smoothly distributed than the visible galaxies.

In a low density open universe, with OmegaLambda0 = 0 and positive OmegaK0, the growth of mass clustering is suppressed at z ltapprox OmegaM0-1 - 1. Thus to agree with the observed low redshift mass distribution density fluctuations at high redshift must be larger in the open model than in the Einstein-de Sitter case. This makes it harder to understand the small 3 K cosmic microwave background anisotropy. In a low density spatially-flat universe with OmegaK0 = 0 and a cosmological constant, the transition from matter-dominated expansion to Lambda-dominated expansion is more recent than the transition from matter-dominated expansion to space-curvature-dominated expansion in an open universe with Lambda = 0 , as one sees from Eq. (11). This makes density fluctuations grow almost as much as in the Einstein-de Sitter model, thus allowing smaller peculiar velocities in the flat-Lambda case, a big help in understanding the observations. 40

An argument for low OmegaM0, with or without Lambda, developed out of the characteristic length scale for structure in the CDM model. In the Friedmann-Lemaître cosmology the mass distribution is gravitationally unstable. This simple statement has a profound implication: the early universe has to have been very close to homogeneous, and the growing departures from homogeneity at high redshift are well described by linear perturbation theory. The linear density fluctuations may be decomposed into Fourier components (or generalizations for open or closed space sections). At high enough redshift the wavelength of a mode is much longer than the time-dependent Hubble length H-1, and gravitational instability makes the mode amplitude grow. Adiabatic fluctuations remain adiabatic, because different regions behave as if they were parts of different homogeneous universes. When the Hubble length becomes comparable to the mode proper wavelength, the baryons and radiation, strongly coupled by Thomson scattering at high redshift, oscillate as an acoustic wave and the mode amplitude for the cold dark matter stops growing. 41 The mass densities in dark matter and radiation are equal at redshift zeq = 2.4 × 104 OmegaM0 h2; thereafter the dark matter mass density dominates and the fluctuations in its distribution start to grow again. The suppressed growth of density fluctuations within the Hubble length at z > zeq produces a bend in the power spectrum of the dark mass distribution, from P(k) propto k at long wavelengths, if scale-invariant (Eq. [40]), to P(k) propto k-3 at short wavelengths. 42 This means that at small scales (large k) the contribution to the variance of the mass density per logarithmic interval of wavelength is constant, and at small k the contribution to the variance of the Newtonian gravitational potential per logarithmic interval of k is constant.

The wavelength at the break in the spectrum is set by the Hubble length at equal radiation and matter mass densities, teq propto zeq-2. This characteristic break scale grows by the factor zeq to lambdabreak ~ zeq teq at the present epoch. The numerical value is (Peebles, 1980a, Eq. [92.47])

Equation 42 (42)

If Lambda is close to constant, or Lambda = 0 and space sections are curved, it does not appreciably affect the expansion rate at redshift zeq, so this characteristic length is the same in flat and open cosmological models. In an Einstein-de Sitter model, with OmegaM0 = 1, the predicted length scale at the break in the power spectrum of CDM mass fluctuations is uncomfortably small relative to structures such as are observed in clusters of clusters of galaxies (superclusters), and relative to the measured galaxy two-point correlation function. That is, more power is observed on scales ~ 100 Mpc than is predicted in the CDM model with OmegaM0 = 1. Since lambdabreak scales as OmegaM0-1, a remedy is to go to a universe with small OmegaM0, either with Lambda = 0 and open space sections or OmegaK0 = 0 and a nonzero cosmological constant. The latter case is now known as LambdaCDM. 43

38 A scorecard is given in Peebles and Silk (1990). Structure formation models that assume all matter is baryonic, and those that augment baryons with hot dark matter such as low mass neutrinos, were already seriously challenged a decade ago. Vittorio and Silk (1985) show that the Uson and Wilkinson (1984) bound on the small-scale anisotropy of the 3 K cosmic microwave background temperature rules out a baryon-dominated universe with adiabatic initial conditions. This is because the dissipation of the baryon density fluctuations by radiation drag as the primeval plasma combines to neutral hydrogen (at redshift z ~ 1000) unacceptably suppresses structure formation on the scale of galaxies. Cold dark matter avoids this problem by eliminating radiation drag. This is one of the reasons attention turned to the hypothetical nonbaryonic cold dark matter. There has not been a thorough search for more baroque initial conditions that might save the baryonic dark matter model, however. Back.

39 The issue is presented in Peebles (1986); the data and history of ideas are reviewed in Peebles (2001). Back.

40 The demonstration that the suppression of peculiar velocities is a lot stronger than the suppression of the growth of structure is in Peebles (1984) and Lahav et al. (1991). Back.

41 At high redshift the dark matter mass density is less than that of the radiation. The radiation thus fixes the expansion rate, which is too rapid for the self-gravity of the dark matter to have any effect on its distribution. Early discussions of this effect are in Guyot and Zel'dovich (1970) and Mészáros (1972). Back.

42 That is, the transfer function in Eq. (40) goes from a constant at small k to T2(k) propto k-4 at large k. Back.

43 The scaling of lambdabreak with OmegaM0 was frequently noted in the 1980s. The earliest discussions we have seen of the significance for large-scale structure are in Silk and Vittorio (1987) and Efstathiou, Sutherland, and Maddox (1990), who consider a spatially-flat universe, Blumenthal, Dekel, and Primack (1988), who consider the open case, and Holtzman (1989), who considers both. For the development of tests of the open case see Lyth and Stewart (1990), Ratra and Peebles (1994), Kamionkowski et al. (1994a), and Górski et al. (1995). Pioneering steps in the analysis of the anisotropy of the 3 K cosmic microwave background temperature in the LambdaCDM model include Kofman and Starobinsky (1985) and Górski, Silk, and Vittorio (1992). We review developments after the COBE detection of the anisotropy (Smoot et al., 1992) in Secs. IV.B.11 and 12. Back.

Next Contents Previous