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3.3 Structure Formation

The Friedmann-Lemaître model is unstable to the gravitational growth of departures from a homogeneous mass distribution. The present large-scale homogeneity could have grown out of primeval chaos, but the initial conditions would be absurdly special. That is, the Friedmann-Lemaître model requires that the present structure - the clustering of mass in galaxies and systems of galaxies - grew out of small primeval departures from homogeneity. The consistency test for an acceptable set of cosmological parameters is that one has to be able to assign a physically sensible initial condition that evolves into the present structure of the universe. The constraint from this consideration in line 3c is discussed by White et al. [44], and in line 3b by Bahcall et al. ([45], [46]). Here I explain the cautious ratings in line 3a.

As has been widely discussed, it may be possible to read the values of Omega and other cosmological parameters from the spectrum of angular fluctuations of the CBR ([47] and references therein). This assumes Nature has kept the evolution of the early universe simple, however, and we have hit on the right picture for its evolution. We may know in the next few years. If the precision measurements of the CBR anisotropy from the MAP and PLANCK satellites match in all detail the prediction of one of the structure formation models now under discussion it will compel acceptance. But meanwhile we should bear in mind the possibility that Nature was not kind enough to have presented us with a simple problem.

Figure 3

Figure 3. Angular fluctuations of the CBR in low density cosmologically flat adiabatic (dashed line) and isocurvature (solid line) CDM models for structure formation. The variance of the CBR temperature anisotropy per logarithmic interval of angular scale theta = pi / l is (Tl)2, as in Eqs. (5) to (7). Data are from the compilation by Ratra [48].

An example of the possible ambiguity in the interpretation of the present anisotropy measurements is shown in Fig. 3. The two models assume the same dynamical actors -cold dark matter (CDM), baryons, three families of massless neutrinos, and the CBR - but different initial conditions. In the adiabatic model the primeval entropy per conserved particle number is homogeneous, the space distribution of the primeval mass density fluctuations is a stationary random process with the scale-invariant spectrum propto k, and the cosmological parameters are Omega = 0.35, lambda = 0.65, and h = 0.625 (following [49]). The isocurvature initial condition in the other model is that the primeval mass distribution is homogeneous - there are no curvature fluctuations - and structure formation is seeded by an inhomogeneous composition. In the model shown here the primeval entropy per baryon is homogeneous, to agree with the standard model for light element production, and the primeval distribution of the CDM has fluctuation spectrum

Equation 16 (16)

The cosmological parameters are Omega = 0.2, lambda = 0.8, and h = 0.7. The lower density parameter produces a more reasonable-looking cluster mass function for the isocurvature initial condition [50]. In both models the density parameter in baryons is OmegaB = 0.03, the rest of Omega is in CDM, and space sections are flat (lambda = 1 - Omega). Both models are normalized to the large-scale galaxy distribution. The adiabatic initial condition follows naturally from inflation, as a remnant of the squeezed field that drove the rapid expansion. A model for the isocurvature condition assumes the CDM is (or is the remnant of) a massive scalar field that was in the ground level during inflation and became squeezed to a classical realization. In the simplest models for inflation this produces m = -3 in Eq. (16). The tilt to m = -1.8 requires only modest theoretical ingenuity [51]. That is, both models have pedigrees from commonly discussed early universe physics.

The lesson from Fig. 3 is that at least two families of models, with different relations between Omega and the value of l at the peak, come close to the measurements of the CBR fluctuation spectrum, within the still substantial uncertainties. An estimate of Omega from the CBR anisotropy measurements thus may depend on the choice of the model for structure formation. Programs of measurement of delta Tl in progress should be capable of distinguishing between the adiabatic and isocurvature models, even given the freedom to adjust the shape of P (k). The interesting possibility is that some other model for structure formation with a very different value of Omega may give an even better fit to the improved measurements.

I assign a failing grade to the Einstein-de Sitter model in line 3a because the adiabatic and isocurvature models both prefer low Omega ([52], [53]). I add question marks to indicate this still is a model-dependent result.

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