3.4. Fluctuations in the Distributions of Mass and the CBR

As noted in Sections 2.3 and 3.1, structure formation on the scale of galaxies and larger is thought to have been dominated by the gravitational growth of small departures from homogeneity present in the very early universe. The nature of the initial conditions is open because we do not have an established theory of what the universe was doing before it was expanding. We do have a consistency condition, that a single set of initial values must match many observational constraints. I discuss here second moments of the large-scale fluctuations in the distributions of galaxies and the thermal cosmic background radiation (the CBR).

It is sensible to try the simplest prescription for initial conditions first. Most widely discussed is the adiabatic cold dark matter (ACDM) model Joe Silk mentions in his introduction. In the simplest case the universe is Einstein-de Sitter and the density fluctuations are scale-invariant (the density contrast / appearing on the Hubble length is independent of time). This case tends to underpredict large-scale density fluctuations; the problem is remedied by lowering H₀ or _m (Blumenthal et al. 1988; Efstathiou et al. 1990). ⁽¹⁾ The wanted value of H₀ is below most estimates of this parameter, so the more commonly accepted interpretation is that _m is less than unity. This leads to the grade in line 2a. It depends on the model for structure formation, of course.

Examples of second moments of the galaxy space distribution and the angular distribution of the CBR are shown in Figures 2 and 3. The power spectrum of the space distribution is

(13)

where the dimensionless galaxy two-point correlation function is

(14)

for the smoothed galaxy number density n(r). The data in Figure 2 are from the IRAS PSC-z (point source catalog) redshift survey (Saunders et al. 1998) of the far infrared-luminous galaxies mentioned in Section 3.1. Since infrared radiation is not strongly affected by dust this promises to be an excellent probe of the large-scale galaxy distribution.

Figure 2. Mass fluctuation spectrum extrapolated to the present in linear perturbation theory for the ACDM model in equation (17) (solid line, from Tegmark 1998b) and the ICDM model in equation (18) (dashed line). The galaxy fluctuation spectrum is from the PSC-z collaboration (Saunders et al. 1998).

Figure 3 shows second moments of the expansion, defined as

(16)

In the approximation of the sum over l as an integral the variance of the CBR temperature per logarithmic interval of l is (T_l)². The T_l data in Figure 3 are from the survey of the measurements by Tegmark (1998a).

Figure 3. Spectrum of angular fluctuations of the CBR. The data are from the compilation by Tegmark (1998a). The ACDM model prediction plotted as the solid line assumes the parameters in equation (17) (Tegmark 1998a). The ICDM model prediction plotted as the dashed line assumes the parameters in equation (18).

The solid curves in Figures 2 and 3 are the prediction (Tegmark 1998a, b) of an ACDM model with a scale-invariant primeval mass fluctuation spectrum and the parameters

(17)

It is impressive to see how well this model fits the two sets of measurements. But at the present accuracy of the measurements there is at least one other viable model, shown as the dashed curves. It assumes the same dynamical actors as in ACDM---cold dark matter, baryons, the CBR, and three families of massless neutrinos---but the isocurvature initial condition is that the primeval mass density and the entropy per baryon are homogeneous, and homogeneity is broken by an inhomogeneous primeval distribution of the CDM. A simple model for the spectrum of primeval CDM fluctuations is P (k) k^m. A rough fit to the measurements has parameters

(18)

Further details and a pedigree within the inflation picture are in Peebles (1999a, b). The solid curve fits the CBR anisotropy measurements better, but it is based on a much more careful search of parameters to fit the data. A bend in P (k) would do wonders for the dashed curve. Hu (1998) gives another example of how the prediction of the CBR angular fluction spectrum depends on the details of the structure formation model.

As mentioned in Section 2.3, the point of this discussion is that reading the values of the cosmological parameters from the CBR anisotropy measurements in Figure 3 is a dangerous operation because it depends on the theory for structure formation as well as the Friedmann-Lemaître model. This applies to other entries in category 2 in Table 2 (and to line 1a: a satisfactory quantitative understanding of galaxy formation would include an understanding of the relation between the distributions of galaxies and mass).

Our knowledge of P (k) and T_l will be considerably improved by work in progress. Redshift surveys to probe P (k) and the large-scale mass distribution include the Century Survey, the Two Degree Field Survey (2dF), and the Sloan Digital Sky Survey (SDSS); precision measurements of the CBR include BOOMERANG, MAP, PLANCK, and other ground, balloon, and satellite projects (Geller et al. 1997; Page 1997; Nordberg & Smoot 1998; Eisenstein et al. 1998; and references therein). If one of the structure formation models now under discussion fits all the bumps and wiggles in the measured spectra it will inspire confidence.

In the Einstein-de Sitter case a scale-invariant ACDM model normalized by the assumption that galaxies trace mass gives quite a good fit to the CBR angular fluctuation spectrum T_l; on this score it would merit a pass in line 2b. But the assumptions that galaxies trace mass and that _m = 1 imply quite unacceptable peculiar velocites. The situation is different from line 1a, where the issue is whether _m = 1 can be saved by the postulate that galaxies do not trace mass. Thus I think it is fair to give the Einstein-de Sitter case separate demerits in lines 1a and 2b, but with a question mark for the latter because it depends on the model for structure formation.

Several authors have concluded that the low density flat ACDM model (with > 0) is a better fit to the T_l measurements than is the low density open case (Gawiser & Silk 1998; Tegmark 1998a). Others note that other treatments of the still quite new measurements can lead to the opposite conclusion (Górski et al. 1998; Ratra 1998). Since the former approach seems to treat the measurements in the more literal way the flat case gets the higher grade in line 2b.