The combination of the COBE normalization, the matter transfer function and the near scale-invariant initial spectrum of fluctuations tells us that by the present fluctuations in the cold dark matter or baryon density fields will have gone non-linear for all scales Mpc. It is a great triumph of the standard cosmological paradigm that there is just enough growth between and to explain structures in the Universe across a wide range of scales.
In particular, since this non-linear scale also corresponds to galaxy clusters and measurements of their abundance yields a robust measure of the power near this scale for a given matter density . The agreement between the COBE normalization and the cluster abundance at low and the observed Hubble constant [Freedman et al, 2001] was pointed out immediately following the COBE result (e.g. [White et al, 1993,Bartlett & Silk, 1993]) and is one of the strongest pieces of evidence for the parameters in the working cosmological model [Ostriker & Steinhardt, 1995,Krauss & Turner, 1995].
More generally, the comparison between large-scale structure and the CMB is important in that it breaks degeneracies between effects due to deviations from power law initial conditions and the dynamics of the matter and energy contents of the Universe. Any dynamical effect that reduces the amplitude of the matter power spectrum corresponds to a decay in the Newtonian potential that boosts the level of anisotropy (see §3.5 and §4.2.1). Massive neutrinos are a good example of physics that drives the matter power spectrum down and the CMB spectrum up.
The combination is even more fruitful in the relationship between the acoustic peaks and the baryon wiggles in the matter power spectrum. Our knowledge of the physical distance between adjacent wiggles provides the ultimate standard ruler for cosmology [Eisenstein et al, 1998]. For example, at very low , the radial distance out to a galaxy is . The unit of distance is therefore Mpc, and a knowledge of the true physical distance corresponds to a determination of . At higher redshifts, the radial distance depends sensitively on the background cosmology (especially the dark energy), so a future measurement of baryonic wiggles at say would be a powerful test of dark energy models. To a lesser extent, the shape of the transfer function, which mainly depends on the matter-radiation scale in Mpc, i.e. , is another standard ruler (see e.g. [Tegmark et al, 2001] for a recent assessment), more heralded than the wiggles, but less robust due to degeneracy with other cosmological parameters.
For scales corresponding to Mpc, density fluctuations are non-linear by the present. Numerical -body simulations show that the dark matter is bound up in a hierarchy of virialized structures or halos (see [Bertschinger, 1998] for a review). The statistical properties of the dark matter and the dark matter halos have been extensively studied in the working cosmological model. Less certain are the properties of the baryonic gas. We shall see that both enter into the consideration of secondary CMB anisotropies.