There is an elegant explanation for the origin of the density fluctuations that seeded structure formation by gravitational instability. Quantum fluctuations are imprinted on a macroscopic scale with a nearly scale-invariant spectral distribution of amplitudes, defined by constant amplitude density fluctuations at horizon crossing. This leads to a bottom-up formation sequence as the smallest subhorizon scales acquire larger amplitudes and are the first to go nonlinear. One can compare the predicted linear fluctuations over scales 10 Mpc with observations via microwave background fluctuations and galaxy number count fluctuations. T/T measures / at last scattering over scales from ~ 100 Mpc up to the present horizon. Temperature fluctuations on smaller scales are progressively damped by radiative diffusion, but a signal is detectable to an angular scale of ~ 10', equivalent to ~ 20 Mpc. The conversion from T/T to / is model-dependent, but can be performed once the transfer function is specified. At these high redshifts, one is well within the linear regime, and if the fluctuations are Gaussian, one can reconstruct the density fluctuation power spectrum.
Deep galaxy surveys yield galaxy number count fluctuations, which are subject to an unknown bias between luminous and dark matter. Moreover, all three dimensional surveys necessarily utilize redshift space. Conversion from redshift space to real space is straightforward if the peculiar velocity field is specified. One normally assumes spherical symmetry and radial motions on large scales, and isotropic motions on scales where virialization has occurred, with an appropriate transition between the linear and nonlinear regimes. On the virialization scale, collapse by of order a factor of 2 has occurred in the absence of dissipation, and correction for density compression must also be incorporated via interpolation or preferably via simulations.
Comparison of models with data is satisfactory only if the detailed shape of the power spectrum is ignored. 24 A two parameter fit, via normalisation at 8 h-1 Mpc and a single shape parameter h, is often used. For example, as defined below, 8 ( / )rms / ( ng / ng)rms, as evaluated at 8 h-1 Mpc, equals unity for unbiased dark matter. COBE normalisation of standard cold dark matter requires 8 1 but the cluster abundance requires 8 0.6. The shape parameter h = 1 for standard cold dark matter, but h 0.3 is favoured for an open universe. One can fit a model to the data with 8 0.6 and h 0.3. However detailed comparison of models and observations reveals that there is no satisfactory fit to the power spectrum shape for an acceptable class of models. There is an excess of large-scale power near 100 Mpc. This is mostly manifested in the APM galaxy and cluster surveys, but is also apparent in the Las Campanas redshift survey. 26