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.
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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} / ( *n*_{g} / *n*_{g})_{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.
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