A model of structure formation has 3 ingredients: (1) Background cosmology, (2) Model for fluctuation generation and (3) Types of dark matter.
The most successful theory of cosmological structure formation is inflationary cold dark matter (a.k.a CDM) in which nearly scale-invariant adiabatic initial fluctuations are set up by a period of inflation in the early universe. Thus the initial conditions are: (a) the fluctuations in the gravitational potential (which can be related to density fluctuation through Poisson's equation) are almost independent of scale and (b) that the fluctuations in the pressure are proportional to those in the density (which keeps the entropy constant, hence the term adiabatic). A direct consequence of the adiabatic assumption is that a cosmic overdense region contains overdensities of all particle species. The alternative mode, where overdensities of one species counterbalance underdensities in another, is known as isocurvature because the spatial curvature is unchanged. Models incorporating isocurvature initial conditions fare very badly when compared to the observations.
Most (though not all) inflation models also predict that the spectrum of fluctuations is gaussian, with zero mean and variance given by the power spectrum. The preponderance of observational evidence suggests that the initial fluctuations which produced the large-scale structure in the universe were gaussian, with non-linear gravitational clustering producing all of the non-gaussianity in the distribution observed today. All the front-running candidates for models of structure formation use gaussian initial conditions.
Viable models of structure formation thus differ mostly in what is assumed for the background cosmological parameters, specifically the density in CDM, spatial curvature or a cosmological constant/dark energy component (and its evolution), baryonic component, exansion rate etc. Of the possible models, the only currently viable one is L(ambda)CDM with roughly 1/3 of the energy in the universe being cold dark matter, 2/3 in a cosmological constant or "dark energy" component, and a few percent being in the form of normal or "baryonic" matter (most of which is also dark). The initial fluctuations have to be almost exactly Gaussian with a close to scale-invariant spectrum which is nearly power-law in scale.
Once the initial spectrum and type of fluctuations is known, the linear evolution (at early times) is determined by the background cosmology and the type of dark matter. Because hot dark matter moves rapidly, it is able to stream out of overdense regions, escaping from the enhanced gravitational potential. This tends to erase fluctuations in the matter distribution on scales smaller than the free-streaming scale (approximately the speed of the HDM particles times the age of the universe). In contrast to this, cold dark matter is able to support perturbations on all scales of cosmological interest. This behaviour is much closer to what is inferred from observations of clustering. The dependence on the background cosmology enters because fluctuations only grow when the universe is matter dominated. Lowering the matter density thus decreases the length of time the perturbations can grow (both at early times, by delaying the dominance of matter over radiation and at late times when the universe becomes curvature or cosmological constant dominated). Some simulations illustrating these principles can be found here.
Once all of these ingredients are specified one can calculate the predictions for the cosmic microwave background anisotropies and the matter power spectrum in linear theory. At this point the absolute normalization of the CMB and matter power spectra is a free parameter, but their relative normalization is not. By forcing the CMB power spectrum to fit the COBE or WMAP data, the overall normalization can be fixed. Then one computes the full evolution of the large-scale structure by means of N-body and/or hydrodynamical simulations, or uses analytic approximations to this evolution where appropriate. The resulting predictions can be compared to the observations.
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