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2.1. Introduction

In studies of structure formation one is particularly interested in the statistical properties of the density contrast of the matter distribution,

Equation 1

which is defined in terms of the density field rho(x, t) and the comoving mean background density rhobar, where x is the comoving position.

A Fourier expansion can be made in a large enough box (of volume V) so that delta(x, t) is periodic within the box. We then have

Equation 2

where k ident | k | is a comoving wavenumber, with the Fourier coefficients being given by

Equation 3

We will define the power spectrum, P(k, t), as

Equation 4

The dispersion of the density contrast is then simply given by

Equation 5

If it is assumed that for any given realization of the volume V the phases of the Fourier coefficients deltak are uncorrelated, the central limit theorem then guarantees that at any point the density contrast delta(x, t) obeys Gaussian statistics. The probability distribution of delta(x, t) at each point is then

Equation 6

This equation implies that there is always some probability of having delta(x, t) < - 1, which, by definition, is not physically possible. Therefore, as a first approximation, it is only valid to consider delta(x, t) as a Gaussian random field if there is only a very small probability of having delta(x, t) < - 1 by the above equation, i.e. if sigma(t) << 1. This condition is also necessary if we wish to use linear perturbation theory to follow the evolution of delta(x, t). Gaussian random fields are very special, since only the power spectrum is required to specify all of the statistical properties of the field, whereas for non-Gaussian fields the full hierarchy of probability distributions is needed.

After matter domination, the power spectrum of the density contrast, delta(x, t), can be written as (56, 57, 67)

Equation 7

where the quantities aH, Omega and lambda ident Lambdac2 / 3H2 are to be calculated at t. The function g(Omega, lambda) accounts for the rate of growth of density perturbations relative to the Einstein-de Sitter case, whose growth is given by the (aH)4 factor. The transfer function, T(k, t), measures the change at t in the amplitude of a perturbation with comoving wavenumber k relative to a perturbation with infinite wavelength, thus in the limit k -> 0 (in practice k -> khor due to gauge ambiguities), one has T(k, t) -> 1. The shape of the transfer function results mostly from the different behaviour of perturbations in the radiation and matter dominated eras, and from sub-horizon damping effects, like Silk damping, which affects baryons, and free-streaming (Landau damping), which acts on hot dark matter perturbations. An oscillatory pattern can also appear in the transfer function if baryons contribute significantly to the matter density in the Universe, due to acoustic oscillations of the photon-baryon fluid on scales below the horizon until decoupling occurs. The calculation of a transfer function not only depends on the type of mechanism responsible for the generation of the density perturbations, but also on the assumed matter and energy content in the Universe. It thus needs to be determined numerically, though nowadays there are several analytical prescriptions which approximate it for the most popular structure formation scenarios [see e.g. (24)].

The quantity deltaH2(k), defined as

Equation 8

specifies the power spectrum of density perturbations at horizon re-entry. In the simplest inflationary models it can be well described by a single power-law,

Equation 9

where n is the so-called spectral index and deltaH(k0) is a normalisation factor at an arbitrary comoving wavelength k0. Since the COBE measurement of the amplitude of the large-angle anisotropies in the temperature of the CMBR became available, the value of deltaH(k0) is usually set so as to reproduce it (though some previous assumption has to be made regarding the contribution of tensor perturbations, i.e. gravitational waves, to the anisotropies). When this is done, in the simplest inflationary models the value of deltaH(k0) then depends essentially only on the values of n, Omega0 and lambda [see e.g. (11)]. A scale-invariant, or Harrison-Zel'dovich, power spectrum corresponds to n = 1. In general most inflationary models give n leq 1, though in some it is possible to have n > 1.

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