Large-Scale Structure of the Universe and Cosmological ...

such initial conditions are likely to happen in very broad classes of models.
The second class of models that have been developed for structure formation
are based on topological defects, of which cosmic strings have been studied in
most detail. In this case the origin of stochasticity lies on thermal fluctuations
of a field that undergoes a phase transition as the universe cools, and is likely
to obey non-Gaussian properties. Note however that these two classes of models
do not necessarily exclude each other. For instance, formation of cosmic
strings are encountered in specific models of inflation [65,66,352]. There are
also models inspired by duality properties of superstring theories, in which an
inflationary phase can be encountered but structure formation is caused by
the quantum fluctuations of the axion field 11 [668,159,111] rather than the
inflaton field. With such a mechanism the initial metric fluctuations will not
obey Gaussian statistics.
3.1.1 Physical Origin of Fluctuations from Inflation
In models of inflation the stochastic properties of the fields originate from
quantum fluctuations of a scalar field, the inflaton. It is beyond the scope
of this review to describe inflationary models in any detail. We instead refer
the reader to recent reviews for a complete discussion [399,400,415]. It is
worth however recalling that in such models (at least for the simplest singlefield
models within the slow-roll approximation) all fluctuations originate from
scalar adiabatic perturbations. During the inflationary phase the energy density
of the universe is dominated by the density stored in the inflaton field.
This field has quantum fluctuations that can be decomposed in Fourier modes
using the creation and annihilation operators a †
k and ak for a wave mode k,
δϕ =
d 3 k
ak ψk(t) exp(ik.x) + a †
k ψ∗ k (t) exp(−ik.x) . (110)
The operators obey the standard commutation relation,
[ak, a †
−k ′] = δD(k + k ′ ), (111)
and the mode functions ψk(t) are obtained from the Klein-Gordon equation
for ϕ in an expanding Universe. We give here its expression for a de-Sitter
metric (i.e. when the spatial sections are flat and H is constant),
ψk(t) =
H
(2 k) 1/2 k
i + k
exp
a H
i k
, (112)
a H
where a and H are respectively the expansion factor and the Hubble constant
that are determined by the overall content of the Universe through the
Friedmann equations, Eqs. (4-5).
11 However, this generally leads to isocurvature fluctuations rather than adiabatic.
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