**3.1 Quantum Fluctuations**

During inflation there are quantum fluctuations in the inflaton field.
Since the total energy density of the universe is dominated by the
inflaton potential energy density, fluctuations in the inflaton field
lead to fluctuations in the energy density. Because of the rapid
expansion of the universe during inflation, these fluctuations in the
energy density are frozen into super-Hubble-radius-size perturbations.
Later, in the radiation or matter-dominated era they will come within
the Hubble radius as if they were *noncausal* perturbations.

The spectrum and amplitude of perturbations depend upon the nature of the inflaton potential. Mukhanov [3] has developed a very nice formalism for the calculation of density perturbations. One starts with the action for gravity (the Einstein-Hilbert action) plus a minimally-coupled scalar inflaton field :

Here *R* is the Ricci curvature scalar. Quantum fluctuations result
in perturbations in the metric tensor

where *g*_{µ}^{FRW} is the
Friedmann-Robertson-Walker metric,
and _{0} *(t)* is the
classical solution for the homogeneous,
isotropic evolution of the inflaton. The action describing the
dynamics of the small perturbations can be written as

i.e., the action in conformal time
(*d*^{2} =
*a*^{2} *(t) dt*^{2}) for
a scalar field in Minkowski space, with mass-squared
*m _{u}*

The simple matter of calculating the perturbation spectrum for a
noninteracting scalar field in Minkowski space will give the amplitude
and spectrum of the density perturbations. The problem is that the
solution to the field equations depends upon the background field
evolution through the dependence of the mass of the field upon *z*.
Different choices for the inflaton potential *V*() results in
different background field evolutions, and hence, different spectra
and amplitudes for the density perturbations.

Before proceeding, now is a useful time to remark that in addition to scalar density perturbations, there are also fluctuations in the transverse, traceless component of the spatial part of the metric. These fluctuations (known as tensor fluctuations) can be thought of as a background of gravitons.

Although the scalar and tensor spectra depend upon *V*(), for most
potentials they can be characterized by *Q _{RMS}^{PS}*
(the amplitude
of the scalar and tensor spectra on large length scales added in
quadrature),

In addition to the primordial spectrum characterized by *n* and *r*,
in order to compare to data it is necessary to specify cosmological
parameters (*H*_{0}, the present expansion rate; _{0}, the ratio
of the present mass-energy density to the critical density - a
spatially flat universe has _{0} = 1; _{B}, the ratio of the
present baryon density to the critical density; _{DM} the
ratio of the present dark-matter density to the critical density; and
, the value of the
cosmological constant), as well as the nature of the dark matter.

The specification of the dark matter is by how ``hot'' the dark matter was when the universe first became matter dominated. If the dark matter was really slow at that time, then it is referred to as cold dark matter. If the dark matter was reasonably hot when the universe became matter dominated, then it is called hot dark matter. Finally, the intermediate case is called warm dark matter. Neutrinos with a mass in the range 1 eV to a few dozen eV would be hot dark matter. Light gravitinos, as appear in gauge-mediated supersymmetry breaking schemes, is an example of warm dark matter. By far the most popular dark matter candidate is cold dark matter. Examples of cold dark matter are neutralinos and axions.