**3.5. Perturbations**

The technology for calculating the production of
inhomogeneities during inflation is by now very well understood (For
an excellent review see
[11]). The
natural variables to follow are the Fourier transformed density contrasts
(*k*) in *comoving
coordinates* (*k*_{co} = *a*
*k*_{phys}). A given mode
(*k*)
behaves very differently depending on its relationship to the Hubble
radius *R*_{H}
1/*H*. During inflation
*H* = const. while
comoving scales are growing exponentially. The modes of interest
start inside *R*_{H} (*k* >>
*R*_{H}^{-1}) and evolve outside
*R*_{H} (*k* <
*R*_{H}^{-1}) during inflation. After inflation,
during the SBB, *R*_{H}
*t* and *a*
*t*^{1/2} or
*t*^{2/3} so
*R*_{H} catches up
with comoving scales and modes are said to ``fall inside'' the Hubble
radius.

It is important to note that *R*_{H} is often called the
``horizon''. This is because in the SBB *R*_{H} is roughly
equal to the
distance light has traveled since the big bang. Inflation changes all
that, but ``Hubble radius'' and ``horizon'' are still often (confusingly)
used interchangeably.

Modes that are relevant to observed structures in the Universe started
out on extremely small scales before inflation. By comparison with
what we see in nature today, the only excitations one expects in these
modes are zero-point fluctuations in the quantum fields. Even if they
start out in an excited state, one might expect that on length-scales much
smaller (or energies-scales much higher) than the scale
(^{1/4})
set by the
energy density the modes would rapidly ``equilibrate'' to the ground
state. These zero-point fluctuations are expanded to cosmic scales
during inflation, and form the ``initial conditions'' that are used to
calculate the perturbations today. Because the perturbations on
cosmologically relevant scales must have been very small over most of
the history of the Universe, linear perturbation theory in GR may be
used. Thus the
question of perturbations from inflation can be treated by a tractable
calculation using well-defined initial conditions.

One can get a feeling for some key features by examining this result which holds for most inflationary models:

Here _{H}^{2}
is roughly the mean squared value of
when
the mode falls below *R*_{H} during the SBB. The ``*''
means evaluate
the quantity when the mode in question goes outside *R*_{H} during
inflation, and *V* is the inflaton potential. Since typically the
inflaton is barely moving during inflation,
_{H}^{2} is nearly
the same for all modes, resulting in a spectrum of perturbations that
is called ``nearly scale invariant''. The cosmological data
require _{H}
10^{-5}.

In typical models the ``slow roll'' condition,

is obeyed, and inflation
ends when *V'M*_{P} / *V*
1. If *V* is dominated
by a single power
of during the relevant period
*V'M*_{P} / *V*
1 means
*M*_{P} at the
end of inflation. Since typically
does not
vary much during inflation
*M*_{P} would apply
throughout. With these values of , and
assuming simple power law forms for *V*, one might guess that the
right side of Eqn. 9 is around unity (as opposed to
10^{-5}!) Clearly some subtlety is involved in constructing a
successful model.

I have noticed people sometimes make rough dimensional arguments by
inserting *V*
*M*_{G}^{4} and
*V'*
*M*_{G}^{3} into Eqn. 9 and conclude
_{H}
(*M*_{G} /
*M*_{P})^{3}. This argument suggests
that _{H} is naturally
small for GUT inflation, and that the adjustments required to get
_{H}
10^{-5} could be
modest. However that argument neglects the fact that the slow roll
condition must be met. When this constraint is factored in, the
challenge of achieving the right amplitude is much greater than the
simplest dimensional argument would indicate.