3.2 Dynamics in the Einstein frame
Let us consider inflationary dynamics in the Einstein frame for the model (3.6) in the absence of matter
fluids (
). The action in the Einstein frame corresponds to (2.32) with a field
defined by
Using this relation, the field potential (2.33) reads [408
, 61, 63]
In Figure 1 we illustrate the potential (3.16) as a function of
. In the regime
the potential is nearly constant (
), which leads to slow-roll inflation. The
potential in the regime
is given by
, so that the field oscillates
around
with a Hubble damping. The second derivative of
with respect to
is
which changes from negative to positive at
.
Since
during inflation, the transformation (2.44) gives a relation between the cosmic
time
in the Einstein frame and that in the Jordan frame:
where
corresponds to
. The end of inflation (
) corresponds to
in the Einstein frame, where
is given in Eq. (3.13). On using Eqs. (3.10) and (3.18),
the scale factor
in the Einstein frame evolves as
where
. Similarly the evolution of the Hubble parameter
is
given by
which decreases with time. Equations (3.19) and (3.20) show that the universe expands quasi-exponentially
in the Einstein frame as well.
The field equations for the action (2.32) are given by
Using the slow-roll approximations
and
during
inflation, one has
and
. We define the slow-roll parameters
For the potential (3.16) it follows that
which are much smaller than 1 during inflation (
). The end of inflation is characterized by the
condition
. Solving
, we obtain the field value
.
We define the number of e-foldings in the Einstein frame,
where
is the field value at the onset of inflation. Since
, it follows that
is identical to
in the slow-roll limit:
. Under the condition
we
have
This shows that
for
. From Eqs. (3.24) and (3.26) together with the approximate
relation
, we obtain
where, in the expression of
, we have dropped the terms of the order of
. The results (3.27) will
be used to estimate the spectra of density perturbations in Section 7.