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]
Figure 1: The field potential (3.16) in the Einstein frame corresponding to the model (3.6). Inflation
is realized in the regime .
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
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
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.