### 3.1 Inflationary dynamics

We consider the models of the form
which include the Starobinsky’s model [564] as a specific case (). In the absence of the matter fluid
(), Eq. (2.15) gives
The cosmic acceleration can be realized in the regime . Under the approximation
, we divide Eq. (3.2) by to give
During inflation the Hubble parameter evolves slowly so that one can use the approximation
and . Then Eq. (3.3) reduces to

Integrating this equation for , we obtain the solution
The cosmic acceleration occurs for , i.e., . When one has , so
that is constant in the regime . The models with lead to super inflation characterized
by and ( is a constant). Hence the standard inflation with decreasing
occurs for .
In the following let us focus on the Starobinsky’s model given by

where the constant has a dimension of mass. The presence of the linear term in eventually causes
inflation to end. Without neglecting this linear term, the combination of Eqs. (2.15) and (2.16) gives
During inflation the first two terms in Eq. (3.7) can be neglected relative to others, which gives
. We then obtain the solution
where and are the Hubble parameter and the scale factor at the onset of inflation (,
respectively. This inflationary solution is a transient attractor of the dynamical system [407]. The
accelerated expansion continues as long as the slow-roll parameter
is smaller than the order of unity, i.e., . One can also check that the approximate relation
holds in Eq. (3.8) by using . The end of inflation (at time
) is characterized by the condition , i.e., . From Eq. (3.11) this
corresponds to the epoch at which the Ricci scalar decreases to . As we will see later,
the WMAP normalization of the CMB temperature anisotropies constrains the mass scale to
be . Note that the phase space analysis for the model (3.6) was carried out
in [407, 24, 131].
We define the number of e-foldings from to :

Since inflation ends at , it follows that
where we used Eq. (3.12) in the last approximate equality. In order to solve horizon and flatness problems
of the big bang cosmology we require that [391], i.e., . The CMB temperature
anisotropies correspond to the perturbations whose wavelengths crossed the Hubble radius around
before the end of inflation.