7.3 The spectra of perturbations in inflation based on f (R) gravity
Let us study the spectra of scalar and tensor perturbations generated during inflation in
metric f (R) gravity. Introducing the quantity
, we have
and
Since the field kinetic term
is absent, one has
in Eqs. (7.42) and (7.49). Under the
conditions
(
), the spectral index of curvature perturbations is given by
.
In the absence of the matter fluid, Eq. (2.16) translates into
which gives
for
. Hence we obtain [315]
From Eqs. (7.50) and (7.60), the amplitude of
is estimated as
Using the relation
, the spectral index (7.57) of tensor perturbations is given by
which vanishes at first-order of slow-roll approximations. From Eqs. (7.58) and (7.63) we obtain the
tensor-to-scalar ratio
7.3.1 The model
(
)
Let us consider the inflation model:
(
). From the discussion given in Section 3.1 the
slow-roll parameters
(
) are constants:
In this case one can use the exact results (7.48) and (7.56) with
and
given in Eqs. (7.42) and
(7.54) (with
). Then the spectral indices are
If
we obtain the scale-invariant spectra with
and
. Even the slight deviation
from
leads to a rather large deviation from the scale-invariance. If
, for example,
one has
, which does not match with the WMAP 5-year constraint:
[367
].
7.3.2 The model 
For the model
, the spectrum of the curvature perturbation
shows some
deviation from the scale-invariance. Since inflation occurs in the regime
and
, one
can approximate
. Then the power spectra (7.63) and (7.58) yield
where we have employed the relation
.
Recall that the evolution of the Hubble parameter during inflation is given by Eq. (3.9). As long as the
time
at the Hubble radius crossing (
) satisfies the condition
, one can
approximate
. Using Eq. (3.9), the number of e-foldings from
to the end of inflation
can be estimated as
Then the amplitude of the curvature perturbation is given by
The WMAP 5-year normalization corresponds to
at the scale
[367
]. Taking the typical value
, the mass
is constrained to be
Using the relation
, it follows that
. Hence the spectral index (7.62) reduces to
For
we have
, which is in the allowed region of the WMAP 5-year constraint
(
at the 68% confidence level [367
]). The tensor-to-scalar ratio (7.65) can be estimated
as
which satisfies the current observational bound
[367]. We note that a minimally coupled field
with the potential
in Einstein gravity (chaotic inflation model [393]) gives rise to a larger
tensor-to-scalar ratio of the order of
. Since future observations such as the Planck satellite are
expected to reach the level of
, they will be able to discriminate between the chaotic inflation
model and the Starobinsky’s f (R) model.
7.3.3 The power spectra in the Einstein frame
Let us consider the power spectra in the Einstein frame. Under the conformal transformation
,
the perturbed metric (6.1) is transformed as
We decompose the conformal factor into the background and perturbed parts, as
In what follows we omit a bar from
. We recall that the background quantities are transformed as
Eqs. (2.44) and (2.47). The transformation of scalar metric perturbations is given by
Meanwhile vector and tensor perturbations are invariant under the conformal transformation (
,
,
).
Using the above transformation law, one can easily show that the curvature perturbation
in f (R) gravity is invariant under the conformal transformation:
Since the tensor perturbation is also invariant, the tensor-to-scalar ratio
in the Einstein frame is
identical to that in the Jordan frame. For example, let us consider the model
.
Since the action in the Einstein frame is given by Eq. (2.32), the slow-roll parameters
and
vanish
in this frame. Using Eqs. (7.49) and (3.27), the spectral index of curvature perturbations is given by
where we have ignored the term of the order of
. Since
in the slow-roll limit
(
), Eq. (7.78) agrees with the result (7.72) in the Jordan frame. Since
in
the Einstein frame, Eq. (7.59) gives the tensor-to-scalar ratio
where the background equations (3.21) and (3.22) are used with slow-roll approximations. Equation (7.79)
is consistent with the result (7.73) in the Jordan frame.
The equivalence of the curvature perturbation between the Jordan and Einstein frames also holds for
scalar-tensor theory with the Lagrangian
[411, 240]. For
the non-minimally coupled scalar field with
[269, 241] the spectral indices of scalar
and tensor perturbations have been derived by using such equivalence [366, 590].