7.2 Tensor perturbations
Tensor perturbations
have two polarization states, which are generally written as
[391]. In terms of polarization tensors
and
they are given by
If the direction of a momentum
is along the
-axis, the non-zero components of polarization tensors
are given by
and
.
For the action (6.2) the Fourier components
(
) obey the following equation [314]
This is similar to Eq. (7.37) of curvature perturbations, apart from the difference of the factor
instead of
. Defining new variables
and
, it follows that
We have introduced the factor
to relate a dimensionless massless field
with a massless scalar
field
having a unit of mass.
If
, we obtain
We follow the similar procedure to the one given in Section 7.1. Taking into account polarization states, the
spectrum of tensor perturbations after the Hubble radius crossing is given by
which should be evaluated at the Hubble radius crossing (
). The spectral index of
is
where
is given in Eq. (7.54). If
, this reduces to
Then the amplitude of tensor perturbations is given by
We define the tensor-to-scalar ratio
For a minimally coupled scalar field
in Einstein gravity, it follows that
,
,
and
.