### 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 .