Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

Revisiting the calculation of inflationary perturbations 243
3.3. THE SPECTRAL INDICES
In general, if the approximation neglecting the right hand sides of
Eqs. (2) is taken into account, expressions for the spectral indices at any
order are easy to be derived noting that in these cases expression (25)
always vanishes:
where, are or and is correspondingly given by Eqs. (26)
and (31). This way it is obtained
An alternative for more general models. In case it comes out
that Eqs. (35), (36) and (37) are also a good approximation for more
general inflationary potentials, then in the fashion it was done in Sec. 2.2,
more general expressions for the spectral indices can be derived.
Under generalized power–law approximation, for the scalar index we
have
and to third order, expression (24) reduces to
Correspondingly, for the tensorial index,
and its obtained,
For the scalar amplitudes of the generalized slow–roll approximation,