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

Revisiting the calculation of inflationary perturbations 239
2.2. THE SPECTRAL INDICES
To derive the expressions for the spectral indices we introduce here
a variation of the standard procedure that has the advantage of being
more comprehensive while dealing with the order of expressions to be
derived using the slow–roll expansion.
First, let us assume the following ansatz for the power spectrum of
general inflationary models,
where may be written as a Taylor expansion while is
a general function which, in principle, can be non–continuous in
Note that any function can be decomposed into this form. We
must also consider the following function of the slow–roll parameters:
In this expression all of the parameters are to be evaluated at values of
corresponding to The functions and can be expanded in
Taylor series,
We proceed with the derivation of the equations for the scalar spectral
index
From Eq. (15) we obtain
where
and After differentiation,