# Frame-dependence of inflationary observables in scalar-tensor gravity

## TL;DR

This paper calculates inflationary spectral indices to third order in slow-roll for scalar-tensor theories, demonstrating frame equivalence under conformal transformations when using invariant quantities, with attention to e-fold definitions.

## Contribution

It provides a third-order slow-roll calculation of spectral indices in scalar-tensor gravity using invariant quantities, clarifying frame equivalence and e-fold considerations.

## Key findings

- Spectral indices computed up to third order in slow-roll.
- Frame equivalence established using invariant quantities.
- Highlights importance of e-fold definition in inflationary models.

## Abstract

By means of the Green's function method, we computed the spectral indices up to third order in the slow-roll approximation for a general scalar-tensor theory in both the Einstein and Jordan frames. Using quantities which are invariant under the conformal rescaling of the metric and transform as scalar functions under the reparametrization of the scalar field, we showed that the frames are equivalent up to this order due to the underlying assumptions. Nevertheless, care must be taken when defining the number of $e$-folds.