# Inflationary spectral tilts as a result of the dilatation symmetry breaking

## TL;DR

This paper introduces a new, rigorous approach to derive spectral indices in single inflation models using dilatation symmetry, providing exact formulas and analyzing the influence of slow roll parameters near horizon crossing.

## Contribution

It develops a novel method based on dilatation transformations and Ward-Takahashi identities to derive exact spectral index expressions, surpassing previous approximate techniques.

## Key findings

- Exact expressions for spectral indices at tree level.
- Spectral indices depend mainly on slow roll behavior around horizon crossing.
- The new method is more general and applicable beyond traditional slow roll assumptions.

## Abstract

We derive the spectral indices and their runnings of single inflation models by a new approach. We perform a dilatation transformation to the linear cosmological perturbations and derive a current (non-)conservation law. Using it, we construct a dilatation charge and a Ward-Takahashi identity for the two-point correlators, and derive two \textit{exact} expressions for the tree-level spectral indices. First, we apply the slow roll expansion to one of the exact expressions. We calculate the spectral indices and their runnings up to the second and the third order of slow roll parameters respectively, with use of the "horizon crossing formalism". By construction, our results are more rigorous and generic than the previous works. Then, we analyze another exact expression to understand how the perturbations and the slow roll parameters contribute to the spectral indices. By a numerical calculation, we confirm that only the behaviors of the slow roll parameters during a few e-folds around the horizon crossing affect significantly to the values of spectral indices. The analysis in this article indicates that if one cannot use the slow roll parameters, regardless of their values, as the expansion parameters around the horizon crossing, then one can no longer apply the slow roll expansion to the spectral indices, and it is thus necessary to apply the more generic method introduced here.