New inflationary exact solution from Lie symmetries

TL;DR

This paper derives a new exact inflationary solution using Lie symmetry analysis, classifies symmetries for the master equation, and compares the resulting spectral indices with observational data.

Contribution

It introduces a novel inflationary solution obtained through Lie symmetries and explicitly derives the scalar field potential.

Findings

New inflationary solution from Lie symmetries

Classification of symmetries for the master equation

Comparison of spectral indices with observations

Abstract

For the inflaton field we determine a new exact solution by using the Lie symmetry analysis. Specifically, we construct a second-order differential master equation for arbitrary scalar field potential by assuming that the spectral index for the density perturbations $n_{s}$ and the scalar to tensor ratio $r$ are related as $n_{s}-1=h\left( r\right) $. Function $h\left( r\right) $ is classified according to the admitted Lie symmetries for the master equation. The possible admitted Lie symmetries form the $A_{2}$, $A_{3,2}$, $A_{3,3}$ and $sl\left( 3,R\right) $ Lie algebras. The new inflationary solution is recovered by the Lie symmetries of the $A_{3,3}$ algebra. Scalar field potential is derived explicitly, while we compare the resulting spectral indices with the observations.