Cyclic cosmology from Cuscuton-Gallileons subjected to Lie point transformations

TL;DR

This paper investigates a modified Cuscuton-Gallileon model, revealing it has only two degrees of freedom, and explores its symmetries and cosmological implications, including damped oscillatory behavior in the equation of state.

Contribution

It extends the Cuscuton model to Galileons, analyzes its symmetries, degrees of freedom, and cosmological dynamics, providing new insights into cyclic cosmology.

Findings

The model has only two degrees of freedom despite higher derivatives.

Lie point symmetries restrict the potential to an exponential form.

Damped oscillatory behavior in the equation of state parameters.

Abstract

Spacetime transformations in any physically viable theory should follow Lie Point symmetry. In this work, we explore the Cuscuton model extended to Galileons, as introduced by de Rham et al in \cite{Rham2017}. We find the true degrees of freedom by converting the model into an equivalent first order model. Despite being a higher derivative model, it possesses only \textit{two} degrees of freedom. We calculate the Noether symmetry parameters corresponding to Lie point transformations, which lead to the vanishing of the original Cuscuton term's coefficient and restrict the potential to an exponential form. Interestingly, the coefficient corresponding to the original Cuscuton term vanish. Additionally, we also use the Killing analysis to find out the charges corresponding to the Killing vectors and the Killing tensors. The cosmological implications are examined through dynamical analysis, revealing that under the condition where the coefficient $a_2$ vanishes, the equation of state parameters exhibit damped oscillatory behavior .