Variable $G$ and $\Lambda$ gravity theory and analytical Cosmological Solutions using Noether symmetry approach

TL;DR

This paper explores scalar field cosmology with variable gravitational and cosmological constants within a quantum gravity framework, identifying symmetries to derive simplified equations and obtain analytical solutions.

Contribution

It introduces a novel approach using Noether symmetry to find analytical cosmological solutions with variable G and Lambda in quantum gravity-modified Einstein-Hilbert action.

Findings

Identified Noether symmetries in the model.

Derived a potential function Lambda(G).

Obtained explicit analytical cosmological solutions.

Abstract

The present work deals with scalar field cosmology in the framework of a quantum gravity modified Einstein-Hilbert Lagrangian with variable $G$ and $\Lambda$. Using Renormalization group, variable $G$ behaves as a minimally coupled filed (not the scalar-tensor theory) and variable $\Lambda$ can be interpreted as a potential function. The point Lagrangian for this model in the background of homogeneous and isotropic flat FLRW space-time model experiences point-like Noether symmetry and equivalent potential function $\Lambda(G)$ is determined. Using a point transformation in the $3D$ augmented space is found that one of the variable become cyclic and as a consequence there is considerable simplification to the physical system. Lastly, the constants of motion can be written in compact form and it is possible to have analytic cosmological solutions in the present context.