Noether analysis of Scalar-Tensor Cosmology

TL;DR

This paper performs a Noether symmetry analysis of scalar-tensor cosmology, identifying conserved quantities and solution structures for models with arbitrary coupling and potential functions, and explores their physical implications.

Contribution

It derives conditions for maximal integrals of motion in scalar-tensor cosmology, linking the mini-superspace metric to specific relations between coupling and potential functions, and analyzes resulting solutions.

Findings

Complete solution space described by autonomous integrals of motion.

Relation between coupling function and potential leads to a flat mini-superspace metric.

Physical implications studied via effective energy-momentum tensor resembling a perfect fluid.

Abstract

A scalar--tensor theory of gravity, containing an arbitrary coupling function $F(\phi)$ and a general potential $V(\phi)$, is considered in the context of a spatially flat FLRW model. The use of reparametrization invariance enables a particular lapse parametrization in which the mini--superspace metric completely specifies the dynamics of the system. A requirement of existence of the maximal possible number of autonomous integrals of motion is imposed. This leads to a flat mini--superspace metric realized by a particular relation between the coupling function and the potential. The space of solutions is completely described in terms of the three autonomous integrals of motion constructed by the Killing fields of the mini--supermetric and an additional rheonomous emanating from the homothetic field. The solutions contain the arbitrary function which remains after the imposition of the relation between $F(\phi)$ and $V(\phi)$. To exemplify the use of the general results, we select some particular cases and study their physical implications through an effective energy--momentum tensor, which tends out to be that of a perfect fluid.