Hojman Symmetry Approach for Scalar-Tensor Cosmology

TL;DR

This paper applies the Hojman symmetry theorem to scalar-tensor cosmology, enabling the determination of exact solutions for the cosmic scale factor and scalar field by exploiting conserved quantities and conformal transformations.

Contribution

It introduces a novel application of Hojman symmetry to derive exact solutions in scalar-tensor cosmology, including transformations between Einstein and Jordan frames.

Findings

Exact solutions for scale factor and scalar field obtained.

Hojman conserved quantities identified in cosmological models.

Conformal transformations relate minimally and non-minimally coupled solutions.

Abstract

Scalar-tensor Cosmologies can be dealt under the standard of the Hojman conservation theorem that allows to fix the form of the coupling $F(\phi)$, of the potential $V(\phi)$ and to find out exact solutions for related cosmological models. Specifically, the existence of a symmetry transformation vector for the equations of motion gives rise to a Hojman conserved quantity on the corresponding minisuperpace and exact solutions for the cosmic scale factor $a$ and the scalar field $\phi$ can be achieved. In particular, we take advantage of the fact that minimally coupled solutions, previously obtained in the Einstein frame, can be conformally transformed in non-minimally coupled solutions in the Jordan frame. Some physically relevant examples are worked out.