Symmetries of Differential Equations in Cosmology

TL;DR

This paper reviews symmetry methods for solving cosmological differential equations, focusing on Lie and Noether symmetries, their integrals, and applications to dark matter/energy models, emphasizing the role of the mini superspace and kinetic metric.

Contribution

It provides a comprehensive overview of symmetry tools in cosmology, linking Lie and Noether symmetries with geometric properties of the kinetic metric and their application to cosmological models.

Findings

Hojman integrals can be associated with Noether symmetries via the Inverse Noether Theorem.

Noether point symmetries are elements of the homothetic algebra of the kinetic metric.

The kinetic metric in mini superspace is crucial for analyzing cosmological models.

Abstract

The purpose of the current article is to present a brief albeit accurate presentation of the main tools used in the study of symmetries of Lagrange equations for holonomic systems and subsequently to show how these tools are applied in the major models of modern cosmology in order to derive exact solutions and deal with the problem of dark matter/energy. The key role in this approach are the first integrals of the field equations. We start with the Lie point symmetries and the first integrals defined by them, that is the Hojman integrals. Subsequently we discuss the Noether point symmetries and the well known method for deriving the Noether integrals. By means of the Inverse Noether Theorem we show that to every Hojman quadratic first integral one is possible to associate a Noether symmetry whose Noether integral is the original Hojman integral. It is emphasized that the point transformation generating this Noether symmetry need not coincide with the point transformation defining the Lie symmetry which produces the Hojman integral. We discuss the close connection between the Lie point and the Noether point symmetries with the collineations of the metric defined by the kinetic energy of the Lagrangian. In particular the generators of Noether point symmetries are elements of the homothetic algebra of that metric. The key point in the current study of cosmological models is the introduction of the mini superspace, that is the space which is defined by the physical variables of the model, which is not the spacetime where the model evolves. The metric in the mini superspace is found from the kinematic part of the Lagrangian and we call it the kinetic metric. The rest part of the Lagrangian is the effective potential.