Symmetries of Differential Equations in Cosmology

Abstract

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, it 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, which is the space that 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. We consider coordinate transformations of the original mini superspace metric in order to bring it to a form where we know its collineations, that is, the Killing vectors, the homothetic vector, etc. Then, we write the field equations of the cosmological model and we use the connection of these equations with the collineations of the mini superspace metric to compute the first integrals and subsequently to obtain analytic solutions for various allowable potentials and finally draw conclusions about the problem of dark energy. We consider the ΛCDM cosmological model, the scalar field cosmology, the Brans-Dicke cosmology, the f (R) gravity, the two scalar fields cosmology with interacting scalar fields and the Galilean cosmology. In each case, we present the relevant results in the form of tables for easy reference. Finally, we discuss briefly the higher order symmetries (the contact symmetries) and show how they are applied in the cases of scalar field cosmology and in the f (R) gravity.

Noether symmetries as a geometric criterion to select theories of gravity

“…(1) We write an ansatz for the generator of the form (27) defined on the configuration space. (2) We expand the symmetry condition (29) to obtain a polynomial depending on ξ(t, q), η i (t, q) and products of the generalized velocities, i.e. (q aqb ...).…”

“…Let us now apply the symmetry condition (29) to the point-like Lagrangian (38). If symmetries exist, such a condition will fix the form of vector X as well as the form f (R, G).…”

F(R,G) Cosmology through Noether Symmetry Approach

“…As it is pointed out in [25], we also note that the classical Noether symmetry approach with a boundary term K constrains the F(R, G) gravity as a selection criterion that can distinguish the F(R, G) models to utilize the existence of non-trivial Noether symmetries. In this study, we found the maximum number of symmetries as five at the non-vacuum case, but it is four at the vacuum case [28].…”

“…However, it is usually required a clever choice of cyclic variables because of that the equations for the change of coordinates have not a unique solution which is also not well defined on the whole space, and thus it is not unique to find those of the cyclic variables (see References [27] for details). Furthermore, we refer to the interested readers the recent review on symmetries in differential equations [28].…”

“…The Noether symmetries (52) have also been obtained in [25] with the symmetry vector (41). In order to determine the invariant functions of the Noether symmetry X 2 given in (52), after solving the Lagrange system [28] dt…”