On the Hojman conservation quantities in Cosmology

TL;DR

This paper examines the application of Hojman's symmetry approach in cosmology, showing its equivalence to Noether's theorem in certain models and analyzing the conditions under which conservation laws arise in various gravitational theories.

Contribution

It demonstrates the equivalence of Hojman's method and Noether's theorem in regular Hamiltonian systems and clarifies the conditions for conservation laws in scalar field and $f(T)$ gravity models.

Findings

Hojman's method is equivalent to Noether's theorem for certain cosmological models.

Conservation laws in scalar field cosmology under a specific Ansatz are equivalent to free particle laws.

In $f(T)$ gravity, conservation laws relate to the existence of a Jacobi Last multiplier.

Abstract

We discuss the application of the Hojman's Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojman's method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noether's Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like $f(R)$ gravity, the application of Hojman's method provide us with the same results with that of Noether's theorem. Moreover we study the special Ansatz. $\phi\left( t\right) =\phi\left( a\left( t\right) \right) $, which has been introduced for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz, the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of Noetherian conservation laws for the free particle. On the other hand, for $f(T)$ teleparallel gravity, it is not the existence of Hojman's conservation laws which provide us with the special function form of $f(T)$ functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation.