On the Hojman conservation quantities in Cosmology

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. $\varphi \left(t\right)=\varphi \left(a\left(t\right)\right)$, 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.

6 pages; minor corrections; accepted for publication by Physics Letters B. arXiv admin note: substantial text overlap with arXiv:1503.08466

References in corpus (6)

• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
• Improved Cosmological Constraints from New, Old and Combined Supernova Datasets
• Using the Noether symmetry approach to probe the nature of dark energy
• Exact Cosmological Solutions of $f(R)$ Theories via Hojman Symmetry
• Hojman Symmetry Approach for Scalar-Tensor Cosmology
• Hojman Symmetry in $f(T)$ Theory

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• Teleparallel Gravity: From Theory to Cosmology
• Noether Symmetries as a geometric criterion to select theories of gravity
• Exact Solutions in Chiral Cosmology
• Conservation laws and Exact Solutions in Brans-Dicke Cosmology with a Scalar Field
• FRW string cosmological solutions via Hojman symmetry
• Quantum cosmology of Bianchi VIII, IX LRS geometries
• Jacobi multipliers and Hojman symmetry
• New analytic solutions in $f\left(R\right)$-Cosmology from Painlevé analysis
• Exact FLRW cosmological solutions via invariants of the symmetry groups