Constraints and analytical solutions off(R)theories of gravity using Noether symmetries

Abstract

We perform a detailed study of the modified gravity f (R) models in the light of the basic geometrical symmetries, namely Lie and Noether point symmetries, which serve to illustrate the phenomenological viability of the modified gravity paradigm as a serious alternative to the traditional scalar field approaches. In particular, we utilize a model-independent selection rule based on first integrals, due to Noether symmetries of the equations of motion, in order to identify the viability of f (R) models in the context of flat FLRW cosmologies. The Lie/Noether point symmetries are computed for six modified gravity models that include also a cold dark matter component. As it is expected, we confirm that all the proposed modified gravity models admit the trivial first integral namely energy conservation. We find that only the f (R) = (R b − 2Λ) c model, which generalizes the concordance Λ cosmology, accommodates extra Lie/Noether point symmetries. For this f (R) model the existence of non-trivial Noether (first) integrals can be used to determine the integrability of the model. Indeed within this context we solve the problem analytically and thus we provide for the first time the evolution of the main cosmological functions such as the scale factor of the universe and the Hubble expansion rate.PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x

Noether symmetry approach in Gauss–Bonnet cosmology

“…As discussed in [52], their existence allows to select physically observable universes and, specifically, to select analytical models related to observations. In particular, it is possible to classify dark energy behaviors related to Noether symmetries and then discriminate between physical and unphysical solutions (see [53,[67][68][69]). These topics will be the argument of a forthcoming, detailed study.…”

ON ANALYTICAL SOLUTIONS OF f(R) MODIFIED GRAVITY THEORIES IN FLRW COSMOLOGIES

“…The existence of Gödel, Einstein and de Sitter universes in modified gravity theories was studied in [49] whereas analytical solutions for spherically symmetric systems were found in [50,51]. Methods of finding analytical solutions of f (R) theories using Noether symmetries were presented in [52].…”

Dynamical analysis in scalar field cosmology

“…The latter implies that in the configuration space (minisuperspace), the first integrals are considered as "selection rules" for scalar field potentials and coupling functions in the case of modified gravity theories. The above features have inspired many authors to propose the admittance of a Noether (point or dynamical) symmetry by the field equations as a selection rule for dark-energy models, including those of modified gravity [8][9][10][11][12][13][14][15][16][17][18][19].…”