Noether Symmetry of Palatini $F(\Re)$ gravity

TL;DR

This paper clarifies the form of Noether symmetry in Palatini $F( ext{ extgreek e})$ gravity, showing it admits $F( ext{ extgreek e}) ext{ extasciitilde} ext{ extgreek e}^{3/2}$ with a specific conserved current, correcting previous claims.

Contribution

It demonstrates that in Palatini $F( ext{ extgreek e})$ gravity, the symmetry admits a specific power-law form, refuting earlier claims of arbitrary exponents.

Findings

$F( ext{ extgreek e}) ext{ extasciitilde} ext{ extgreek e}^{3/2}$ is the symmetry form.

The conserved current is explicitly derived as $ ext{ extSigma}_0 a[rac{ ext{ extdot}a}{a} + rac{ ext{ extdot} ext{ extPhi}}{2 ext{ extPhi}}]$.

Previous claims of arbitrary power-law forms are shown to be incorrect.

Abstract

In metric formalism, Noether symmetry of F(R) theory of gravity in vacuum and in the presence of pressureless dust yields $F(R)\propto R^\frac{3}{2}$ along with the conserved current $\frac{d}{dt} (a\sqrt R)$ in Robertson-Walker metric and nothing else. However, Roshan et.al. had claimed in $\mathrm{Phys. Lett. \textbf{B668}, 238 (2008)}$ \cite{o} that Noether symmetry in the context of Palatini $F(\Re)$ theory of gravity admits $F(\Re)\propto \Re^{s}$, (where $s$ is an arbitrary constant) in matter domain era in Friedmann- Robertson-Walker universe. But, it has been shown that the conserved current obtained under the process does not satisfy the field equations in general. Here, it is shown that Noether Symmetry admits $F(\Re)\propto\Re^\frac{3}{2}$ along with a conserved current $\Sigma_{0} {a}[\frac{\dot a}{a}+ \frac{\dot\Phi}{2\Phi}]$ in Palatini gravity. Thus, their claim is not right.