Canonization of F (R) theory of gravity to explore Noether symmetry is performed treating R−6(ä a +ȧ 2 a 2 + k a 2 ) = 0 as a constraint of the theory in Robertson-Walker space-time, which implies that R is taken as an auxiliary variable. Although it yields correct field equations, Noether symmetry does not allow linear term in the action, and as such does not produce a viable cosmological model. Here, we show that this technique of exploring Noether symmetry does not allow even a non-linear form of F (R) , if the configuration space is enlarged by including a scalar field in addition, or taking anisotropic models into account. Surprisingly enough, it does not reproduce the symmetry that already exists in the literature [15] for scalar tensor theory of gravity in the presence of R 2 term. Thus, R can not be treated as an auxiliary variable and hence Noether symmetry of arbitrary form of F (R) theory of gravity remains obscure. However, there exists in general, a conserved current for F (R) theory of gravity in the presence of a non-minimally coupled scalar-tensor theory [26,27]. Here, we briefly expatiate the non-Noether conserved current and cite an example to reveal its importance in finding cosmological solution for such an action, taking F (R) ∝ R 3 2 .PACS 04.50.+h IntroductionIncreasing interest in F (R) theory of gravity (see [1] for a recent review) initiates to probe deeply into it and to explore its merits-demerits from different angles. Cosmological consequences of F (R) theory of gravity have been explored [2] taking into account different sets of field equations which appear through the metric variation and Palatini variation formalisms. Although, both set of field equations lead to late time cosmic acceleration without the requirement of dark energy in the form of scalar fields, different results have emerged following these two different techniques. For example, unification of early inflation and late time acceleration [3] on one hand, and violent instability [4], cosmologically non-viability [5], big bang nucleosynthesis and fifth-force constraints altogether [6], on the other, are the outcome of metric formalism. On the contrary, no instability appears as such in Palatini formalism [7] and it leads to correct Newtonian limit [8], while curvature corrections are found to induce effective pressure gradients which create problem in the formation of large-scale structure [9]. Further, even density perturbation in the two techniques, produce different results [10]. Nevertheless, it has been shown that under conformal transformation the two formalisms are the same dynamically [11] and also are equivalent for a large class of theories [12]. Despite such contrasting results on one hand and equivalence of the two formalisms on the other, at the end it is required to choose some particular form of F (R) out of indefinitely many. This is only possible by imposing Noether symmetry. In the present work we discuss this issue in the metric variation formalism.Noether symmetry when applied for the first time in ...
Whether Jordan's and Einstein's frame descriptions of F(R) theory of gravity are physically equivalent, is a long standing debate. However, practically none questioned on true mathematical equivalence, since classical field equations may be translated from one frame to the other following a transformation relation. Here we show that, neither Noether symmetries, Noether equations, nor may quantum equations be translated from one to the other. The reason being, -conformal transformation results in a completely different system, with a different Lagrangian. Field equations match only due to the presence of diffeomorphic invariance. Unless a symmetry generator is found which involves Hamiltonian constraint equation, mathematical equivalence between the two frames appears to be vulnerable. In any case, in quantum domain Mathematical and therefore physical equivalence can't be established.
Noether symmetry of F (R) theory of gravity in vacuum or in matter dominated era yields F (R) ∝ R 3 2 . We show that this particular curvature invariant term is very special in the context of isotropic and homogeneous cosmological model as it makes the first fundamental form hij cyclic. As a result, it allows a unique power law solution, typical for this particular fourth order theory of gravity, both in the vacuum and in the matter dominated era. This power law solution has been found to be quite good to explain the early stage but not so special and useful to explain the late stage of cosmological evolution. The usefulness of Palatini variational technique in this regard has also been discussed.
Noether symmetry in teleparallel f (T ) gravity, where T is the torsion scalar, has been studied in the background of Robertson-Walker space-time. It is found that Noether symmetry admits f (T ) ∝ T 3 2 and the associated conserved current is Σ = aȧT 1 2 , in matter dominated era. In the process, the recent claim by Wei et.al.[1] that Noether symmetry admits f (T ) ∝ T n , (where n is an arbitrary constant) is found not to be correct, since the conserved current satisfies the field equations only for a special choice of n = 3 2 . Further, correspondence between f (R) and f (T ) theories of gravity has also been established.
Noether gauge symmetry for F(R) theory of gravity has been explored recently. The fallacy is that, even after setting gauge to vanish, the form of F(R) \propto R^n (where n \neq 1, is arbitrary) obtained in the process, has been claimed to be an outcome of gauge Noether symmetry. On the contrary, earlier works proved that any nonlinear form other than F(R) \propto R^3/2 is obscure. Here, we show that, setting gauge term zero, Noether equations are satisfied only for n = 2, which again does not satisfy the field equations. Thus, as noticed earlier, the only admissible form that Noether symmetry is F(R) \propto R^3/2 . Noether symmetry with non-zero gauge has also been studied explicitly here, to show that it does not produce anything new.Comment: 9 pages, To appear in Astrophysics Space Scienc
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