Scalar-multi-tensorial equivalence for higher orderf(R,∇μR,∇μ1

Abstract

The equivalence between theories depending on the derivatives of R, i.e. f (R, ∇R, ..., ∇ n R), and scalar-multi-tensorial theories is verified. The analysis is done in both metric and Palatini formalisms. It is shown that f (R, ∇R, ..., ∇ n R) theories are equivalent to scalar-multi-tensorial ones resembling Brans-Dicke theories with kinetic terms ω0 = 0 and ω0 = − 3 2 for metric and Palatini formalisms respectively. This result is analogous to what happens for f (R) theories. It is worthy emphasizing that the scalar-multi-tensorial theories obtained here differ from Brans-Dicke ones due to the presence of multiple tensorial fields absent in the last. Furthermore, sufficient conditions are established for f (R, ∇R, ..., ∇ n R) theories to be written as scalar-multi-tensorial theories. Finally, some examples are studied and the comparison of f (R, ∇R, ..., ∇ n R) theories to f (R, R, ... n R) theories is performed.

Kinetic Scalar Curvature Extended $f(R)$ Gravity

“…The kinetic Ricci f (R) gravity we considered in the previous sections, belongs to a general class of models of the form f (R, ∇ µ R, , ..., ∇ n R), studied in Ref. [47]. In this section we shall demonstrate the equivalence of the theory at hand with a scalar-tensor theory.…”

“…Essentially, the field equations ( 49), (51) with the definitions (47), can be used as a reconstruction method in which, given the Hubble rate and one of the functions f (R) and X(R), it is possible to determine the other function, and hence determine the kinetic Ricci modified f (R) gravity which realizes the given cosmological evolution. Apparently, the most interesting case is by choosing f (R) = R, so the theory is standard Einstein-Hilbert gravity with scalar curvature kinetic corrections.…”

“…Following the formalism firstly proposed in Ref. [47], we make the following substitutions, we can use the substitution…”

“…Note that the condition (93) ensures that ξ = ξ(φ, φ µ ) and ξ ν = ξ ν (φ, φ µ ). The scalar potential U (φ, φ µ ) [47] is reduced to,…”

“…It should be noted here that we obtained a Brans-Dicke-like theory without a kinetic term for the corresponding Brans-Dicke scalar. By variation of the action (102), one gets the field equations [47],…”