Scalar-multi-tensorial equivalence for higher order $f\left( R,\nabla_{\mu} R,\nabla_{\mu_{1}}\nabla_{\mu_{2}}R,...,\nabla_{\mu_{1}}...\nabla_{\mu_{n} }R\right)$ theories of gravity

TL;DR

This paper demonstrates the equivalence of higher-order $f(R, abla R, ..., abla^n R)$ gravity theories to scalar-multi-tensorial theories in both metric and Palatini formalisms, extending known results for $f(R)$ theories.

Contribution

It establishes the scalar-multi-tensorial representation of complex higher-derivative gravity theories and provides conditions for their equivalence, including comparisons with $f(R, ox R, ..., ox^n R)$ models.

Findings

Higher-order $f(R, abla R, ..., abla^n R)$ theories are equivalent to scalar-multi-tensorial theories.

Theories resemble Brans-Dicke models with specific kinetic parameters in different formalisms.

Conditions are identified under which these theories can be rewritten as scalar-multi-tensorial models.

Abstract

The equivalence between theories depending on the derivatives of $R$, i.e. $f\left( R,\nabla R,...,\nabla^{n}R\right) $, and scalar-multi-tensorial theories is verified. The analysis is done in both metric and Palatini formalisms. It is shown that $f\left( R,\nabla R,...,\nabla^{n}R\right) $ theories are equivalent to scalar-multi-tensorial ones resembling Brans-Dicke theories with kinetic terms $\omega_{0}=0$ and $\omega_{0}= - \frac{3}{2}$ 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\left( R,\nabla R,...,\nabla^{n}R\right) $ theories to be written as scalar-multi-tensorial theories. Finally, some examples are studied and the comparison of $f\left( R,\nabla R,...,\nabla^{n}R\right) $ theories to $f\left( R,\Box R,...\Box^{n}R\right) $ theories is performed.