Scalar-multi-tensorial equivalence for higher order f (R,μR,μ1μ2R,...,μ1...μnR) theories of gravity

The equivalence between theories depending on the derivatives of R, i.e. f(R,R,...,nR), and scalar-multi-tensorial theories is verified. The analysis is done in both metric and Palatini formalisms. It is shown that f(R,R,...,nR) theories are equivalent to scalar-multi-tensorial ones resembling Brans-Dicke theories with kinetic terms ω0=0 and ω0=-32 for metric and Palatini formalisms respectively. This result is analogous to what happens for f(R) theories. It is worth 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,...,nR) theories to be written as scalar-multi-tensorial theories. Finally, some examples are studied and the comparison of f(R,R,...,nR) theories to f(R,□R,...,□nR) theories is performed.