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In Cuzinatto et al. [Phys. Rev. D 93, 124034 (2016)], it has beendemonstrated that theories of gravity in which the Lagrangian includes termsdepending on the scalar curvature $R$ and its derivatives up to order $n$, i.e.$f\left(R,\nabla_{\mu}R,\nabla_{\mu_{1}}\nabla_{\mu_{2}}R,\dots,\nabla_{\mu_{1}}\dots\nabla_{\mu_{n}}R\right)$theories of gravity, are equivalent to scalar-multitensorial theories in theJordan frame. In particular, in the metric and Palatini formalisms, thisscalar-multitensorial equivalent scenario shows a structure that resembles thatof the Brans-Dicke theories with a kinetic term for the scalar field with$\omega_{0}=0$ or $\omega_{0}=-3/2$, respectively. In the present work, theaforementioned analysis is extended to the Einstein frame. The conformaltransformation of the metric characterizing the transformation from Jordan's toEinstein's frame is responsible for decoupling the scalar field from the scalarcurvature and also for introducing a usual kinetic term for the scalar field inthe metric formalism. In the Palatini approach, this kinetic term is absent inthe action. Concerning the other tensorial auxiliary fields, they appear in thetheory through a generalized potential. As an example, the analysis of anextension of the Starobinsky model (with an extra term proportional to$\nabla_{\mu}R\nabla^{\mu}R$) is performed and the fluid representation for theenergy-momentum tensor is considered. In the metric formalism, the presence ofthe extra term causes the fluid to be an imperfect fluid with a heat fluxcontribution; on the other hand, in the Palatini formalism the effectiveenergy-momentum tensor for the extended Starobinsky gravity is that of aperfect fluid type. Finally, it is also shown that the extra term in thePalatini formalism represents a dynamical field which is able to generate aninflationary regime without a graceful exit.

physical review. d/physical review. d.

$f\left(R,\nabla_{\mu_{1}}R,\dots,\nabla_{\mu_{1}}\dots\nabla_{\mu_{n}}R\right)$  theories of gravity in Einstein frame: A higher order modified Starobinsky  inflation model in the Palatini approach