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

In Cuzinatto et al. [Phys. Rev. D 93, 124034 (2016)], it has been demonstrated that theories of gravity in which the Lagrangian includes terms depending 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 the Jordan frame. In particular, in the metric and Palatini formalisms, this scalar-multitensorial equivalent scenario shows a structure that resembles that of 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, the aforementioned analysis is extended to the Einstein frame. The conformal transformation of the metric characterizing the transformation from Jordan's to Einstein's frame is responsible for decoupling the scalar field from the scalar curvature and also for introducing a usual kinetic term for the scalar field in the metric formalism. In the Palatini approach, this kinetic term is absent in the action. Concerning the other tensorial auxiliary fields, they appear in the theory through a generalized potential. As an example, the analysis of an extension of the Starobinsky model (with an extra term proportional to $\nabla_{\mu}R\nabla^{\mu}R$) is performed and the fluid representation for the energy-momentum tensor is considered. In the metric formalism, the presence of the extra term causes the fluid to be an imperfect fluid with a heat flux contribution; on the other hand, in the Palatini formalism the effective energy-momentum tensor for the extended Starobinsky gravity is that of a perfect fluid type. Finally, it is also shown that the extra term in the Palatini formalism represents a dynamical field which is able to generate an inflationary regime without a graceful exit.