Unifying Einstein and Palatini gravities

We consider a novel class of $f(\mathcal{R})$ gravity theories where the connection is related to the conformally scaled metric ${\widehat{g}}_{\mu \nu }=C(\mathcal{R})g_{\mu \nu }$ with a scaling that depends on the scalar curvature $\mathcal{R}$ only. We call them $C$ theories and show that the Einstein and Palatini gravities can be obtained as special limits. In addition, $C$ theories include completely new physically distinct gravity theories even when $f(\mathcal{R})=\mathcal{R}$. With nonlinear $f(\mathcal{R})$, $C$ theories interpolate and extrapolate the Einstein and Palatini cases and may avoid some of their conceptual and observational problems. We further show that $C$ theories have a scalar-tensor formulation, which in some special cases reduces to simple Brans-Dicke–type gravity. If matter fields couple to the connection, the conservation laws in $C$ theories are modified. The stability of perturbations about flat space is determined by a simple condition on the Lagrangian.