First-order formalism treatment of R+$R^2$ gravity

We show that the field equations of R+$R^2$ gravity formulated via the first-order formalism are different from those derived in the second-order formalism. These new field equations lead to a new set of connection coefficients that are conformally metric and in general not fully metric compatible. This conformally metric theory is unique in the sense that the conformal gauge field is readily identifiable in terms of the trace of the energy-momentum tensor and its derivatives. This identification leads to a new set of differential equations that can be interpreted as evolution equations for cosmological quantities. The importance of these equations in the early Universe is discussed.