| Summary: | Abstract In the framework of the mimetic approach, we study the $$f(R,R_{\mu \nu }R^{\mu \nu })$$ f ( R , R μ ν R μ ν ) gravity with the Lagrange multiplier constraint and the scalar potential. We introduce field equations for the discussed theory and overview their properties. By using the general reconstruction scheme we obtain the power law cosmology model for the $$f(R,R_{\mu \nu }R^{\mu \nu })=R+d(R_{\mu \nu }R^{\mu \nu })^p$$ f ( R , R μ ν R μ ν ) = R + d ( R μ ν R μ ν ) p case as well as the model that describes symmetric bounce. Moreover, we reconstruct model, unifying both matter dominated and accelerated phases, where ordinary matter is neglected. Using inverted reconstruction scheme we recover specific $$f(R,R_{\mu \nu }R^{\mu \nu })$$ f ( R , R μ ν R μ ν ) function which give rise to the de-Sitter evolution. Finally, by employing the perfect fluid approach, we demonstrate that this model can realize inflation consistent with the bounds coming from the BICEP2/Keck array and the Planck data. We also discuss the holographic dark energy density in terms of the presented $$f(R,R_{\mu \nu }R^{\mu \nu })$$ f ( R , R μ ν R μ ν ) theory. Thus, it is suggested that the introduced extension of the mimetic regime may describe any given cosmological model.
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