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The class of covariant gravity theories which have nice ultraviolet behaviorand seem to be (super)-renormalizable is proposed. The apparent breaking ofLorentz invariance occurs due to the coupling with the effective fluid which isinduced by Lagrange multiplier constrained scalar field. Spatially-flat FRWcosmology for such covariant field gravity may have accelerating solutions.Renormalizable versions of more complicated modified gravity which depends onRiemann and Ricci tensor squared may be constructed in the same way.

A proposal for covariant renormalizable field theory of gravity