Ghost free theory in unitary gauge: a new candidate

We propose an algebraic analysis using a 3+1 decomposition to identify conditions for a clever cancellation of the higher derivatives, which plagued the theory with Ostrogradsky ghosts, by exploiting some existing degeneracy in the Lagrangian. We obtain these conditions as linear equations (in terms of coefficients of the higher derivative terms) and demand that they vanish, such that the existence of nontrivial solutions implies that the theory is degenerate. We find that, for the theory under consideration, no such solutions exist for a general inhomogeneous scalar field, but that the theory is degenerate in the unitary gauge. We, then, find modified FLRW equations and narrow down conditions for which there could exist a de Sitter inflationary epoch. We further find constraints on the coefficients of the remaining higher-derivative interaction terms, based on power-counting renormalizability and tree-level unitarity up to the Planck scale.