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We show a new class of interaction terms with higher derivatives that can beadded to every low derivative real scalar, such that the theory is degenerate,and the equation of motion remains of second order. In contrast to previoussetups, the necessary constraints that eliminate the ghosts also have a clearphysical motivation: they impose and preserve at all times the low derivativedynamics of the real scalar as a background, on top of which, the fluctuationsinduced by these higher derivative terms are degenerate, and ghost-free. Wesummarize the setup for these constrained ghost-free real scalars in a two-stepprescription. In contrast to some models with first-order derivativeinteractions with applications for dark energy and inflation, these theorieswith necessarily constrained second-order derivative self-interactions do notmodify the speed of propagation, neither the cone of influence for the fieldequations, which are prescribed by the low derivative background that is alsoessential to eliminate the ghost; hence, avoiding the potential superluminalityissues of the former.

Ghost condensation and Ostrogradskian instability on low derivative backgrounds