Ghost condensation and Ostrogradskian instability on low derivative backgrounds

We show a new class of interaction terms with higher derivatives that can be added to every low derivative real scalar, such that the theory is degenerate, and the equation of motion remains of second order. In contrast to previous setups, the necessary constraints that eliminate the ghosts also have a clear physical motivation: they impose and preserve at all times the low derivative dynamics of the real scalar as a background, on top of which, the fluctuations induced by these higher derivative terms are degenerate, and ghost-free. We summarize the setup for these constrained ghost-free real scalars in a two-step prescription. In contrast to some models with first-order derivative interactions with applications for dark energy and inflation, these theories with necessarily constrained second-order derivative self-interactions do not modify the speed of propagation, neither the cone of influence for the field equations, which are prescribed by the low derivative background that is also essential to eliminate the ghost; hence, avoiding the potential superluminality issues of the former.