Hamiltonian analysis of mimetic scalar gravity revisited

We perform the Hamiltonian analysis of several mimetic gravity models and compare our results with those obtained previously by different authors. We verify that, for healthy mimetic scalar-tensor theories, the condition for the corresponding part of the Hamiltonian to be bounded from below is the positive value of the mimetic field energy density λ. We show that, for mimetic dark matter possessing a shift symmetry, the mimetic energy density remains positive in time, provided appropriate boundary conditions are imposed on its initial value, while in models without shift symmetry, the positive energy density can be maintained by simply replacing λ→eλ. The same result also applies to mimetic f(R) gravity, which is healthy if the usual stability conditions of the standard f(R) gravity are assumed and λ>0. In contrast, if we add mimetic matter to an unhealthy seed action, the resulting mimetic gravity theory remains, in general, unstable. As an example, we consider a scalar-tensor theory with the higher-derivative term (□φ)2, which contains an Ostrogradski ghost. We also revisit results regarding stability issues of linear perturbations around the FLRW background of the mimetic dark matter in the presence of ordinary scalar matter. We find that the presence of conventional matter does not revive dynamical ghost modes (at least in the UV limit). The modes, whose Hamiltonian is not positive definite, are nonpropagating (have zero sound speed) and are associated with the mimetic matter itself. They are already present in the case in which the ordinary scalar fluid is absent, causing a growth of dust overdensity.