Extended mimetic gravity: Hamiltonian analysis and gradient instabilities

We propose a novel class of degenerate higher-order scalar-tensor theories as an extension of mimetic gravity. By performing a noninvertible conformal transformation on seed scalar-tensor theories which may be nondegenerate, we can generate a large class of theories with at most three physical degrees of freedom. We identify a general seed theory for which this is possible. Cosmological perturbations in these extended mimetic theories are also studied. It is shown that either of tensor or scalar perturbations is plagued with gradient instabilities, except for a special case where the scalar perturbations are presumably strongly coupled, or otherwise there appear ghost instabilities.