  We study cosmological perturbations in mimetic gravity in the presence of
classified higher derivative terms which can make the mimetic perturbations
stable. We show that the quadratic higher derivative terms which are
independent of curvature and the cubic higher derivative terms which come from
curvature corrections are sufficient to remove instabilities in mimetic
perturbations. The advantage of working with the classified higher derivative
terms is that we can control both the background and the perturbation equations
allowing us to construct the higher derivative extension of mimetic dark matter
and the mimetic nonsingular bouncing scenarios. The latter can be thought as a
new higher derivative effective action for the loop quantum cosmology scenario
in which the equations of motion coincide with those suggested by loop quantum
cosmology. We investigate a possible connection between the mimetic cosmology
and the Randall-Sundrum cosmology.
