Minimally modified gravity with an auxiliary constraint: A Hamiltonian construction

Working directly with a general Hamiltonian for the spacetime metric $3+1$ decomposition and keeping only spatial covariance, we investigate possibility of reducing number degrees freedom by introducing an auxiliary constraint. The constraint is considered as part definition theory. Through analysis, find conditions well constraint, under which theory propagates two tensorial only. class theories satisfying these can be viewed new construction type-II minimally modified gravity theories, propagate same but are not equivalent to relativity in vacuum. We also illustrate our formalism concrete example, derive dispersion relation gravitational waves, constrained observations.