The Self-consistent Matter Coupling of a Class of Minimally Modified Gravity Theories

The self-consistent matter coupling is found in a broad class of minimally modified gravity theories which was discovered recently. All constraints in the theories remain first class and thus a graviton has only 2 local degrees of freedom. The cosmological solution of one of the examples in this class, the so-called square root gravity, exhibits a singularity freeness at high energy limit. At low energy limit, the theory smoothly connects to GR. A general feature of the theories in this class, with the self-consistent matter coupling discovered in our current work, is the non-trivial interaction among different components of matter sector. We have also checked the Hamiltonian structure of a scalar QED coupling to the square root gravity in the same manner. All constraints in the theory are first class too and thus the local U(1) gauge symmetry in scalar QED is preserved.