Generalized unimodular gravity as a new form of k -essence

We consider modifications of general relativity characterized by a special noncovariant constraint on metric coefficients, which effectively generates a perfect-fluid type of matter stress tensor in Einstein equations. Such class of modified gravity models includes recently suggested generalized unimodular gravity (GUMG), its simplest version-unimodular gravity-and self-gravitating media theories. We make these gravity models covariant by introducing four Stueckelberg fields and show that in the case of generalized unimodular gravity three out of these fields dynamically decouple. This means that the covariant form of generalized unimodular gravity is dynamically equivalent a special form of k -essence theory with translationally noninvariant kinetic term in the Lagrangian which can be reconstructed from the parameters of GUMG theory. We provide the examples, where such reconstruction can be done explicitly, and briefly discuss theories beyond GUMG, related to self-gravitating media models. Also we compare GUMG k inflation with cuscuton models of dynamically inert k -essence field and discuss motivation for GUMG coming from effective field theory.