REMARKS ON AND COSMOLOGICAL EXTENSIONS OF COVARIANT RENORMALIZABLE GRAVITY

Covariant renormalizable gravity is a Hořava-like extension of general relativity, enjoying full diffeomorphism invariance. However, the price to pay in order to maintain both covariance and renormalizability is the presence of an unknown fluid, whose non-standard coupling dynamically breaks Lorentz invariance. In this brief work, we identify and explain the nature of this fluid, which we note describes the conformal mode of gravity, and arises naturally in frameworks such as that of mimetic gravity. Finally, we lay out extensions of the covariant Hořava-like model, which can serve as a guide for future model-building in this area. Covariant renormalizable gravity (CRG hereafter) is an extension of Hořava gravity in which, unlike the latter, diffeomorphism invariance is preserved at the level of the action, only to be broken via a dynamical symmetry breaking. It in turn requires a new degree of freedom which might be thought of as playing a similar role to the Higgs field in the Standard Model. Covariant renormalizable gravity was introduced in 2010 by Nojiri and Odintsov [1]. Renormalizability is attained a la Hořava [2] by means of a modification to the graviton propagator, which in the UV is made to scale with spatial momenta k as . 1 2z k For a renormalizable theory of gravity in 1 3 + dimensions, 3 = z is required 1 . However, whereas in [2] such a modified behavior is obtained by the introduction of terms explicitly breaking diffeomorphism invariance, the model 1 Here z measures the anisotropy between space and time. Specifically, the theory is constructed so that it is compatible with anisotropic scaling with dynamical critical exponent . , : t b t bx x z z → → REMARKS ON AND COSMOLOGICAL EXTENSIONS OF COVARIANT ... 121 proposed in [1] enjoys instead full diffeomorphism invariance. The breaking of Lorentz invariance which leads to the modified behavior of the propagator is established dynamically, via non-standard coupling to a perfect fluid, in a “spontaneous symmetry breaking”-like fashion. The model was reformulated in [3] by introducing an extra scalar field (whose gradient norm is constrained by a Lagrange multiplier in the action), the gradient of which plays the role of fourvelocity of the unknown fluid. The action of the theory for 2 2 + = n z then takes the form: