**Marco de Cesare**

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B 5A3, Canada; marco.de\_cesare@unb.ca

Received: 1 April 2019; Accepted: 4 May 2019; Published: 7 May 2019

**Abstract:** We illustrate a general reconstruction procedure for mimetic gravity. Focusing on a bouncing cosmological background, we derive general properties that must be satisfied by the function *f*(*φ*) implementing the limiting curvature hypothesis. We show how relevant physical information can be extracted from power-law expansions of *f* in different regimes, corresponding e.g., to the very early universe or to late times. Our results are then applied to two specific models reproducing the cosmological background dynamics obtained in group field theory and in loop quantum cosmology, and we discuss the possibility of using this framework as providing an effective field theory description of quantum gravity. We study the evolution of anisotropies near the bounce, and discuss instabilities of scalar perturbations. Furthermore, we provide two equivalent formulations of mimetic gravity: one in terms of an effective fluid with exotic properties, the other featuring *two* distinct time-varying gravitational "constants" in the cosmological equations.

**Keywords:** mimetic gravity; limiting curvature; bouncing cosmology; quantum gravity; effective field theory
