**8. Discussion**

We conclude by reviewing the main results obtained in this work and indicating directions for future studies.

In Section 3 we illustrated in complete generality the theory reconstruction procedure for the function *f*(*χ*) in mimetic gravity. In the case of bouncing backgrounds, the implementation of the limiting curvature hypothesis requires that *f*(*χ*) be multivalued. This case was carefully examined and we gave general prescriptions to ensure continuity of *f*(*χ*) and its first derivative along the cosmic trajectory; in particular, by imposing suitable matching conditions at the branching points, both the energy density *ρ*˜ and pressure *P*˜ of the effective fluid are continuous throughout cosmic history. We showed that local properties of the function *f*(*χ*) are directly related to physically relevant quantities characterizing the evolution of the cosmic background, such as the critical energy density and the bounce duration, as well as the equation of state of the effective fluid. In particular, the latter was shown to approach a constant value at late times, which is determined by the dominant matter species and the leading order term in the asymptotic expansion of *f*(*χ*) in that regime.

In Section 4 we focused on a specific model obtained in Ref. [22], where the function *f*(*χ*) was suitably reconstructed in order to reproduce the background evolution obtained from GFT condensates in Ref. [5,6]. Quantities of physical interest were derived from local analysis of the two branches *f*B, *f*L, using the results of Section 3. The special case corresponding to the effective dynamics of LQC for a flat, isotropic universe was studied in detail. As an application, we studied the evolution of anisotropies near the bounce in a Bianchi I universe for the model of Ref. [22]: our results generalize those obtained in Ref. [21] and show that anisotropies do not grow significantly during the bounce, and therefore do not spoil the smoothness of the bounce. It would be interesting to compare the results obtained in the effective approach considered here, with those of Ref. [64], where the dynamics of GFT condensates of anisotropic quanta was studied (see also Ref. [65]). As discussed in Ref. [11,29], the evolution of anisotropies is qualitatively similar in LQC and the corresponding mimetic gravity theory. It is, therefore, natural to ask whether an analogous statement can be made for GFT cosmology and the related model in mimetic gravity. We leave this question for future work. Spherically symmetric geometries are also of interest and can be studied in the present framework by extending the analysis of Refs. [27,28].

In Section 6 we showed that there is an interesting reformulation of mimetic gravity involving two distinct time-varying effective gravitational constants *Geff* F and *Geff* R , featuring respectively in the Friedmann and the Raychaudhuri equations. Consistency of such a description with the Bianchi identities is ensured by Equation (67), which is identically satisfied in mimetic gravity by all choices of the function *f*(*χ*). We derived the time evolution of the effective gravitational constants during the phase of decelerated expansion for *f*(*χ*) ∼ *χp*, with *p* > 2. We showed that the predicted time variation is too small to be observed if the limiting curvature hypothesis is realized. It would be of interest to further explore the consequences of the time variation of *Geff* F and *Geff* R in a more general and model independent setting.

Our discussion of perturbative instabilities in Section 7 highlights some serious limitations of bouncing models in mimetic gravity, which may hinder the possibility of using the simplest framework with the covariant action (1) for an effective description of quantum gravity in inhomogeneous spacetimes. The presence of gradient or ghost instabilities, which is a distinctive feature of mimetic gravity, seems to be even more serious in bouncing cosmologies; in fact, in such models the infinite age of the universe would offer no chance to keep instabilities under control. Remarkably, this issue has not been much appreciated in the literature on bouncing cosmologies in mimetic gravity. Based on the analogy with LQC (see Ref. [58]), we expect the bounce to be accompanied by a short-lived gradient instability around the bounce affecting short-wavelength modes; however, there should be no instabilities away from the bounce. Some proposals to cure the instabilities by means of further modification of the mimetic gravity action have been made in Refs. [66,67]; however, their correspondence with the effective dynamics of quantum gravity models is ye<sup>t</sup> to be established and shall be investigated in future work.

**Funding:** This work was partially supported by the Atlantic Association for Research in the Mathematical Sciences (AARMS) and by the Natural Sciences and Engineering Research Council of Canada (NSERC).

**Acknowledgments:** It is a pleasure to thank Sabir Ramazanov and Edward Wilson-Ewing for helpful discussions on instabilities in mimetic gravity.

**Conflicts of Interest:** The authors declare no conflict of interest.
