**7. Instabilities**

Our presentation of mimetic gravity would not be complete without a discussion of perturbative instabilities. Instabilities of cosmological perturbations for the mimetic gravity theory with action (1) have been studied in Refs. [36,52] for a generic *f*(*χ*); for earlier studies focused on the case of a quadratic *f* see Ref. [53,54].<sup>10</sup> Compared to general relativity, the theory has one extra propagating scalar degree of freedom, whose speed of sound is given by

$$c\_s^2 = \frac{1}{2} \frac{f\_{\chi\chi}}{1 - \frac{3}{2} f\_{\chi\chi}} \,. \tag{73}$$

Depending on the sign of the speed of sound, the theory has a ghost instability (for *c*2*s* > 0) or a gradient instability (for *c*2*s* < 0), see references above. The propagation speed of tensor perturbations is not affected by the term *f*(*χ*) in the action (1).<sup>11</sup>

In the following we will assume that the analytic properties of the function *f*(*χ*) are such as to accommodate for a bouncing background. Some general conclusions can then be drawn on the profile of the speed of sound as a function of the expansion, based on the results derived in Section 3.2. In fact, around the bounce *f*(*χ*) must admit the expansion (19). Moreover, since *χ*˙ > 0 in a neighborhood of the bounce, Equation (40) implies that we must have *ϑ* > 23 , provided that ordinary matter fields satisfy the NEC. Thus, at the bounce we have

$$
\omega\_s^2 = \frac{\theta}{2 - 3\theta} < 0\,,\tag{74}
$$

which corresponds to a gradient instability. The expansion rate attains its extremum at |*χ*| = *χ*m, where two different branches of the multivalued function *f*(*χ*) are joined together; at that point the second derivative *fχχ* is divergent, whereby the speed of sound squared takes the universal value *c*2*s* = −13 . We conclude that a generic feature of bouncing models in mimetic gravity is that the bounce is always accompanied by a gradient instability of scalar perturbations, which extends beyond the onset of the standard decelerated expansion. The possibility that *c*2*s* may turn to positive values at a later stage is not excluded, but depends on the details of the model, and specifically on the functional form of the branch *f*L(*χ*) corresponding to a large universe.

It is interesting to study the behavior of *c*2*s* in the models examined in Section 4, where a bouncing background is explicitly realized. To begin with, let us start from the special case *α* = 0, which reproduces the LQC effective dynamics for the cosmological background. The two branches *f*B, *f*L in this case are given by Equations (37) and (38), respectively. We find, using Equations (73) and (41)

$$c\_s^2 = -\frac{1}{3} \left( 1 \pm \sqrt{1 - q^2} \right) = -\frac{2}{3} \frac{\rho}{\rho\_c} \,. \tag{75}$$

<sup>10</sup> It must be noted that the quadratic case is equivalent with the IR limit of projectable Hoˇrava-Lifshitz gravity [54], see also Ref. [55].

<sup>11</sup> The situation is different in other versions of mimetic gravity, see e.g., [23] for a general analysis based on the DHOST formulation of (extended) mimetic gravity theories.

In the second step of (75), the upper sign corresponds to *f*B, whereas the lower one corresponds to *f*L. We note that the speed of sound squared is always negative, has a minimum at the bounce *c*2 *s* min = −2 3 when *ρ* = *ρ<sup>c</sup>*, and approaches zero from below as *ρ* → 0. Given Equation (75), and recalling that maximal expansion rate in this model is reached at *ρ* = *ρc* 2 , it is straightforward to check the general feature *c*2 *s*(<sup>±</sup> *χ* m) = −1 3 . We observe that *c*2 *s* is negative throughout cosmic history for the model with *α* = 0, and approaches zero from below in the large universe branch as *χ* tends to zero (cf. Ref. [25]). It is interesting to compare these results with those obtained in Ref. [56] for a model based on generalized Galileons [57], where the speed of sound squared becomes negative—although only for a short period—around the bounce; see also Ref. [58,59] for a comparison between such effective models and the dynamics of perturbations in LQC. In the models cited above gradient instabilities arise due to the violation of the NEC at the bounce (see also [60] and references therein). Recently, the possibility of establishing a theoretical no-go theorem regarding the realization of a healthy non-singular bounce (i.e., free of pathologies such as gradient instabilities) has been discussed in the context of generalized Galileons, see Refs. [61–63].

The example examined above is just a particular case of the model reproducing the background dynamics of GFT condensates, studied in Section 4. In the general case, i.e., for *α* = 0, we have at the bounce 

$$\left(c\_s^2\right)\_{\text{min}} = -\frac{2}{3}\left(1 + \frac{a}{V\_{\text{B}}}\right) \,. \tag{76}$$

In the large universe branch instead and for *χ* 0 we have, to leading order in *χ*

$$
\varepsilon\_s^2 \simeq \frac{3\alpha}{V\_\*} \sqrt{2 + \frac{6\alpha}{V\_\*}} \frac{|\chi|}{\chi\_{\text{m}}} \,. \tag{77}
$$

Thus, *c*2 *s* and *α* have the same sign in this regime. Therefore, for *α* < 0 the situation is qualitatively similar to the *α* = 0 case examined above, with a gradient instability extending also to the large universe branch. For *α* > 0 the situation is different: there is a cross-over from *c*2 *s* < 0 near the bounce to *c*2 *s* > 0 when the universe is large. Such a cross-over must necessarily take place after the universe enters the phase of decelerated expansion, since *c*2 *s* = −1 3 when *χ*˙ = 0 (see above). Thus, while the bounce is always accompanied by a gradient instability, the late universe branch would be characterized by a ghost instability for *α* > 0. We remark that the cross-over point where *c*2 *s* = 0 corresponds to a regime of strong coupling [54].
