**3. Theory Reconstruction**

We henceforth consider a spatially flat, homogeneous, and isotropic universe, as described by the FLRW line element d*s*<sup>2</sup> = d*t*<sup>2</sup> − *<sup>a</sup>*<sup>2</sup>(*t*)*<sup>δ</sup>ij*d*xi*d*xj*. The proper time gauge *N* = 1 will be used throughout. The spacetime geometry is then fully characterized by the evolution of a single degree of freedom: the scale factor *<sup>a</sup>*(*t*). Given a theory of gravity with second order field equations, cosmological solutions can be represented as trajectories in the plane (*a*, *<sup>χ</sup>*). In general relativity, the trajectories are determined by the Friedmann equation

$$\frac{1}{3}\chi^2 = \sum\_{i} \rho\_i \,. \tag{12}$$

Here the quantities *ρi* denote the energy density of different matter species. For the sake of simplicity, we can assume that all matter species are non-interacting and have constant equation of state parameters *wi*. Thus, we have *ρi* = *ciV*−(*wi*+<sup>1</sup>), where *ci* are constants depending on the initial

<sup>3</sup> See also Refs. [39,40].

conditions and *V* = *a*3 is the proper volume of a unit comoving cell. It is convenient to introduce a new variable *η* = *<sup>V</sup>*−1, so that the Friedmann equation can be re-expressed as

$$\frac{1}{3}\chi^2 = \sum\_{i} c\_i \,\eta^{w\_i+1}\,. \tag{13}$$

Such a parametrization is particularly useful in bouncing cosmologies, where *η* has a bounded range. In the following, we will denote by Γ the trajectory in the (*η*, *χ*) plane given by Equation (13).

Despite the derivation given above, based on the standard Friedmann equation, Equation (13) has a broader applicability. In fact, it also holds in a more general class of modified gravity theories and quantum cosmological models, provided that the corrections to the standard Friedmann equation can be described—at an effective level—as perfect fluids. Such effective fluids may have exotic properties and, depending on the model, can violate the energy conditions. This is the case, for instance, in the effective dynamics of both LQC and GFT condensate cosmology. In fact, having a bounce requires that both the weak and the null energy conditions must be violated due to the effective fluids. The former violation is necessary to accommodate for a vanishing expansion, see Equation (13). The latter violation follows instead from the requirement that *χ*˙ > 0 at the bounce, and from the Raychaudhuri equation including effective fluids contributions

$$
\dot{\chi} = -\frac{3}{2} \sum\_{i} (\rho\_i + P\_i) \,. \tag{14}
$$

It is important to observe that in general Equation (13) allows us to define *χ* as a function of *η* only locally. In fact, in bouncing models, the function *χ*(*η*) has (at least) two branches. More branches are possible if one allows, e.g., for intermediate recollapse eras; we shall disregard this possibility in the following for simplicity. For a universe undergoing a single bounce, the trajectory Γ has the profile depicted in Figure 1. The bounce is represented by the point *B* = (*η*max, <sup>0</sup>), where Γ and the *η* axis intersect orthogonally. Since we are assuming a flat spatial geometry, both endpoints of Γ will have *η* = 0 if the weak energy condition is satisfied for a large universe. The value of *χ* at the endpoints is determined by the equation of state of the dominant matter species in such a regime: for *w* > −1 one has that *χ* vanishes as *η* tends to zero, for *w* = −1 (cosmological constant) *χ* approaches a constant value. We note that for *w* + 1 > 0 the two endpoints coincide with the origin; moreover, for −1 < *w* < 1 the trajectory Γ intersects the *η* axis orthogonally at the origin, whereas for *w* ≥ 1 it has a cusp.

**Figure 1.** Trajectories Γ in the (*η*, *χ*) plane for a (symmetric) bouncing universe. The upper half-plane corresponds to the expanding phase, whereas the lower half-plane describes the contracting phase. The bounce is represented by the point *B*, where the expansion rate vanishes, and the scale factor attains its minimum (correspondingly *η* is maximized). The left figure shows the trajectory Γ for a universe filled with a scalar field (thick green line), or dust (dashed orange line); parameters are chosen so that the two trajectories are characterized by the same critical density *ρc* and limiting expansion rate *χ*m. The right figure shows the two integration contours *γ*1, *γ*2 used in Equation (17).

### *3.1. Reconstruction Procedure*

Given a background evolution as specified by the trajectory Γ, it is possible to apply a reconstruction procedure that allows us to uniquely determine the function *f*(*χ*) in the mimetic gravity action (1). The method illustrated in this section extends to a generic background evolution the procedure applied in Refs. [22,25,26] and ensures that appropriate matching conditions are implemented at the branching points.<sup>4</sup> We start by rewriting Equation (7), using Equation (9), as

$$\frac{\chi^2}{3}\left[1-3\frac{\mathrm{d}}{\mathrm{d}\chi}\left(\frac{f}{\chi}\right)\right]=\rho\,.\tag{15}$$

The solution to this equation can be obtained by quadrature, and is given by

$$f(\chi) = \frac{\chi}{3} \int\_{A}^{P} d\chi \,\left(1 - \frac{3\rho(\eta(\chi))}{\chi^2}\right) + \mathbb{E}\chi\,. \tag{16}$$

where *c*¯ is an integration constant. The integral is computed along an arc of curve *γ* ⊆ Γ with endpoints *A* and *P*, representing a fixed reference point and a generic point on Γ, respectively.

In bouncing cosmologies, the background dynamics is characterized by the existence of a limiting curvature scale, which is attained at the bounce. In turn, this scale is related to the existence of a maximum expansion rate, which will be denoted by *χ*m ≡ max Γ *χ*, see Figure 1. In this class of models, it is convenient to take the bounce as a reference point, i.e., *A* ≡ *B* in Equation (16). Since the energy density of matter is given as a function of the inverse volume, i.e., *ρ* = *ρ*(*η*), the explicit computation of the integral (16) requires the determination of the inverse function *η*(*χ*). In general, such an inverse function exists only locally. This implies that in bouncing models the function *f*(*χ*) given by Equation (16) must be multivalued as a function of *<sup>χ</sup>*.<sup>5</sup> More precisely, in models with a single bounce *f*(*χ*) has two branching points where *χ* attains its extrema, one in the expanding phase, the other in the contracting phase. For a generic bouncing background *f*(*χ*) would have three branches, each corresponding to one of the three branches of the inverse functions *η*(*χ*). Thus, one branch *f*B corresponds to the bounce phase, and two (a priori distinct) branches *f c*L , *f e*L correspond to the regions away from the bounce in the contracting and expanding phase, respectively. We will refer to the latter as the large universe branches, characterized by *χ*˙ < 0. As shown in Section 3.2, for symmetric bounces the two branches *f c*L , *f e*L can be identified, provided that an appropriate choice is made for the integration constant in Equation (16).

We remark that our solution for *f* is continuous on Γ by construction. The derivative *fχ* is also continuous, except at the origin *χ* = *η* = 0.<sup>6</sup> This ensures that the energy density of the effective fluid, Equation (9), is continuous throughout cosmic history. Thus, the matching conditions prescribed in Ref. [22] are automatically implemented in Equation (16). As a general property of this class of models *fχχ* diverges at the branching points, see discussion in Section 4.

After computing the integral in Equation (16), the reconstructed action for mimetic gravity can then be obtained by replacing *χ* → *φ* in the result. Clearly, the value of the integration constant *c*¯ has no influence on the equations of motion, since the linear term contributes a total divergence to the action (1).

<sup>4</sup> We note that a different version of mimetic gravity is considered in Ref. [26] that agrees at the background level with the one presently considered. However, the two theories will differ in general at the level of perturbations.

<sup>5</sup> We observe that for models entailing a single bounce, the solution (16) is single-valued if regarded as a function of the pair (*<sup>χ</sup>*, *<sup>χ</sup>*˙).

<sup>6</sup> The fact that the origin is a singular point in the parametrization adopted here should not be too surprising: in fact, it corresponds to the infinite volume limit of both contracting and expanding branches. In a flat universe these are clearly two disconnected regimes.

### *3.2. Bounce Asymptotics*

For a symmetric bounce model, the function *f*(*χ*) is even, provided that an appropriate choice of the integration constant is made in Equation (16). In fact, defining *P*1 = (*η*, *χ*) and *P*2 = (*η*, <sup>−</sup>*<sup>χ</sup>*), with *η* and *χ* satisfying the background equation, one has

$$\int\_{\frac{B}{\gamma\_1}}^{P\_1} \mathrm{d}\chi \; \left(1 - \frac{3\rho(\eta(\chi))}{\chi^2} \right) = -\int\_{\frac{B}{\gamma\_2}}^{P\_2} \mathrm{d}\chi \; \left(1 - \frac{3\rho(\eta(\chi))}{\chi^2} \right) \; . \tag{17}$$

Thus, the integral is odd. The curves *γ*1 and *γ*2 are depicted in Figure 1. Using Equations (17) and (16), it is then straightforward to show that setting *c*¯ = 0 leads to *f*(*χ*) = *f*(−*<sup>χ</sup>*). In the following, we shall restrict our attention to symmetric bounce models and assume that *f*(*χ*) be even, unless otherwise stated.

The value of the function *f* at the bounce is independent from all other details of cosmic history. It can be computed as a limit of Equation (16). Denoting by *f*B the bounce branch of the multivalued function *f* , we have

$$f\_{\mathbb{B}}(0) = \lim\_{\mathbb{P} \to \mathbb{B}} \frac{\chi}{3} \int\_{B}^{\mathbb{P}} \mathrm{d}\chi \, \left(1 - \frac{3\rho(\eta(\chi))}{\chi^2} \right) = \lim\_{\chi \to 0} \frac{\chi}{3} \int\_{0}^{\chi} \mathrm{d}\chi \, \left(1 - \frac{3\rho(\eta(\chi))}{\chi^2} \right) = \rho\_{\mathbb{C}^\times} \tag{18}$$

where *ρ*c is the critical density, i.e., the maximum of the energy density of matter, which is attained at the bounce. Since *f*B is even by hypothesis, we have for *χ* 0

$$f\_{\mathbb{B}}(\chi) = \rho\_{\mathbb{c}} + \frac{1}{2!} \theta \ \chi^2 + \mathcal{O}(\chi^4) \ . \tag{19}$$

where we introduced the notation *ϑ* = (*f*B)*χχ*0 . Hence, it follows that the energy density of the effective fluid at the bounce is given by *ρ*˜ = <sup>−</sup>*ρ*c. The sign of the second derivative can be determined by the requirement that the effective fluid must also violate the null energy condition (NEC) at the bounce. In fact, using Equation (10) we have

$$
\not{p} + \not{p} = -(f\_{\mathbb{B}})\_{\chi\chi} \not{\chi} < 0 \tag{20}
$$

Since *χ*˙ > 0 at the bounce, we conclude *ϑ* > 0.

NEC violation also allows us to derive an upper bound for the duration of the bounce in limiting curvature mimetic gravity. To prove such a statement, let us assume that at the bounce the most relevant contributions to the energy density are due to the effective fluid and to a perfect fluid with equation of state parameter *w*. The condition *χ*˙ > 0, which must be valid in a neighborhood of the bounce, implies

$$
\rho + p + \overline{\rho} + \overline{P} < 0 \,. \tag{21}
$$

In turn, Equation (21) implies

$$(1+w)\rho\_{\mathbb{C}} - (f\_{\mathbb{B}})\_{\chi\chi}\dot{\chi} < 0 \,. \tag{22}$$

For first-order bounce models<sup>7</sup> during the bounce phase the expansion *χ* is well approximated by a linear function of time. We can estimate the time derivative of *χ* at the bounce as *χ*˙ ∼ *χ*m*T* , where *T* is the bounce duration. Therefore, in this case we obtain from Equation (22)

$$T \lesssim \frac{\vartheta \, \chi\_{\rm m}}{\rho\_{\rm c} (1 + w)} \,. \tag{23}$$

Typically, *ρc* ∼ *χ*2m and *ϑ* ∼ O(1), so that *T χ*<sup>−</sup><sup>1</sup> m . When such an approximation applies, the number of e-folds of expansion during the bounce phase is *N* = log *a*(*T*) *a*B O(1). These considerations also apply to the models studied in Section 4 (see Equation (33) for the corresponding expansion of *f* near the bounce). In fact, the estimate (23) agrees with the upper bound for the number of e-folds obtained in Ref. [42] for the so-called non-interacting model. We mention that the so-called fast-bounce models, considered e.g., in Ref. [43], are first-order bounces whose duration is much shorter than the time-scale linked to the maximum expansion rate, i.e., such that *T χ*<sup>−</sup><sup>1</sup> m ; such a scenario can be realized in mimetic gravity by requiring (*f*B)*χχ f*B *χ*=0 *χ*<sup>−</sup><sup>2</sup> m .

### *3.3. Late Time Asymptotics*

Considerations on the evolution of the universe at late times allow us to put restrictions on the leading order terms of the branch *f*L around *χ* 0. In fact, we observe that the effective fluid is characterized by a time-dependent equation of state parameter *w*˜, given by

$$d\vec{v} = \frac{\mathcal{P}}{\tilde{\rho}} = -\left(1 + \frac{f\_{\chi\chi}}{\tilde{\rho}}\dot{\chi}\right) \, , \tag{24}$$

where we used Equations (9) and (10). It is interesting to examine the case where the universe at late times is dominated by matter with equation of state *w* and the effective fluid is sub-dominant, with *w*˜ approaching a constant value as *χ* → 0. Clearly, consistency of such assumptions requires *w* < *w*˜. The leading order term in the expansion of *f*L(*χ*) around *χ* 0 is then given by

$$f\_{\mathbb{L}}(\chi) \simeq \lambda \propto^{2\left(\frac{\mathbb{N}+1}{w+1}\right)},\tag{25}$$

where *λ* is a constant. In fact, since by hypothesis we must leading order *χ* ∼ *η* 1+*w*2 , Equation (25) implies *ρ*˜ ∼ *η*<sup>1</sup>+*w*˜ , consistently with our assumptions.

### **4. Effective Approach to Quantum Gravitational Bouncing Cosmologies**

In Ref. [22] the reconstruction procedure outlined in Section 3 was successfully applied to the cosmological dynamics obtained from group field theory condensates in [5,6]. The evolution equation for such a model can be expressed in relational form by introducing a minimally coupled massless scalar field *ψ* [44]. In fact, provided that its momentum be non-vanishing *pψ* = 0, *ψ* is a monotonic function of *t* and thus represents a perfect clock. For definiteness, we will assume *pψ* > 0. Using the relational clock *ψ* as time, the FLRW line element can be expressed as

$$\mathrm{d}s^2 = N^2(\psi)\,\mathrm{d}\psi^2 - a^2(\psi)\delta\_{i\bar{j}}\mathrm{d}x^i\mathrm{d}x^j \,,\tag{26}$$

where the lapse function reads as

$$N(\psi) = (\psi)^{-1} = p\_{\psi}^{-1} a^3(\psi) \,. \tag{27}$$

<sup>7</sup> The order of the bounce is defined as the positive integer *n* such that *<sup>a</sup>*(<sup>2</sup>*n*)(*<sup>t</sup>*B) > 0 is the lowest-order non-vanishing derivative of the scale factor at the bounce [41].

*Universe* **2019**, *5*, 107

We can define a relational Hubble rate as H = *a a* , where a prime denotes differentiation with respect to *ψ*. The expansion *χ* is related to H as follows

$$
\chi = 3 \, p\_{\psi} \frac{\mathcal{H}}{a^3} \,. \tag{28}
$$

The relational Friedmann equation governing the dynamics of GFT condensates reads as (recall *V* = *a*3)

$$
\mathcal{H}^2 = \frac{1}{6} + \frac{a}{V} - \frac{\beta}{V^2} \,'\,\tag{29}
$$

where *α* and *β* > 0 are parameters depending on the details of the microscopic model, see Ref. [5,6].<sup>8</sup> An effective Friedmann equation with the same form as Equation (29) was obtained in the GFT models of Refs. [45,46]. The first term in Equation (29) is the contribution of the massless scalar field *ψ*, whereas the remaining two terms represent quantum gravitational corrections; in particular, the *α* term represents a correction to the effective dynamics of LQC. It must be stressed that for simplicity, we are neglecting interactions between GFT quanta, which would contribute additional terms to Equation (29). The cosmological consequences of interactions were considered in Ref. [42].

Changing time parametrization back to proper time and recalling *η* = *<sup>V</sup>*−1, we have

$$\frac{1}{3}\chi^2 = p\_{\Psi}^2 \left(\frac{1}{2}\eta^2 + 3a\,\eta^3 - 3\beta\,\eta^4\right). \tag{30}$$

The first term to the r.h.s. of Equation (30) gives the energy density *ρψ* of the scalar *ψ*; the quantum gravitational corrections (second and third terms) correspond instead to effective fluids with equation of state parameter *w* = 2 , 3. The third term becomes important for large values of *η* (i.e., small values of the scale factor); moreover, since *β* > 0 such a term violates both the weak and the NECs, and is therefore responsible for the bounce. It must be noted that the bounce is symmetric for any choice of parameters in this model. The equation for *χ*˙ is

$$\dot{\chi} = -\frac{3}{2} p\_{\psi}^{2} \left( \eta^{2} + 9a \,\eta^{3} - 12\beta \,\eta^{4} \right) \,. \tag{31}$$

For further details on the effective fluid description of quantum gravity corrections in the effective Friedmann equation arising in the GFT approach, including interactions between quanta, the reader is referred to Refs. [42,47]. For a large universe (i.e., small *η*) the first term in Equation (30) becomes the dominant one: the standard Friedmann evolution is thus recovered, and the quantum gravity corrections are sub-leading.

The background evolution (30) can be exactly reproduced in mimetic gravity if the function *f*(*χ*) is given by [22]

$$f(\chi) = \rho\_{\Psi}(\chi) + \frac{1}{3}\chi^2 + \frac{p\_{\Psi}}{3\sqrt{\beta}}|\chi| \left[ \arctan\left(\frac{1}{\sqrt{\beta}}\frac{\mathbf{d}|\mathcal{H}|}{\mathbf{d}\eta}\right) + \frac{\pi}{2} \right].\tag{32}$$

By construction, the different branches of the multivalued function in Equation (32) satisfy matching conditions at the branching points, to ensure the regularity of cosmological evolution. Around the bounce the following expansion holds

$$f\_{\mathbb{B}}(\chi) = \rho\_{\varepsilon} + \frac{1}{3} \left( \frac{2V\_{\mathbb{B}} + 3a}{V\_{\mathbb{B}} + 3a} \right) \chi^2 + \mathcal{O}(\chi^4) \,. \tag{33}$$

<sup>8</sup> It is worth remarking that *α* and *β* are defined only up to arbitrary constant rescalings of the comoving volume *V*0, which was set equal to one above. We have in general *V* = *V*0 *a*3. Under the transformation *V*0 → *kV*0 with constant *k*, *α* and *β* transform according to *α* → *k α*, *β* → *<sup>k</sup>*<sup>2</sup>*β*. Thus, the scale invariance property of the standard Friedmann equation is preserved by the quantum corrections. In the GFT formalism such rescaling properties correspond to the invariance of the dynamics under constant rescalings of the number of quanta, cf. Ref. [5,6].

where *V*B = −3*<sup>α</sup>* + <sup>9</sup>*α*<sup>2</sup> + 6*β* is the volume at the bounce and *ρc* = *p*2*ψ* 2*V*2B . For the asymptotic expansion of *f*(*χ*) around the branching points at maximum expansion rate |*χ*| = *χ*m, see Ref. [22]. Both *f* and *fχ* are continuously matched at the branching points. However, the second derivative *fχχ* has a discontinuity there: this is a general property of mimetic gravity theories with a limiting curvature scale. Nevertheless, the effective pressure *P* ˜(*χ*) is guaranteed to be finite even when *fχχ* diverges, since Equation (8) implies

$$\mathcal{P}(\pm\chi\_{\mathfrak{m}}) = -(\rho + P) = -(w+1)\rho \,. \tag{34}$$

When the universe is large (i.e., in the regime *χ* , *η* ∼ 0) one has the expansion (disregarding the linear term, which does not affect the equations of motion)

$$f\_{\mathbb{L}}(\chi) = \sqrt{\frac{2}{3}} \frac{a}{p\_{\mathbb{\varphi}}} |\chi|^3 - \frac{4}{p\_{\mathbb{\varphi}}^2} \left(a^2 + \frac{1}{9}\beta\right) \chi^4 + \mathcal{O}(|\chi|^5) \,. \tag{35}$$

which can be rewritten as

$$f\_{\mathbb{L}}(\chi) = \frac{a}{2V\_{\ast}} \sqrt{2 + \frac{6a}{V\_{\ast}}} \frac{|\chi|^3}{\chi\_{\mathbb{m}}} - \frac{(V\_{\ast} + 3a)(V\_{\ast}^2 + 9aV\_{\ast} + 108a^2)}{36V\_{\ast}^3} \frac{\chi^4}{\chi\_{\mathbb{m}}^2} + \mathcal{O}(|\chi|^5) \,,\tag{36}$$

where *χ*m = *pψ* 2*V*∗ 43 + 9*αV*∗ , and *V*∗ = 12 <sup>81</sup>*α*<sup>2</sup> + 48*β* − 9*α* is the volume at *χ* = *χ*m. Please note that if *α* = 0, the next non-vanishing term in the expansion is <sup>O</sup>(*χ*<sup>6</sup>).

Once the function *f*(*χ*) has been reconstructed from a given background evolution, one can also consider different matter species coupled to gravity. It must be pointed out that when matter species other than a minimally coupled massless scalar field are considered, parameters such as *pψ* and *V*B in Equation (32) lose their usual interpretation. This is to be expected, since the relation between *χ* and *η* will be different from Equation (30) in the general case. Nevertheless, the values of the critical energy density *ρc* and the maximum expansion rate *χ*m are not affected by the different matter species, and represent universal features of the model.

Let us now assume hydrodynamic matter with constant equation of state parameter *w*. Comparing Equations (36) and (25), at late times we obtain a simple description of the effective fluid corresponding to the mimetic gravity corrections as a sum of perfect fluid contributions, each with a constant equation of state. Specifically, we find for the third order term in Equation (36) *w*˜ 3 = 12 (<sup>3</sup>*w* + <sup>1</sup>), whereas for the fourth order term we have *w*˜ 4 = 2*w* + 1. Clearly, for a massless scalar field *w* = 1 one recovers the effective fluid corrections given in Equation (30).

### *A Special Case: Reproducing the LQC Effective Dynamics*

The case *α* = 0 is special and deserves being discussed separately. In fact, in this case one recovers the model of Ref. [21], which reproduces the effective dynamics of LQC for a spatially flat, isotropic universe. After locally inverting *χ* = *<sup>χ</sup>*(*η*), one finds the two branches of the function *f*(*χ*)

$$f\_{\rm B} = \frac{2}{3} \chi\_{\rm m}^2 \left\{ 1 + \frac{1}{2} q^2 + \sqrt{1 - q^2} + q \arcsin(q) \right\} \,, \tag{37}$$

$$f\_{\perp} = \frac{2}{3} \chi\_{\text{m}}^2 \left\{ 1 + \frac{1}{2} q^2 - \sqrt{1 - q^2} - |q| \left( \arcsin|q| - \pi \right) \right\},\tag{38}$$

where *χ*m = *<sup>p</sup>ψ*4 348*β* and we defined *q* = *χχ*m to make the notation lighter. It must be noted that Equations (37) and (38) do not make any reference to the scalar field *ψ*, which was assumed as the only matter species coupled to gravity in the derivation of Equation (29) in Ref. [5,6]. Thus, for *α* = 0 the effective Friedmann equation will take the same universal form regardless of the matter species considered. Using Equation (18), the critical energy density is determined as *ρc* = *f*B(0) = 43*χ*2m. The energy density of the effective fluid can be computed using Equation (9); the result is *ρ*˜ = −*ρ<sup>c</sup>*2 1 − *q*22 ± 1 − *q*2, where the upper sign corresponds to the bounce branch and the lower one corresponds to a large universe. After some straightforward algebraic manipulations, the Friedmann Equation (7) can then be recast in the following form

$$
\frac{1}{3}\chi^2 = \rho \left(1 - \frac{\rho}{\rho\_c}\right) \tag{39}
$$

where *ρ* denotes the total energy density of all matter species that are present. Similarly, using Equations (8) and (10) we can obtain the equation for *χ*˙. We have, for a general *f*(*χ*)

$$\left(1 - \frac{3}{2} f\_{\chi\chi} \right) \dot{\chi} = -\frac{3}{2} (\rho + P) \,. \tag{40}$$

The bracket to the r.h.s. of Equation (40) can be evaluated using Equations (37) and (38)

$$1 - \frac{3}{2} f\_{\chi\chi} = \mp \frac{1}{\sqrt{1 - q^2}} = \left( 1 - \frac{2\rho}{\rho\_c} \right)^{-1},\tag{41}$$

where we used Equation (39) in the last equality. Finally, we have

$$
\dot{\chi} = -\frac{3}{2} (\rho + P) \left( 1 - \frac{2\rho}{\rho\_c} \right) \,. \tag{42}
$$

Thus, the time derivative of the expansion is positive for *ρc*2 < *ρ* ≤ *ρc* (super-inflation). This is to be contrasted with general relativity, where one always has *χ*˙ < 0 for matter satisfying the NEC. Equations (39) and (42) coincide with the effective dynamics of (flat, isotropic) LQC, see e.g., Ref. [4].

It is important to observe that one must change branch of *f*(*χ*) when *χ*˙ = 0 [24]. This happens when the density reaches the value *ρc*2 , see Equation (42), whereby the expansion attains its extremum *χ*2 = *χ*2m. It must be noted that in both branches, as given by Equations (37) and (38), *fχχ* diverges as |*χ*| → *χ*m; however, the effective pressure *P*˜ is continuous in the limit since *P*˜ = − *ρρc*(*ρ* + <sup>2</sup>*<sup>P</sup>*).

Exact solutions of the effective Friedmann Equation (39) can be derived for hydrodynamic matter (see Ref. [21])

$$a(t) = a\_{\mathbb{B}} \left( 1 + \frac{3}{4} \rho\_c (w+1)^2 (t - t\_{\mathbb{B}})^2 \right)^{\frac{1}{3(1+w)}},\tag{43}$$

where the origin of time has been set to have the bounce at *t* = 0. Provided that matter satisfies the NEC, one finds for the bounce duration (defined to have *χ*(*T*) = *χ*m)

$$T = \frac{1}{\chi\_{\text{m}}(1+w)} \, ' \tag{44}$$

which is in good agreemen<sup>t</sup> with the estimate given by Equation (23).

Finally, the expansions (33) and (36) for *α* = 0 become, respectively

$$f\_{\mathbb{B}}(\chi) = \rho\_{\varepsilon} + \frac{2}{3}\chi^2 + \mathcal{O}(\chi^4) \,, \tag{45}$$

and

$$f\_{\rm L}(\chi) = -\frac{1}{36} \frac{\chi^4}{\chi\_{\rm m}^2} + \mathcal{O}(\chi^6) \,. \tag{46}$$

### **5. Anisotropies near the Bounce**

In this Section we generalize the analysis of Ref. [21], studying the evolution of anisotropies near the bounce in a non-singular Bianchi I spacetime, for the model of Section 4 and in the presence of hydrodynamic matter with generic equation of state.

The line element of Bianchi I in proper time gauge is

$$\mathrm{d}s^2 = \mathrm{d}t^2 - a^2(t) \sum\_{i} \mathrm{e}^{2\mathcal{G}\_{(i)}(t)} (\mathrm{d}x^i)^2,\tag{47}$$

where *a*(*t*) is the mean scale factor, and the variables *β*(*i*) representing the anisotropies satisfy ∑*i β*(*i*) = 0. We will assume hydrodynamical matter with barotropic equation of state. Using the field Equations (3), it can be shown that the *β*(*i*) evolve according to

$$
\beta\_{(i)} + \chi \beta\_{(i)} = 0 \,. \tag{48}
$$

The solution of Equation (48) gives

$$
\dot{\beta}\_{(i)} = \frac{\lambda\_{(i)}}{a^3(t)},
\tag{49}
$$

with *<sup>λ</sup>*(*i*) integration constants satisfying ∑*i <sup>λ</sup>*(*i*) = 0. The field equations lead to an effective Friedmann equation for the mean scale factor, which includes the contribution of anisotropies

$$
\frac{1}{3}\chi^2 = \rho + \vec{\rho} + \frac{1}{2}\sum\_{i} \dot{\beta}\_{(i)}^2 \,. \tag{50}
$$

The last term of Equation (50) represents the effective energy density of anisotropies (cf. e.g., Ref. [48]), which will be denoted by *ρ*<sup>Σ</sup>. Using Equation (49), we have

$$
\rho\_{\Sigma} = \frac{\Sigma^2}{2a^6} \tag{51}
$$

having defined the shear scalar as Σ<sup>2</sup> = ∑*i <sup>λ</sup>*<sup>2</sup>(*i*). Thus, the contribution of anisotropies to the modified Friedmann equation is described as a perfect fluid with stiff equation of state *w* = 1, as in general relativity.

The evolution of anisotropies, as represented by the *β*(*i*), is obtained by integrating Equation (49)

$$
\beta\_{(i)}(t) = \lambda\_{(i)} \int \frac{\mathbf{d}t}{a^3(t)}\,'\,\tag{52}
$$

where *a*(*t*) in the integrand is a solution of Equation (50). In the remainder of this Section, we will determine the evolution of anisotropies during the bounce phase for the function *f*(*χ*) given by Equation (32). Since we are only interested in the region around the bounce, it is convenient to use the expansion (33). The energy density of the effective fluid then reads as

$$
\bar{\rho} \simeq -\rho\_{\mathfrak{c}} + \frac{1}{3} \left( \frac{2V\_{\mathfrak{B}} + 3\mathfrak{a}}{V\_{\mathfrak{B}} + 3\mathfrak{a}} \right) \chi^2 \,. \tag{53}
$$

The effective Friedmann equations in this regime can then be recast as

$$\frac{\chi^2}{\mathfrak{Z}} \simeq \left(\frac{V\_{\mathbb{B}} + 3\mathfrak{a}}{V\_{\mathbb{B}}}\right) \left(\rho\_{\mathbb{C}} - \rho - \rho\_{\mathbb{E}}\right) \,,\tag{54}$$

$$
\dot{\chi} \simeq \frac{3}{2} \left( \frac{V\_{\text{B}} + 3\alpha}{V\_{\text{B}}} \right) (\rho + p + 2\rho\_{\text{E}}) \;. \tag{55}
$$

At the bounce, the scale factor attains its minimum *a*B, and the r.h.s. of Equation (54) must vanish. We can use this condition to determine the energy density of matter at the bounce *ρB* (not to be confused with the critical energy density *ρ<sup>c</sup>*, which includes the contribution of anisotropies). We have

*ρB* = *ρc* − *ρ*<sup>Σ</sup>,<sup>B</sup> , (56)

with *ρ*<sup>Σ</sup>,<sup>B</sup> = Σ<sup>2</sup> <sup>2</sup>*a*6B being the energy density of anisotropies at the bounce. The r.h.s. of Equation (54) can be expanded around *a*B; taking into account that *ρ* = *ρB aBa* <sup>3</sup>(*w*+<sup>1</sup>), this gives

$$\frac{\chi^2}{3} \simeq 3 \left( \frac{V\_{\rm B} + 3a}{V\_{\rm B}} \right) \left( \rho\_{\rm B}(w+1) + 2\rho\_{\rm \Sigma, \rm B} \right) \left( \frac{a}{a\_{\rm B}} - 1 \right) \,. \tag{57}$$

Taking into account Equation (56), we can rewrite Equation (57) as

$$\frac{\chi^2}{3} \simeq 3(w+1) \left(\frac{V\_{\mathbb{B}} + 3\alpha}{V\_{\mathbb{B}}}\right) \left(\rho\_c - \frac{w-1}{w+1} \rho\_{\mathbb{Z},\mathbb{B}}\right) \left(\frac{a}{a\_{\mathbb{B}}} - 1\right) \,. \tag{58}$$

The solution is

$$a(t) \simeq a\_{\mathbb{B}} \left( 1 + \frac{1}{4} \Omega^2 t^2 \right) \,, \tag{59}$$

where we defined

$$
\Omega^2 = (w+1) \left( \frac{V\_{\mathbb{B}} + 3a}{V\_{\mathbb{B}}} \right) \left( \rho\_{\mathbb{C}} - \frac{w-1}{w+1} \rho\_{\mathbb{E}, \mathbb{B}} \right) \,. \tag{60}
$$

The solution (59) for the scale factor shows that regardless of the presence of anisotropies, the model features a first-order bounce, according to the definition given in Ref. [41]. From Equation (59), we find that the mean expansion rate evolves as

$$
\chi(t) \simeq \frac{3}{2} \Omega^2 t \,. \tag{61}
$$

Finally, using Equations (59) and (52) we find that the *β*(*i*) evolve linearly during the bounce

$$\mathcal{B}\_{(i)}(t) \simeq \mathcal{B}\_{(i)}^0 + \frac{\lambda\_{(i)}}{a\_{\rm B}^3} \mathbf{t} \,, \tag{62}$$

where *β*0(*i*) are integration constants. Our solution (62) shows that anisotropies stay bounded during the bounce, and can be kept under control by means of a suitable choice of parameters for the model. It is interesting to compare this result with a similar one obtained in Ref. [48] for a non-singular bouncing model based on kinetic gravity braiding theories [49].

### **6. Effective Gravitational Constant(s)**

The cosmological background equations of mimetic gravity, Equations (7) and (8), can be recast in an alternative form which makes no reference to perfect fluids. The effects introduced by the function *f*(*χ*) in the action (1) are then included in two effective gravitational "constants" *Geff* F and *Geff* R , representing respectively the effective coupling of matter to gravity in the Friedmann and the Raychaudhuri equations

1 3*χ*2 = 8*π Geff* F (*χ*)*ρ* , (63)

$$\dot{\chi} = -12\pi \,\mathrm{G}\_{\mathrm{R}}^{eff}(\chi)(\rho + P)\,. \tag{64}$$

The effective couplings are functions of the expansion rate, and are defined as

$$8\pi \,\mathrm{G}\_{\mathrm{F}}^{eff}(\chi) = \left(1 - 3\frac{\mathrm{d}}{\mathrm{d}\chi} \left(\frac{f}{\chi}\right)\right)^{-1},\tag{65}$$

$$8\pi \, G\_{\text{R}}^{eff}(\chi) = \left(1 - \frac{3}{2} f\_{\chi \chi} \right)^{-1} . \tag{66}$$

It is worth remarking that variable gravitational constants arise in this framework despite of the fact that the action (1) contains no dilaton couplings. In fact, the reformulation provided here hinges on the presence of a function of the expansion rate *f*(*χ*).

From Equations (63) and (64), and the continuity equation for matter, we find the following equation relating the change of *Geff* F over time to the difference between the two gravitational constants

$$\mathcal{G}\_{\rm F}^{eff} \rho = \chi (\mathcal{G}\_{\rm F}^{eff} - \mathcal{G}\_{\rm R}^{eff}) (\rho + p) \,. \tag{67}$$

We observe that *Geff* F = *Geff* R if and only if *f*(*χ*) = *k*1 *χ* + *k*22 *χ*2. In this case, the linear term in *χ* has no effect, while the quadratic one leads to a finite redefinition of the Newton constant 8*π Geff* F = 1 − 32 *<sup>k</sup>*2−<sup>1</sup> (see Ref. [37]); thus, in a large universe we must require *k*2 < 23 to ensure that the gravitational interaction remains attractive.<sup>9</sup> In the general case, both *Geff* F and *Geff* R will evolve with *χ*. For instance, assuming that in the large universe branch one has *f*(*χ*) *k χ<sup>p</sup>* with *p* > 2 to leading order in *χ*, leads to

$$8\pi \, G\_{\text{F}}^{\prime f \dagger}(\chi) \simeq 1 + 3k(p-1)\chi^{p-2},\tag{68}$$

$$8\pi \,\mathrm{G}\_{\mathrm{R}}^{eff}(\chi) \simeq 1 + \frac{3}{2}k \, p(p-1)\chi^{p-2} \,. \tag{69}$$

If we assume that the universe (away from the bounce) is dominated by hydrodynamic matter with equation of state parameter *w*, we have

$$8\pi \, G\_{\rm F}^{eff}(t) \simeq 1 + 3k(p - 1) \left(\frac{2}{(w + 1)t}\right)^{p - 2},\tag{70}$$

$$8\pi \, G\_{\rm R}^{eff}(t) \simeq 1 + \frac{3}{2}k \, p(p-1) \left(\frac{2}{(w+1)t}\right)^{p-2}.\tag{71}$$

The reformulation of the cosmological equations of mimetic gravity offered by Equation (63) and (64) suggests that the coefficients of the leading order terms in the expansion of the branch *f*L can be constrained using observational bounds on the time variation of the gravitational constant. We have from Equation (70), for a small *k* and retaining only the main contribution (corresponding to the radiation dominated era, *w* = 13)

$$\frac{\Delta G\_{\rm F}^{eff}}{G\_{\rm F}^{eff}} = 1 - \frac{G\_{\rm F}^{eff}(t\_{\rm BBN})}{G\_{\rm F}^{eff}(t\_0)} \simeq 3k(p-1)\left(\frac{3}{2}\right)^{p-2} (t\_{\rm BBN})^{2-p} \,. \tag{72}$$

where *t*0 is the age of the universe and *t*BBN is the time of nucleosynthesis. Bounds on the time variation of the gravitational constant *Geff* F can be derived from primordial nucleosynthesis: −0.10 < Δ*Geff* F *Geff* F < 0.13 [50,51]. For a given *p* > 2, such a bound can be translated into a constraint on *k*. However, such a constraint is very weak for bouncing models. In fact, if the limiting curvature

<sup>9</sup> This must be contrasted with the case of bouncing models examined in Sections 3.2 and 4, where the coefficient of the quadratic term must satisfy an opposite inequality in order to guarantee that gravity becomes repulsive at the bounce.

hypothesis is made, dimensional arguments sugges<sup>t</sup> that *k* ∼ *χ*2−*<sup>p</sup>* m . This is in fact the case for the models considered in Section 4, see Equations (46) and (36). Moreover, typically one has for the limiting value of the expansion rate *χ*m ∼ *<sup>t</sup>*−<sup>1</sup> Pl , where *t*Pl is Planck time. Therefore, the time variation of the gravitational constant is extremely small in such models Δ*Geff* F *Geff* F∼ *t*Pl *t*BBN *<sup>p</sup>*−2.

A more detailed investigation of the phenomenological consequences of the time variation of *Geff* F and *Geff* R is beyond the scope of the present article and will be left for future work.
