**1. Introduction**

The resolution of spacetime singularities is one of the main expected consequences of quantum gravity. In cosmology, the realization of such a possibility would lead to the replacement of the Big Bang singularity by a smooth spacetime region, e.g., a bounce, with profound implications for our understanding of the earliest stages of cosmic expansion and of the initial conditions for our universe. Non-singular bouncing cosmologies have been extensively studied and may represent an alternative to the inflationary scenario [1] with specific observational signatures (see also [2]). Resolution of the initial singularity in cosmology has been achieved in various approaches based on a loop quantization of the gravitational field, such as loop quantum cosmology (LQC) [3,4], group field theory (GFT) condensate cosmology [5,6], and quantum reduced loop gravity [7]; more specifically, both in LQC and in GFT the initial singularity is replaced by a regular bounce, marking the transition from a contracting phase to an expanding one.

One of the main open problems that is common to all background-independent approaches to quantum gravity is the derivation of an effective field theory taking into account effects due to the underlying discreteness of spacetime at the Planck scale. In fact, at present very little is known about quantum gravity beyond perfect homogeneity, although efforts to include inhomogeneities in the description of an emergen<sup>t</sup> universe from full quantum gravity are underway [8–10]. One possible alternative approach then consists of considering modifications of general relativity that can reproduce known features of a given quantum gravity theory. The hope is that by doing so, we can gain insight (at least qualitatively) into the consequences of quantum gravitational effects in different regimes. In this work, we adopt the framework of limiting curvature mimetic gravity and examine in detail the problem of reconstructing the theory from the evolution of the cosmological background, with particular attention to the case of a bouncing background. Such a theory should then be regarded as a toy model for an effective description of quantum gravity [11,12] and can be used to study its phenomenological consequences. Possible applications include e.g., the dynamics of inhomogeneous and anisotropic degrees of freedom in cosmology, and black holes.

The idea of limiting curvature as a possible solution to the singularities of general relativity was first envisaged in Ref. [13–16], and subsequently implemented in modifications of the Einstein-Hilbert action including higher-order curvature invariants in Refs. [17–20]. An alternative proposal for constructing a gravitational theory with a built-in limiting curvature scale was put forward in Ref. [21] as an extension of mimetic gravity. This is achieved by including in the action functional a (multivalued) potential term *f* depending on the d'Alembertian of a scalar field *φ*. Upon closer inspection, such a potential turns out to depend on the expansion scalar *χ* of a privileged irrotational congruence of time-like geodesics, singled out by the so-called mimetic constraint [22]. On a cosmological spacetime, *f*(*χ*) reduces to a function of the Hubble rate [23]. Multivaluedness of the potential is necessary for a consistent realization of bouncing cosmologies in this framework [22,24–26]. Non-singular black hole solutions have been studied in Refs. [27,28].

The particular model proposed in Ref. [21] exactly reproduces the effective dynamics obtained in (flat, isotropic) homogeneous LQC. Thus, all curvature invariants are bounded throughout spacetime by a limiting curvature scale, which is in turn related to the existence of a critical value for the energy density of matter at the bounce. From the point of view of quantum gravity, it is natural to require that the limiting curvature scale be Planckian. In Ref. [12] a broader class of theories was identified in the degenerate higher-order scalar tensor theories (DHOST) family, all reproducing the effective dynamics of LQC; these models can be further extended by the inclusion of a term corresponding to the spatial curvature. The relation between the model of Ref. [21] and effective LQC was further investigated in Refs. [11,29] from a Hamiltonian perspective, showing that the equivalence holds in the spatially flat, homogeneous, and isotropic sector; however, the correspondence is lost in the anisotropic case. Nevertheless, even for anisotropic cosmologies the solutions of the two models are qualitatively similar [11,29]. The mimetic model of Ref. [21] has been recently generalized in Ref. [22], where a limiting curvature mimetic gravity theory was reconstructed so as to exactly reproduce the background evolution obtained from GFT condensates in Ref. [5,6]; the effective dynamics of homogeneous LQC is then recovered as a particular case for some specific choice of the parameters of the model.

This paper has two main goals. The first one is to give a general account of theory reconstruction in mimetic gravity, showing how essential information about background evolution (e.g., the critical energy density, the bounce duration, and the equation of state of effective fluids) is encoded in the function *f*(*χ*), particularly in its asymptotic behavior in regimes of physical interest. The case of a generic bouncing background is examined in detail, although our methods have a much broader applicability. We provide general prescriptions for the matching of the different branches of the multivalued function *f*(*χ*), which are necessary in order to obtain a smooth evolution of the universe, thus generalizing the analysis of matching conditions in Ref. [22]. Our second goal is to study in detail the properties of mimetic gravity theories with the same background evolution as obtained in non-perturbative approaches to quantum gravity. Specifically, we analyze the model of Ref. [22] reproducing the background evolution obtained from GFT condensates, and compare it to the special case corresponding to the LQC effective dynamics. We study the evolution of anisotropies near the bounce in a Bianchi I spacetime, including the effects of hydrodynamic matter with generic equation of state, thus extending the results of Ref. [21]. As in the model of Ref. [21], our more general results also show that the smooth bounce is not spoiled by anisotropies, which stay bounded during the bounce era. Instabilities in the inhomogeneous sector are also discussed. Moreover, given its relevance and simplicity, the particular case corresponding to the effective dynamics of LQC is analyzed separately.

The plan of the paper is as follows. The formulation of mimetic gravity is briefly reviewed in Section 2. In Section 3 we discuss the reconstruction procedure. In Section 4 we focus on the model of Ref. [22]: we discuss the background evolution, exhibit the form of the function *f*(*χ*) and derive its expansion in its two branches, corresponding to the region around the bounce and to a large universe. The model of Ref. [22], which can be obtained as a particular case from our more general model, is discussed separately due to its relevance and simplicity. Section 5 is devoted to the study of anisotropies in a bouncing background. In Section 6 we provide an alternative description of the cosmological dynamics of mimetic gravity in terms of two effective gravitational "constants", both depending on the expansion rate of the universe. In Section 7 we discuss instabilities of scalar perturbations. We conclude with a discussion of our results in Section 8.

We choose units such that 8 *π G* = 1. Landau-Lifshitz conventions for the metric signature (+ − −−) are adopted.

### **2. Mimetic Gravity and Its Cosmology**

The version of mimetic gravity considered in Ref. [21] is based on the action

$$S[\underline{\mathbf{g}}\_{\mu\nu}, \boldsymbol{\Phi}, \boldsymbol{\lambda}, \boldsymbol{\Psi}] = \int \mathbf{d}^4 \mathbf{x} \sqrt{-\underline{\mathbf{g}}} \left( -\frac{1}{2} \mathbf{R} + \lambda (g^{\mu\nu} \partial\_{\mu} \boldsymbol{\Phi} \partial\_{\nu} \boldsymbol{\Phi} - \mathbf{1}) + f(\boldsymbol{\chi}) + L\_{\mathbf{m}}(\boldsymbol{\Psi}, \boldsymbol{\mathcal{g}}\_{\boldsymbol{\mu}\nu}) \right), \tag{1}$$

with *χ* = *φ*. The gravitational sector consists of the metric *gμν* and the scalar field *φ*. The Lagrange multiplier *λ* enforces the mimetic constraint

$$
\mathfrak{g}^{\mu\upsilon}\partial\_{\mu}\phi\partial\_{\upsilon}\phi = 1\,\,.\tag{2}
$$

We have included a matter Lagrangian *L*m, where *ψ* represents a generic matter field, coupled to *gμν* only and not to *φ*. Due to the term *f*(*χ*), the action (1) represents a higher-derivative extension of the original mimetic gravity theory of Ref. [30].<sup>1</sup>

Due to the mimetic constraint, the vector field *uμ* = *gμν∂νφ* has unit norm and generates an irrotational congruence of time-like geodesics (see Ref. [22] for more details). Thus, the theory admits a preferred foliation<sup>2</sup> with time function *t* = *φ* and time-flow vector field *uμ ∂ ∂xμ* = *∂ ∂t* . The quantity *χ*, defined above, can be expressed as *χ* = <sup>∇</sup>*μuμ* and represents the expansion of the geodesic congruence generated by *<sup>u</sup>μ*. In FLRW spacetime, one has *χ* = 3*H*, where *H* denotes the Hubble rate. It is for this reason that the term *f*(*χ*) in the action (1) plays an important role in the cosmological applications of the model, since for a homogenous and isotropic background *f*(*χ*) reduces to a function of the Hubble rate only. This is a crucial property of the model, which allows for a straightforward theory reconstruction procedure, starting from a given cosmological background evolution. This aspect will be analyzed in detail in Section 3.

It is worth stressing that although the action for mimetic gravity includes higher-derivative terms through *f*(*χ*), the equations of motion are second order. In fact, mimetic gravity is a particular case of so-called DHOST, which are characterized by the absence of Ostrogradski ghost [23,35]. Nevertheless, compared to general relativity, the mimetic gravity theory described by (1) has an extra propagating scalar degree of freedom if *fχχ* = 0 [33,36]. Importantly, this is always a source of instabilities in the theory, as discussed in Section 7.

The field equations read as [21]

$$G\_{\mu\nu} = T^{\Psi}\_{\mu\nu} + \mathcal{T}\_{\mu\nu} \tag{3}$$

where the matter stress-energy tensor is defined as usual

<sup>1</sup> The original formulation of mimetic gravity of Ref. [30] relied on a singular disformal transformation [31] (see also Ref. [23]).An equivalent formulation with a Lagrange multiplier implementing the constraint (2) was given in Ref. [32]. The latter represents the starting point for further generalizations of the model considered in Refs. [21,33]. See also the review [34].

<sup>2</sup> Such a gauge choice corresponds to unit lapse and vanishing shift, i.e., *N* = 1 and *N<sup>i</sup>* = 0.

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

$$T^{\psi}\_{\mu\nu} = \frac{2}{\sqrt{-g}} \frac{\delta S\_{\text{m}}}{\delta g^{\mu\nu}} \, \text{ } \tag{4}$$

and the extra term in Equation (3) is an effective stress-energy tensor arising from the *φ*-sector of the action (1)

$$\mathcal{T}\_{\mu\nu} = 2\lambda \partial\_{\mu}\phi \partial\_{\nu}\phi + \mathcal{g}\_{\mu\nu}(\chi f\_{\chi} - f + \mathcal{g}^{\rho\sigma}\partial\_{\rho}f\_{\chi}\partial\_{\sigma}\phi) - (\partial\_{\mu}f\_{\chi}\partial\_{\nu}\phi + \partial\_{\nu}f\_{\chi}\partial\_{\mu}\phi) \,. \tag{5}$$

The Lagrange multiplier *λ* can be eliminated by solving the following equation

$$
\Box f\_{\lambda} - 2\nabla^{\mu}(\lambda \partial\_{\mu}\phi) = 0 \,,\tag{6}
$$

which can be obtained by varying the action with respect to *φ*. Equation (6) can be interpreted as a conservation law for the Noether current associated with the global shift-symmetry of the action (1), see Refs. [22,37]

Considering a flat FLRW model 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 field Equation (3) lead to a modification of the Friedmann and Raychaudhuri equations

1 3*χ*2 = *ρ* + *ρ*˜ + *M* , (7)

$$\dot{\chi} = -\frac{3}{2} \left[ (\rho + P) + (\vec{\rho} + \vec{P}) + M \right] \,. \tag{8}$$

The quantities introduced in Equations (7) and (8) are defined as follows: *ρ* and *P* denote the energy density and pressure of ordinary matter, whereas *ρ*˜ and *P* ˜ represent the corresponding quantities for the effective fluid, given by

$$
\tilde{\rho} = \chi f\_{\chi} - f\_{\text{-}\chi} \tag{9}
$$

$$
\mathcal{P} = -\left(\vec{\rho} + f\_{\chi\chi}\vec{\chi}\right). \tag{10}
$$

The properties of the effective fluid for a quadratic *f*(*χ*) were studied in Ref. [37]. Lastly, we have *M* = *C a*3 , where *C* is an integration constant for Equation (6). The quantity *M* represents the energy density of so-called mimetic dark matter [30]. We note that for vanishing *f* the action (1) describes irrotational dust minimally coupled to gravity, corresponding to a particular case of the Brown-Kuchaˇr action [38].<sup>3</sup> Finally, we observe that the effective fluid satisfies the continuity equation

$$
\dot{\rho} + \chi(\dot{\rho} + \dot{P}) = 0 \,. \tag{11}
$$
