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 emergent universe from full quantum gravity are underway [
8,
9,
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,
14,
15,
16], and subsequently implemented in modifications of the Einstein-Hilbert action including higher-order curvature invariants in Refs. [
17,
18,
19,
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
. Upon closer inspection, such a potential turns out to depend on the expansion scalar
of a privileged irrotational congruence of time-like geodesics, singled out by the so-called mimetic constraint [
22]. On a cosmological spacetime,
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,
25,
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
, 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
, 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
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 . 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
with
. The gravitational sector consists of the metric
and the scalar field
. The Lagrange multiplier
enforces the mimetic constraint
We have included a matter Lagrangian
, where
represents a generic matter field, coupled to
only and not to
. Due to the term
, the action (1) represents a higher-derivative extension of the original mimetic gravity theory of Ref. [
30].
1Due to the mimetic constraint, the vector field
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
2 with time function
and time-flow vector field
. The quantity
, defined above, can be expressed as
and represents the expansion of the geodesic congruence generated by
. In FLRW spacetime, one has
, where
H denotes the Hubble rate. It is for this reason that the term
in the action (1) plays an important role in the cosmological applications of the model, since for a homogenous and isotropic background
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
, 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
[
33,
36]. Importantly, this is always a source of instabilities in the theory, as discussed in
Section 7.
The field equations read as [
21]
where the matter stress-energy tensor is defined as usual
and the extra term in Equation (
3) is an effective stress-energy tensor arising from the
-sector of the action (1)
The Lagrange multiplier
can be eliminated by solving the following equation
which can be obtained by varying the action with respect to
. 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
, the field Equation (
3) lead to a modification of the Friedmann and Raychaudhuri equations
The quantities introduced in Equations (
7) and (
8) are defined as follows:
and
P denote the energy density and pressure of ordinary matter, whereas
and
represent the corresponding quantities for the effective fluid, given by
The properties of the effective fluid for a quadratic
were studied in Ref. [
37]. Lastly, we have
, 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-Kuchař action [
38].
3 Finally, we observe that the effective fluid satisfies the continuity equation
3. Theory Reconstruction
We henceforth consider a spatially flat, homogeneous, and isotropic universe, as described by the FLRW line element
. The proper time gauge
will be used throughout. The spacetime geometry is then fully characterized by the evolution of a single degree of freedom: the scale factor
. Given a theory of gravity with second order field equations, cosmological solutions can be represented as trajectories in the plane
. In general relativity, the trajectories are determined by the Friedmann equation
Here the quantities
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
. Thus, we have
, where
are constants depending on the initial conditions and
is the proper volume of a unit comoving cell. It is convenient to introduce a new variable
, so that the Friedmann equation can be re-expressed as
Such a parametrization is particularly useful in bouncing cosmologies, where
has a bounded range. In the following, we will denote by
the trajectory in the
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
at the bounce, and from the Raychaudhuri equation including effective fluids contributions
It is important to observe that in general Equation (
13) allows us to define
as a function of
only locally. In fact, in bouncing models, the function
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
has the profile depicted in
Figure 1. The bounce is represented by the point
, where
and the
axis intersect orthogonally. Since we are assuming a flat spatial geometry, both endpoints of
will have
if the weak energy condition is satisfied for a large universe. The value of
at the endpoints is determined by the equation of state of the dominant matter species in such a regime: for
one has that
vanishes as
tends to zero, for
(cosmological constant)
approaches a constant value. We note that for
the two endpoints coincide with the origin; moreover, for
the trajectory
intersects the
axis orthogonally at the origin, whereas for
it has a cusp.
3.1. Reconstruction Procedure
Given a background evolution as specified by the trajectory
, it is possible to apply a reconstruction procedure that allows us to uniquely determine the function
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.
4 We start by rewriting Equation (
7), using Equation (
9), as
The solution to this equation can be obtained by quadrature, and is given by
where
is an integration constant. The integral is computed along an arc of curve
with endpoints
A and
P, representing a fixed reference point and a generic point on
, 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
, see
Figure 1. In this class of models, it is convenient to take the bounce as a reference point, i.e.,
in Equation (
16). Since the energy density of matter is given as a function of the inverse volume, i.e.,
, the explicit computation of the integral (
16) requires the determination of the inverse function
. In general, such an inverse function exists only locally. This implies that in bouncing models the function
given by Equation (
16) must be multivalued as a function of
.
5 More precisely, in models with a single bounce
has two branching points where
attains its extrema, one in the expanding phase, the other in the contracting phase. For a generic bouncing background
would have three branches, each corresponding to one of the three branches of the inverse functions
. Thus, one branch
corresponds to the bounce phase, and two (a priori distinct) branches
,
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
. As shown in
Section 3.2, for symmetric bounces the two branches
,
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
by construction. The derivative
is also continuous, except at the origin
.
6 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
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
in the result. Clearly, the value of the integration constant
has no influence on the equations of motion, since the linear term contributes a total divergence to the action (1).
3.2. Bounce Asymptotics
For a symmetric bounce model, the function
is even, provided that an appropriate choice of the integration constant is made in Equation (
16). In fact, defining
and
, with
and
satisfying the background equation, one has
Thus, the integral is odd. The curves
and
are depicted in
Figure 1. Using Equations (
17) and (
16), it is then straightforward to show that setting
leads to
. In the following, we shall restrict our attention to symmetric bounce models and assume that
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
the bounce branch of the multivalued function
f, we have
where
is the critical density, i.e., the maximum of the energy density of matter, which is attained at the bounce. Since
is even by hypothesis, we have for
where we introduced the notation
. Hence, it follows that the energy density of the effective fluid at the bounce is given by
. 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
Since
at the bounce, we conclude
.
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
, which must be valid in a neighborhood of the bounce, implies
In turn, Equation (
21) implies
For first-order bounce models
7 during the bounce phase the expansion
is well approximated by a linear function of time. We can estimate the time derivative of
at the bounce as
, where
T is the bounce duration. Therefore, in this case we obtain from Equation (
22)
Typically,
and
, so that
. When such an approximation applies, the number of e-folds of expansion during the bounce phase is
. 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
; such a scenario can be realized in mimetic gravity by requiring
.
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
around
. In fact, we observe that the effective fluid is characterized by a time-dependent equation of state parameter
, given by
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
approaching a constant value as
. Clearly, consistency of such assumptions requires
. The leading order term in the expansion of
around
is then given by
where
is a constant. In fact, since by hypothesis we must leading order
, Equation (
25) implies
, 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
[
44]. In fact, provided that its momentum be non-vanishing
,
is a monotonic function of
t and thus represents a perfect clock. For definiteness, we will assume
. Using the relational clock
as time, the FLRW line element can be expressed as
where the lapse function reads as
We can define a relational Hubble rate as
, where a prime denotes differentiation with respect to
. The expansion
is related to
as follows
The relational Friedmann equation governing the dynamics of GFT condensates reads as (recall
)
where
and
are parameters depending on the details of the microscopic model, see Ref. [
5,
6].
8 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
, whereas the remaining two terms represent quantum gravitational corrections; in particular, the
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
, we have
The first term to the r.h.s. of Equation (
30) gives the energy density
of the scalar
; the quantum gravitational corrections (second and third terms) correspond instead to effective fluids with equation of state parameter
. The third term becomes important for large values of
(i.e., small values of the scale factor); moreover, since
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
is
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
) 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
is given by [
22]
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
where
is the volume at the bounce and
. For the asymptotic expansion of
around the branching points at maximum expansion rate
, see Ref. [
22]. Both
f and
are continuously matched at the branching points. However, the second derivative
has a discontinuity there: this is a general property of mimetic gravity theories with a limiting curvature scale. Nevertheless, the effective pressure
is guaranteed to be finite even when
diverges, since Equation (
8) implies
When the universe is large (i.e., in the regime
) one has the expansion (disregarding the linear term, which does not affect the equations of motion)
which can be rewritten as
where
, and
is the volume at
. Please note that if
, the next non-vanishing term in the expansion is
.
Once the function
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
and
in Equation (
32) lose their usual interpretation. This is to be expected, since the relation between
and
will be different from Equation (
30) in the general case. Nevertheless, the values of the critical energy density
and the maximum expansion rate
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)
, whereas for the fourth order term we have
. Clearly, for a massless scalar field
one recovers the effective fluid corrections given in Equation (
30).
A Special Case: Reproducing the LQC Effective Dynamics
The case
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
, one finds the two branches of the function
where
and we defined
to make the notation lighter. It must be noted that Equations (
37) and (
38) do not make any reference to the scalar field
, which was assumed as the only matter species coupled to gravity in the derivation of Equation (
29) in Ref. [
5,
6]. Thus, for
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
. The energy density of the effective fluid can be computed using Equation (
9); the result is
, 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
where
denotes the total energy density of all matter species that are present. Similarly, using Equations (
8) and (
10) we can obtain the equation for
. We have, for a general
The bracket to the r.h.s. of Equation (
40) can be evaluated using Equations (
37) and (
38)
where we used Equation (
39) in the last equality. Finally, we have
Thus, the time derivative of the expansion is positive for
(super-inflation). This is to be contrasted with general relativity, where one always has
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
when
[
24]. This happens when the density reaches the value
, see Equation (
42), whereby the expansion attains its extremum
. It must be noted that in both branches, as given by Equations (
37) and (
38),
diverges as
; however, the effective pressure
is continuous in the limit since
.
Exact solutions of the effective Friedmann Equation (
39) can be derived for hydrodynamic matter (see Ref. [
21])
where the origin of time has been set to have the bounce at
. Provided that matter satisfies the NEC, one finds for the bounce duration (defined to have
)
which is in good agreement with the estimate given by Equation (
23).
Finally, the expansions (
33) and (
36) for
become, respectively
and
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
where
is the mean scale factor, and the variables
representing the anisotropies satisfy
. We will assume hydrodynamical matter with barotropic equation of state. Using the field Equations (
3), it can be shown that the
evolve according to
The solution of Equation (
48) gives
with
integration constants satisfying
. The field equations lead to an effective Friedmann equation for the mean scale factor, which includes the contribution of anisotropies
The last term of Equation (
50) represents the effective energy density of anisotropies (cf. e.g., Ref. [
48]), which will be denoted by
. Using Equation (
49), we have
having defined the shear scalar as
. Thus, the contribution of anisotropies to the modified Friedmann equation is described as a perfect fluid with stiff equation of state
, as in general relativity.
The evolution of anisotropies, as represented by the
, is obtained by integrating Equation (
49)
where
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
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
The effective Friedmann equations in this regime can then be recast as
At the bounce, the scale factor attains its minimum
, 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
(not to be confused with the critical energy density
, which includes the contribution of anisotropies). We have
with
being the energy density of anisotropies at the bounce. The r.h.s. of Equation (
54) can be expanded around
; taking into account that
, this gives
Taking into account Equation (
56), we can rewrite Equation (
57) as
The solution is
where we defined
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
Finally, using Equations (
59) and (
52) we find that the
evolve linearly during the bounce
where
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
in the action (1) are then included in two effective gravitational “constants”
and
, representing respectively the effective coupling of matter to gravity in the Friedmann and the Raychaudhuri equations
The effective couplings are functions of the expansion rate, and are defined as
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 .
From Equations (
63) and (
64), and the continuity equation for matter, we find the following equation relating the change of
over time to the difference between the two gravitational constants
We observe that
if and only if
. In this case, the linear term in
has no effect, while the quadratic one leads to a finite redefinition of the Newton constant
(see Ref. [
37]); thus, in a large universe we must require
to ensure that the gravitational interaction remains attractive.
9 In the general case, both
and
will evolve with
. For instance, assuming that in the large universe branch one has
with
to leading order in
, leads to
If we assume that the universe (away from the bounce) is dominated by hydrodynamic matter with equation of state parameter
w, we have
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
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,
)
where
is the age of the universe and
is the time of nucleosynthesis. Bounds on the time variation of the gravitational constant
can be derived from primordial nucleosynthesis:
[
50,
51]. For a given
, 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 hypothesis is made, dimensional arguments suggest that
. 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
, where
is Planck time. Therefore, the time variation of the gravitational constant is extremely small in such models
.
A more detailed investigation of the phenomenological consequences of the time variation of and is beyond the scope of the present article and will be left for future work.
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
; for earlier studies focused on the case of a quadratic
f see Ref. [
53,
54].
10 Compared to general relativity, the theory has one extra propagating scalar degree of freedom, whose speed of sound is given by
Depending on the sign of the speed of sound, the theory has a ghost instability (for
) or a gradient instability (for
), see references above. The propagation speed of tensor perturbations is not affected by the term
in the action (1).
11In the following we will assume that the analytic properties of the function
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
must admit the expansion (
19). Moreover, since
in a neighborhood of the bounce, Equation (
40) implies that we must have
, provided that ordinary matter fields satisfy the NEC. Thus, at the bounce we have
which corresponds to a gradient instability. The expansion rate attains its extremum at
, where two different branches of the multivalued function
are joined together; at that point the second derivative
is divergent, whereby the speed of sound squared takes the universal value
. 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
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
corresponding to a large universe.
It is interesting to study the behavior of
in the models examined in
Section 4, where a bouncing background is explicitly realized. To begin with, let us start from the special case
, which reproduces the LQC effective dynamics for the cosmological background. The two branches
,
in this case are given by Equations (
37) and (
38), respectively. We find, using Equations (
73) and (
41)
In the second step of (
75), the upper sign corresponds to
, whereas the lower one corresponds to
. We note that the speed of sound squared is always negative, has a minimum at the bounce
when
, and approaches zero from below as
. Given Equation (
75), and recalling that maximal expansion rate in this model is reached at
, it is straightforward to check the general feature
. We observe that
is negative throughout cosmic history for the model with
, and approaches zero from below in the large universe branch as
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,
62,
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
, we have at the bounce
In the large universe branch instead and for
we have, to leading order in
Thus,
and
have the same sign in this regime. Therefore, for
the situation is qualitatively similar to the
case examined above, with a gradient instability extending also to the large universe branch. For
the situation is different: there is a cross-over from
near the bounce to
when the universe is large. Such a cross-over must necessarily take place after the universe enters the phase of decelerated expansion, since
when
(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
. We remark that the cross-over point where
corresponds to a regime of strong coupling [
54].
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
in mimetic gravity. In the case of bouncing backgrounds, the implementation of the limiting curvature hypothesis requires that
be multivalued. This case was carefully examined and we gave general prescriptions to ensure continuity of
and its first derivative along the cosmic trajectory; in particular, by imposing suitable matching conditions at the branching points, both the energy density
and pressure
of the effective fluid are continuous throughout cosmic history. We showed that local properties of the function
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
in that regime.
In
Section 4 we focused on a specific model obtained in Ref. [
22], where the function
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
,
, 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
and
, 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
. We derived the time evolution of the effective gravitational constants during the phase of decelerated expansion for
, with
. 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
and
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 yet to be established and shall be investigated in future work.