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Mimetic gravity is a Weyl-symmetric extension of General Relativity, related to the latter by a singular disformal transformation, wherein the appearance of a dust-like perfect fluid can mimic cold dark matter at a cosmological level. Within this framework, it is possible to provide a unified geometrical explanation for dark matter, the late-time acceleration, and inflation, making it a very attractive theory. In this review, we summarize the main aspects of mimetic gravity, as well as extensions of the minimal formulation of the model. We devote particular focus to the reconstruction technique, which allows the realization of any desired expansionary history of the universe by an accurate choice of potential or other functions defined within the theory (as in the case of mimetic

The past decade has seen the astounding confirmation of the “dark universe” picture, wherein the energy budget of our universe is dominated by two dark components: dark matter and dark energy [

A particularly interesting theory of modified gravity which has emerged in the past few years is mimetic gravity [

This review is structured as follows: in this section we will provide a historical and technical introduction to modified gravity, which shall justify our subsequent endeavour in mimetic gravity. In Section

General Relativity (GR henceforth), first formulated by Einstein in 1915 [

Confirmations of GR abound (see, e.g., [

Nonetheless, theoretical motivation for modifying the gravitational action came quite soon. The underlying reason is that GR is nonrenormalizable and thus not quantizable in the way conventional Quantum Field Theories are quantized. However, it was proven that 1-loop renormalization requires the addition of higher order curvature terms to the Einstein-Hilbert action. In fact, it was later demonstrated that, while actions constructed from invariants quadratic in curvature are renormalizable [

With the emergence of the “dark universe” picture in recent years, the limits of GR have been fully exposed, and further motivations to modify this theory have emerged. A series of experiments and surveys, including but not limited to CMB experiments, galaxy redshift surveys, cluster surveys, supernovae surveys, lensing experiments, and quasar surveys, have depicted a peculiar picture of our universe [

The late-time acceleration of our universe, however, is most likely not the only period of accelerated expansion that our universe has experienced. A period of accelerated (exponential) expansion during the very early universe, prior to the conventional radiation and matter domination epochs, is required to solve the horizon, flatness, and monopole problems. This period of accelerated expansion is known as inflation (see, e.g., [

It thus appears that concordance cosmology requires at least three extra (possibly dark) cosmological components: one or more dark matter components, some form of dark energy, and one or more inflaton fields. There is no shortage of ideas as to what might be the nature of each of these components. Nonetheless, adding these three or more components opens another set of questions, which include but are not limited to the compatibility with the current SM and the consistency of formulation. On the other hand, gravity is the least understood of the four fundamental interactions and the most relevant one on cosmological and astrophysical scales. If so, it could be that our understanding of gravity on these scales is inadequate or incomplete, and modifying our theory of gravitation could indeed be the answer to the dark components of the universe. One could argue that this solution is indeed more economical and possibly the one to pursue in the spirit of Occam’s razor. In other words, modifications to Einstein’s theory of General Relativity might provide a consistent description of early and late-time acceleration and of the dark matter which appears to pervade the universe. Modified theories of gravity not only can provide a solution to the “dark universe riddle” but also possess a number of alluring features such as unification of the various epochs of acceleration and deceleration (matter domination) of the universe’s evolution, transition from nonphantom to phantom phase being transient (and thus without Big Rip), solution to the coincidence problem, and also interesting connections to string theory.

Having presented some motivation to modify our theory of gravitation, we now proceed to briefly discuss systematic ways by means of which this purpose can be achieved.

Essentially all attempts to modify General Relativity are guided by Lovelock’s theorem [

Presence of other fields apart from or in lieu of the metric tensor

Work in a number of dimensions different from 4

Accept metric derivatives of degree higher than 2 in the field equations

Giving up locality or Lorentz invariance

Therefore, we can imagine broadly classifying the plethora of modified gravity theories according to which of the above assumptions is broken.

Relaxing the first assumption leads to what is probably the largest class of modified gravity theories. Theories corresponding to the addition of

One can instead choose to add

Theories where

Broadly speaking, mimetic gravity belongs to the class of theories of modified gravity where an additional scalar degree of freedom is added. Caution is needed with this identification though because, as we shall see later, mimetic gravity does not possess a proper scalar degree of freedom, but rather a constrained one.

Relaxing the second assumption instead brings us to consider models with extra dimensions, the prototype of which is constituted by Kaluza-Klein models (e.g., [

The most famous and studied example of a theory falling within this category is undoubtedly represented by

If we choose to relax the assumption of locality (we have already seen cases where the assumption of Lorentz invariance is relaxed above), we can consider nonlocal gravity models whose action contains the inverse of differential operators of curvature invariants, such as

Broadly speaking, mimetic gravity belongs to the class of theories of modified gravity where an additional scalar degree of freedom is added. Caution is needed with this identification though because mimetic gravity does not possess a proper scalar degree of freedom. Instead, the would-be scalar degree of freedom is constrained by a Lagrange multiplier, which kills all higher derivatives. As such, the mimetic field cannot have oscillating solutions and the sound speed satisfies

The expression “mimetic dark matter” was first coined in a 2013 paper by Chamseddine and Mukhanov [

By varying the action with respect to the physical metric one obtains the equations for the gravitational field. However, this process must be done with care, for the (variation of the) physical metric can be written in terms of the (variation of the) auxiliary metric and the (variation of the) mimetic field. Taking this dependency into account, variation of the action with respect to the physical metric yields [

Let us examine the structure of the mimetic stress-energy tensor. Recall that the stress-energy tensor of a perfect fluid whose energy density is

A remark is in order here. Actions such as (

Before we can make further progress in exploring solutions in mimetic gravity, generalizing the theory, or studying connections to other theories, we need to touch on two very important points: first, why the seemingly innocuous parametrization given by (

It might appear puzzling at first that, only by rearranging parts of the metric, one is faced with a different model altogether. A first explanation appeared in [

Another explanation was presented in [

As we anticipated above, mimetic gravity and the appearance of the extra degree of freedom which can mimic cosmological dark matter are rooted into the role played by singular disformal transformations. As was shown by Bekenstein [

To make progress, it is useful to contract the two equations of motion with

The situation is quite different if the determinant in (

The parametrization (

There actually exists a third route to mimetic gravity, apart from disformal transformations and Lagrange multiplier, whose starting point is the singular Brans-Dicke theory. Namely, by starting from the action (

Is mimetic gravity stable? In other words, does its spectrum contemplate the presence of states with negative norm, or fields whose kinetic term has the wrong sign (corresponding to negative energy states), which could possibly destabilize the theory? This is an important question which has yet to find a definitive answer. Recall that the original mimetic theories formulated in 2010 were found to suffer from a tachyonic instability [

If we formulate the theory of mimetic gravity using the physical metric

The analysis of [

A possible solution to these instability issues was presented in [

Another recent work confirmed in all generality that the original mimetic gravity theory suffers from ghost instability [

Having discussed the underlying physical foundation of mimetic gravity, and its stability, we can now proceed to study solutions and extensions of this theory.

Recall that, in a cosmological setting, the mimetic field plays the role of “clock.” Therefore, one can imagine making the mimetic field dynamical by adding a potential for such field to the action. A field-dependent potential corresponds to a time-dependent potential which, by virtue of the Friedmann equation, corresponds to a time-varying Hubble parameter (and correspondingly scale factor). Therefore, by adding an appropriate potential for the mimetic field, one can in principle reconstruct any desired expansion history of the universe. This is the idea behind the minimal extension of mimetic gravity first proposed in [

To study cosmological solutions, it is useful to consider a flat FLRW background (

Let us consider the following potential [

We can consider the case where mimetic matter is a subdominant energy component in the universe, which is instead dominated by another form of matter with EoS

We can consider an arbitrary power-law potential:

One can always reconstruct the appropriate potential for the mimetic field which can provide an inflationary solution. The method is very simple: choose a desired expansion history of the universe [encoded in the Hubble parameter

Another interesting possibility is given by an exponential potential [

As we have already seen in previous cases, one can easily construct bouncing solutions in mimetic gravity. Let us work through one further example here. Consider a potential of the form:

In the case we have just examined, the bounce occurs at the Planck scale, and hence the classical analysis we provided might not be valid as quantum gravity effects would be playing an important role. However, a minimal modification allows lowering the scale of the bounce and correspondingly increases the duration of the bounce (which now lasts more than a Planckian time). The corresponding potential which can provide this behaviour is given by [

The next step which was performed by Nojiri and Odintsov is to generalize mimetic gravity to mimetic

The equations of motion of the theory are slightly more complicated than that of conventional mimetic gravity. Varying with respect to the metric gives the gravitational field equations [

As we have mentioned previously, in mimetic

Let us proceed to study some of the properties of mimetic

The above equations put on a quantitative footing the statement we previously made: namely, that by tuning the behaviour of either or both the two additional scalar degrees of freedom, we can reconstruct any possible expansion history of the universe [

Three further recent studies by Odintsov and Oikonomou [

In addition, [

So far we have discussed mimetic gravity and variants thereof at early times, that is, at the epoch when primordial curvature perturbations were generated. However, it is also interesting to consider late-time evolution in mimetic gravity. The equations of motion are incredibly complex and in principle do not allow for analytical solutions. However, this complexity can be bypassed by means of the method of dynamical analysis (see, e.g., [

A detailed dynamical analysis of mimetic

As we mentioned above, the conclusions reached about the late-time evolution in mimetic

In [

The question of constructing a theoretically motivated but at the same time simple model for mimetic

A further extension of mimetic

As we have seen, mimetic gravity provides a geometric explanation for dark matter in the universe, with dark matter emerging as an integration constant as a result of gauging local Weyl invariance, without the need for additional fluids. An older theory, known as unimodular gravity [

In order to combine mimetic gravity and unimodular gravity it is necessary to enforce two constraints. The first is the constraint on the gradient of the mimetic field (

The equations of motion for the gravitational field are obtained by varying the action with respect to the metric and are given by [

Having made this consideration, let us consider a few examples where the reconstruction technique is applied. Let us consider the following simple potential [

Two further comments are in order here. First, it is possible to provide an effective fluid description of unimodular mimetic gravity [

The second comment is related to the fact that the unimodular constraint is enforced in a noncovariant way (cf. the action given by (

A minimal extension of the unimodular mimetic gravity framework we have discussed so far is to consider unimodular mimetic

One can further consider more general scalar-tensor theories, which can be “mimetized” according to the procedures we have described so far, namely, through a singular disformal transformation or through a Lagrange multiplier term in the action enforcing the mimetic constraint. In fact, analogously to GR, one can show that the most general scalar-tensor model is invariant under disformal transformations, provided the latter is invertible. This has been shown in all generality in [

Of course, the considerations made above can be applied in the case of a specific scalar-tensor model, namely, Horndeski gravity [

The mimetic version of the above Horndeski model has been studied in a variety of papers recently (e.g., [

To conclude, we report on the following specific case of mimetic Horndeski model which was studied in [

In closing, we comment on some connections between mimetic gravity and other theories of modified gravity, such connections having been identified recently: namely, the scalar Einstein-aether theory, Hořava-Lifshitz gravity, and a covariant realization of the latter, that is, covariant renormalizable gravity.

An interesting connection which can be identified is that between mimetic gravity and Einstein-aether theories [

To be precise, mimetic gravity is in correspondence with a particular version of the Einstein-aether theory, namely, the scalar Einstein-aether theory [

Recall that Hořava-Lifshitz gravity [

Previous work has shown that the IR limit of the nonprojectable version of Hořava-Lifshitz gravity can be obtained from the Einstein-aether theory (the vector version) by requiring that the aether be hypersurface orthogonal: that is,

Further requiring that

A complete proof of the equivalence between mimetic gravity and the IR limit of projectable Hořava-Lifshitz gravity was presented in [

Recall that HLG achieves power-counting renormalizability by breaking diffeomorphism invariance. However, this breaking appears explicitly at the level of the action. This has been at the center of criticism, which has related this explicit breaking to the appearance of unphysical modes in the theory which is coupled strongly in the IR [

An example of such theory has been presented by Nojiri and Odintsov [

Following the initial proposal by Nojiri and Odintsov, other CRG-like models were studied in recent years. For instance, one particular CRG-like model was studied by Cognola et al. in [

In the continuation of our review, after a brief interlude on perturbations in mimetic gravity, we shall consider a case study of a mimetic-like theory. Our choice of case study will fall upon the CRG-like model of Cognola et al. as defined by the action in (

Before proceeding to the case study of a specific mimetic-like model, it is mandatory to discuss the issue of perturbations in mimetic gravity. Recall in Section

However, it is clear that the Lagrange multiplier kills the wave-like parts of the scalar degree of freedom: in other words, given that the constraint takes out any higher derivative, it is not possible to have oscillating (wave-like solutions). As a consequence, we can already envisage that the sound speed in the minimal mimetic gravity model will satisfy

The property of vanishing sound speed in mimetic gravity can be rigorously demonstrated, by considering small longitudinal perturbations around a flat background [

In order to have a theory whose quantum perturbations can be defined in a sensible way, the minimal action for mimetic gravity has to be modified, for instance, by introducing higher derivative (HD) terms. As an example, consider the following action [

In concluding this brief interlude, let us also spare a few words on further modifications of mimetic gravity involving higher derivative terms. These have been studied in [

The suppression of power on small scales is particularly intriguing in the light of the observation that the collisionless cold dark matter paradigm appears to suffer from a number of shortcomings on subgalactic scales. The core-cusp problem refers to the discrepancy between

Several approaches to solving these problems exist in the literature. If one insists that dark matter is cold and collisionless, then an important role must be played by the baryonic content of the universe. In fact, it has been argued in several works that baryonic feedback processes (see, e.g., [

A different mechanism, but with similar outcomes, occurs in mimetic gravity. Namely, the suppression of small-scale power, operated by the higher derivative terms, has the potential to solve the missing satellite problem and the too big to fail problem, as shown in [

Having discussed in detail the physics behind mimetic gravity, and many of its extensions, we now provide a detailed case study of a specific mimetic model. Our choice falls on the covariant Hořava-like theory of gravity first discussed by Cognola et al. [

Let us start from the action of the CRG-like model first discussed by Cognola et al. [

Let us now consider cosmological solutions, in particular considering a flat FLRW metric (

Let us begin by considering the tensor

In the case where we set

Let us manipulate (

Let us provide one final example with a potential given by

In this section, we will consider the scalar perturbations around the FLRW metric (

Finally, we can obtain a closed equation for

With our goal being that of addressing the problem of scalar perturbations, we modify the model given by (

If we consider the perturbed metric (

In this section, we will explore static spherically symmetric solutions (SSS) in mimetic gravity. To do so, let us return to the general formulation of mimetic gravity with action given by (

In this chapter we will consider pseudo-SSS space-times, whose general topological formulation is given by

The

We can also rewrite (

In this subsection we set

The spherical case (

In general, we observe that when

Finally, the topological case

In this subsection, we will consider the case

In considering other solutions, we will take the spherical case

Let us consider a linear modification to the Schwarzschild metric:

The metric under investigation reduces to the usual Schwarzschild space-time for short distances, while at large distances its 00-component behaves as

As a second example of reconstruction procedure, we consider the following ansatz:

The above metric (

Given

The above correspond to the traversability conditions [

In the next section we will use a different approach to fix

So far we have seen that, within mimetic gravity, dark matter emerges as a geometrical effect at a cosmological level, that is, in the form of a perfect fluid whose energy density decays as

The first solution to this problem was found in [

For our purpose, it is convenient to redefine

At small distances, the metric leads to a classical Newtonian term

At very large distances, the “cosmological constant” term

At intermediate distance the linear term

At intermediate galactic scales, we can safely assume that the

The situation is different for sufficiently extended galaxies, for instance, large high surface brightness (HSB) galaxies. For these galaxies the Newtonian contribution might be sufficient to complete with the rising linear term,

Let us turn to the question of reproducing such behaviour in mimetic gravity. In order to reconstruct the complete form of the metric (

In the limit

Finally, in the limit

To fix the values of

As per the analysis of [^{−2}, suggesting that it is most important on scales of large galaxies or clusters.

We previously mentioned that the idea adopted to fit rotation curves resembles that of MOND, that is, to introduce a new scale in the theory, which could be a scale intrinsically present in the data. Let us elaborate on this point more quantitatively. Considering the measured distance

We conclude this chapter by considering, for completeness, the case of a general power-law correction:

The non-Newtonian correction we have considered has recently been studied in the context of

Let us conclude by making an important remark. Although the potential leading to the chosen non-Newtonian corrections has only been given implicitly, the form of

Mimetic gravity has emerged as an interesting and viable alternative to General Relativity, wherein the dark components of the universe (underlying dark matter, the late-time acceleration, and inflation) can find unified geometrical explanation and interpretation. The theory is related to General Relativity by a singular disformal transformation, which is the reason behind its exhibiting a wider class of solutions. Here, we have reviewed the main aspects of mimetic gravity, beginning by placing it in the wider context of theories of modified gravity. After having reviewed the underlying theory behind mimetic gravity, we have studied some of its solutions and extensions, such as mimetic

The dark components of our universe remain as mysterious as ever. It is possible that we might shed light on the nature of dark matter, dark energy, and inflation, as more data from experiments and surveys pours in the coming years. Thus far, theories of modified gravity, despite their

The authors declare that there is no conflict of interests regarding the publication of this paper.

Sunny Vagnozzi thanks the Niels Bohr Institute, where the majority of this work was completed, for hospitality. The authors have benefited from many discussions with Sergei Odintsov and Sergio Zerbini, whom we wish to thank.

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