# Inflation in \(f(R,\phi )\)-theories and mimetic gravity scenario

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## Abstract

We investigate inflation within \(f(R,\phi )\)-theories, where a dynamical scalar field is coupled to gravity. A class of models which can support early-time acceleration with the emerging of an effective cosmological constant at high curvature is studied. The dynamics of the field allow for exit from inflation leading to the correct amount of inflation in agreement with cosmological data. Furthermore, the spectral index and tensor-to-scalar ratio of the models are carefully analyzed. A generalization of the theory to incorporate dark matter in the context of mimetic gravity, and further extensions of the latter, are also discussed.

## Keywords

Dark Matter Scalar Field Spectral Index Ricci Scalar High Derivative Term## 1 Introduction

Over the past years, interest in inflationary cosmology has grown considerably, as a consequence of the great amount of data from recent cosmological surveys [1, 2, 3, 4]. The inflationary paradigm was first introduced in 1981 by Guth [5] and Sato [6, 7] to explain the thermalization of the observable Universe inferred from observations of the CMB. It also allows one to address some of the problems associated to the initial conditions of a Friedmann universe. Moreover, quantum fluctuations during the inflationary epoch presumably seeded the perturbations which grew under gravitational instability into the structures we see today [8]. For reviews on inflation, see e.g. [9, 10, 21, 22, 23].

An early-time period of acceleration should presumably be supported by a repulsive gravitational force. At the same time, a mechanism which allows one to quickly exit this stage and enter the radiation dominated era is necessary. The arena of inflationary models is quite vast. In the scalar field formulation or chaotic inflation [24], a scalar field (the inflaton) is subject to a potential and drives accelerated expansion when its magnitude is negative and very large: at the end of inflation, it settles down in a minimum of the potential and begins oscillating, giving rise to the reheating mechanism responsible for particle production. Other implementations of inflation include for instance natural inflation (see e.g. [11, 12, 13, 14]), *k*-inflation [15], brane inflation [16], and many others.^{1} In the context of modified gravity (see Refs. [25, 26, 27, 28, 29] for reviews), a modification to Einstein’s gravitational action emerges at high curvature and supports the early-time acceleration (see Ref. [30]). This can be realized for instance in the so-called Starobinsky model [31], which provides a correction quadratic in the Ricci scalar.

A model for inflation is viable only if it is able to reproduce the inferred spectral index and the tensor-to-scalar ratio at the origin of cosmological perturbations in the Friedmann universe. The evaluation of these indices depends on the theory under investigation. In [32, 33] an unified description has been derived in the context of \(f(R,\phi )\)-gravity, where a scalar field subject to a potential is coupled to gravity (a non-minimal coupling in the kinetic part of the field is also present). In this work, we will analyze \(f(R,\phi )\)-inflation by working through some simple examples based on modified gravity models which mimic the “false vacuum” of the primordial universe: in fact, we will study a class of models (exponential models and power-law models) describing an effective cosmological constant at high curvature and whose exit from inflation is induced by the coupling of a scalar field to gravity. We will show that the model can yield values for the spectral index and the tensor-to-scalar ratio (the magnitude of the last one in \(f(R,\phi )\)-gravity is particularly small) in agreement with those inferred from observations.

In the last part of the work we embed this model within the framework of mimetic gravity, which additionally endows it with a dark matter candidate. We then discuss extensions which can address the controversies of cold dark matter on small scales within the mimetic gravity scenario, such extensions being theoretically driven by the correspondence between mimetic gravity and the scalar formulation of the Einstein-aether theories. We then speculate on possible further extensions of the scenario depicted.

The paper is organized as follows. In Sect. 2 we will revisit the form of the spectral indices and the tensor-to-scalar ratio in \(f(R,\phi )\)-gravity by deriving some useful relations. In Sect. 3 we study inflation in two different \(f(R,\phi )\) models. Early-time acceleration takes place at high curvature in agreement with the latest Planck data and the field allows for a quick exit from this stage recovering Friedmann evolution of Einstein’s gravity. In Sect. 4 we formulate the model within the mimetic gravity framework. Section 5 is devoted to our conclusions and final remarks.

We use units where \(k_{\mathrm {B}} = c = \hbar = 1\) and denote the gravitational constant, \(G_N\), by \(\kappa ^2\equiv 8 \pi G_{N}\), such that \(G_{N}=M_{\mathrm {Pl}}^{-2}\), \(M_{\mathrm {Pl}} =1.2 \times 10^{19}\) GeV being the Planck mass.

## 2 Inflation in \(f(R, \phi )\)-gravitational models

*R*and the scalar field \(\phi \) is subject to the potential \(V(\phi )\). In the above, \(\omega (\phi )\) is in principle a function of \(\phi \) which represents a non-minimal coupling of the kinetic term of the field. In a flat Friedmann–Robertson–Walker universe, with metric given by

*t*, the equations of motion of the theory read

^{2}

*r*defined as [32, 33],

^{3}As a consequence, the spectral index and tensor-to-scalar ratio read

## 3 Viable inflation in \(f(R,\phi )\)-gravity describing an effective cosmological constant

In order to reproduce the “false vacuum” of inflation, one possibility is to introduce a large effective cosmological constant (whose value is close to the Planck scale) within Einstein’s framework. In this way, it is easy to obtain the repulsive gravity required to support the early-time acceleration. However, one of the main problems faced by the inflationary paradigm is the realization of a mechanism to gracefully exit this stage.

*f*(

*R*)-modified gravity which can successfully realize the current acceleration of our universe have been investigated. These models feature what can be viewed as a “switching on” cosmological constant and assume the following form [37, 38, 39, 40, 41, 42]:

*b*is a dimensionless number of order unity. In this way, the scale at which the cosmological constant appears is a sort of “running scale”, which varies as the field does. We consider the Lagrangian

*e*-folds \(\mathcal N\), which is defined by

*e*-folds must be \(55<\mathcal N<65\) in order for the observable universe to thermalize. Accounting for the fact that the Ricci scalar changes considerably slower than the field itself, we can write in our case that

*r*is required to be very small but not vanishing, by taking into account that the most likely value for

*r*is \(r\sim 0.06\), we require that \(\Lambda ^2\sim 10^{-6}\times \phi _0^2/\kappa ^2\).

*f*(

*R*) model which mimics the behavior of (3.1), namely the Hu–Sawicki model [43],

*n*is a positive fixed parameter and the cosmological constant emerges when \(1\ll (R/R_0)^n\). To exit from the accelerated phase, one can make the substitution (3.2) in the above expression to obtain

*e*-folds can be written as

*n*must be large. In the limit of \(1\ll n\) one has

*r*.

## 4 Mimetic gravity

Here, we stress that it must be \(0<F(R,\phi )\) to have a positive defined effective Newton constant in \(\kappa ^2\equiv 8\pi G_N\), such that \(G_N^{\text {eff}}=G_N/F(R,\phi )\). In this respect, the \(f(R,\phi )\) models (3.4) and (3.20) avoid the “antigravity” in the corresponding *f*(*R*) models (3.1) and (3.19) at small curvatures after the end of inflation^{4} when \(\phi \rightarrow 0^{-}\) and \(F(R,\phi )\simeq 1/\kappa ^2\).

*a priori*, one is faced with a wider class of solutions, as opposed to the simpler case where \(F(R,\phi )G _{\mu \nu } = (T _{\mu \nu } ^{\phi } + T _{\mu \nu } ^{\text {MG}})\). A particularly interesting case arises when one considers a Friedmann–Robertson–Walker metric (2.2), which combined with (4.2) yields:

The mimetic gravity framework offers an alternative approach to solving some of the outstanding problems in modern cosmology. On cosmological scales, mimetic dark matter behaves precisely as collisionless cold dark matter, and as such is affected by gravitational instability [44]. On the other hand, it is known that the collisionless cold dark matter paradigm suffers from a number of shortcomings on small scales: the core-cusp problem [65, 66, 67, 68], the missing satellite problem [69, 70], and the “too-big-to-fail” problem [71, 72, 73, 74], just to mention a few (see e.g. [75] for a recent review on the subject and more references). While in the particle dark matter framework these issues might be addressed by positing that dark matter is self-interacting and collisional (see e.g. [76, 77, 78, 79, 80, 81] and references therein for further discussions), in the mimetic dark matter framework a possible solution is instead the addition of higher derivative (HD) terms for the scalar field \(\varphi \) to the action. The HD terms (which are encoding UV physics) provide mimetic dark matter with a non-vanishing sound speed, and they are effectively dissipation terms^{5} [82, 83, 84]. In fact, these HD terms might be crucial to avoid caustic singularities, from which the original mimetic dark matter framework suffers [82]. More importantly, they have the effect of suppressing power on small scales, which has the potential to address the shortcomings of the collisionless cold dark matter framework on galactic and subgalactic scales [82, 84].

Another important observation is that mimetic gravity is equivalent to a class of Lorentz-violating generally covariant extensions of Einstein’s General Relativity, known as Einstein-aether theories (EA hereafter; see [103] for the original formulation of the theory and [104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124] for more recent extensive discussions on this class of theories). In EA, Lorentz invariance is broken by a dynamical unit timelike vector \(u ^{\mu }\) (the “*aether*”), which fixes a preferred rest frame at each space-time point. In particular, mimetic gravity corresponds to the scalar formulation of the EA theory [125, 126] (see [127] for further discussions), where the aether vector is identified with the gradient of a scalar function, \(u _{\mu } = \partial _{\mu }\varphi \). This scalar function corresponds to the scalar field encoding the conformal mode in mimetic gravity. The most general action for the scalar field, through the inclusion of the aforementioned HD terms, is constructed in [125]. Recently it was also noted that mimetic gravity can be identified and incorporated into the framework of covariant renormalizable gravity [128].

Another small-scale open question in mimetic gravity is whether this framework is able to account for the inferred flat rotation curves of spiral galaxies [129, 130]. The symmetries of the theory allow for the addition of a non-minimal coupling between matter and gravity, in the form of a coupling between the aether vector and a matter hydrodynamic flux. In the Newtonian limit such a term yields the phenomenology of MOND (see e.g. [131, 132, 133, 134, 135, 136] for comprehensive reviews), and hence reproduces flat rotation curves and the Tully–Fisher relation [137, 138], among others. The phenomenology and constraints on such a coupling remain to be explored.

The mimetic gravity scenario can be successfully integrated with *f*(*R*) gravity, and also with \(f(R,\phi )\). Provided that the \(f(R,\phi )\) (non-mimetic) theory is ghost-free, the corresponding mimetic formulation should also presumably be ghost-free, since the addition of a Lagrange multiplier term to the action is not expected to spoil such property (although it should be remarked that such a statement remains to be checked) [55, 56, 57]. One can then see how \(f(R,\phi )\)-gravity can successfully be incorporated in the mimetic gravity framework to additionally introduce a dark matter component to the theory, a crucial element in our current understanding of cosmology. Furthermore, it is possible to extend this picture by including a potential for the mimetic scalar field \(\varphi \). Given that in a Friedmann–Robertson–Walker universe \(\varphi \) can be identified with time, such an addition effectively introduces a time-dependent energy density, allowing one to realize any given evolutionary history of the Universe. Any potential \(V(\varphi )\) can be used to reconstruct a function \(f(R,\phi )\) which gives the corresponding evolution, as shown in [55, 56, 57]. In particular, by suitably choosing the form of the potential it is possible to construct an unified and consistent description of inflation with graceful exit, the current epoch of acceleration presumably sourced by dark energy, and a bouncing non-singular universe.

In a first solution, note that when \(\tilde{T}_{\mu \nu }=0\), \(f(R,\phi )\)-mimetic gravity leads to Eqs. (2.3)–(2.4) and we recover the same results illustrated in the preceding section. Inflation ends when \(\phi \rightarrow 0^-\) and the models (3.4) and (3.20) feature a “phase transition” from a high curvature regime (\(f(R,\phi )\simeq R-2\Lambda \)) to a low curvature one (\(f(R,\phi )\simeq R\)). One may assume that while inflation is realized at \(\tilde{T}_{\mu \nu }=0\), the Friedmann universe belongs to the sector \(T_{\mu \nu }\ne 0\) of the theory, and dark matter emerges only at the end of inflation. Another possible way of protecting dark matter from decay during inflation is to couple the two scalar fields \(\varphi \) and \(\phi \) through a term of the form \(\varphi F(\phi )\), as discussed in [44].

Another open question concerns generating the observed radiation and baryonic content in the universe, including the observed baryon-antibaryon asymmetry. This can presumably be generated by gravitational particle production following the end of inflation, through direct coupling of other fields to the scalar field \(\varphi \), or through fluctuation–dissipation dynamics inherent to the scalar fields \(\phi \) and \(\varphi \) (see [139, 140] for further discussions on the topic and the implementation of a model of dissipative leptogenesis). We plan to explore these and other ideas in a forthcoming paper.

## 5 Conclusions

In the present paper, we have studied inflation in the context of \(f(R,\phi )\)-theories of gravity, where a scalar field is coupled to gravity. This class of theories is very interesting, given that one can use the *f*(*R*)-gravity sector to reproduce a variety of cosmological scenarios (in our specific case, accelerated cosmology at high curvatures and Einsteins gravity at low curvatures), while a dynamical scalar field allows for one to move between one scenario and another. We note that the *f*(*R*)-formulations of the models under investigation (namely, exponential gravity and the so called Hu–Sawiki model with power-law corrections to Einstein’s gravity) belong to a class of viable models for the dark energy epoch which the universe undergoes today, where the appearance of an effective cosmological constant easily supports an (eternal) accelerated de Sitter expansion. Within the same models (perhaps in the attempt to unify the inflationary scenario with the dark energy epoch), one may reproduce the false vacuum of inflation by an effective cosmological constant, but a mechanism to make inflation unstable is necessary. In this respect, the introduction of a dynamical field induces a phase transition in the models and inflation ends when the effective cosmological constant disappears.

We have explicitly calculated the spectral indices and tensor-to-scalar ratio in the given models, starting from their first principle formulation in these kind of theories. We find that for the theory to give the correct amount of inflation (namely, a number of *e*-folds sufficiently large to allow for thermalization of the entire observable universe) and at the same time generate a spectral index in agreement with cosmological data, the magnitude of the tensor-to-scalar ratio (which is quadratic in one of the slow-roll parameter) is particularly small (note that the same occurs in pure modified gravity but not in scalar field inflation within Einstein’s framework) but non-vanishing: this occurs by virtue of the fact that the energy scale of inflation is sub-Planckian, while the magnitude of the scalar field can exceed the Planck scale. For recent work on \(f(R,\phi )\)-inflation see also [141, 142, 143, 144, 145].

The minimal formulation we have considered does not contain a dark matter candidate. To address this point, we have then considered extensions of such a model within the mimetic gravity framework, where dark matter appears as an integration constant of the equations of motion. In the minimal mimetic gravity formulation, mimetic dark matter behaves precisely as collisionless cold dark matter. In the light of the issues which collisionless cold dark matter faces on small scales, we have discussed possible extensions of the mimetic gravity framework which allow one to deal with these shortcomings, and at the same time explain a number of observations, the origin of which is usually attributed to particle dark matter (for instance, the inferred flat rotation curves and the Tully–Fisher relation). The extensions we have discussed were theoretically motivated by the equivalence between the original formulation of mimetic gravity and the Einstein-aether class of Lorentz-violating theories of gravity. We have further expounded how the extension of \(f(R,\phi )\) inflation within the non-minimal mimetic gravity framework allows one to realize basically any evolutionary history of the Universe. Finally, we have commented on possible ways of protecting dark matter from decay during inflation, and generating the observed baryonic and radiation content of the Universe.

To conclude, the model we have explored introduces two additional scalar degrees of freedom to the framework of *f*(*R*) gravity. A first one allows one to move between two different cosmological scenarios (accelerated expansion and Einstein gravity at high and low curvature, respectively), while the second one endows the model with a natural dark matter candidate (which can address some of the small-scale tensions with collisionless cold dark matter), and can be used to reproduce any desired background cosmological expansion. While introducing extra degrees of freedom might seem a high price to pay, we have shown that if used appropriately such degrees of freedom allow a more natural implementation of a unified expansion history, while at the same time providing a candidate for the missing dark matter in the Universe.

## Footnotes

- 1.
- 2.
If \(\dot{\omega }(\phi )=0\), we obtain directly \(|\omega (\phi )\dot{\phi }^2/F(R,\phi )H^2|\ll 1\) from \(|\epsilon _4|\ll 1\).

- 3.
If \(\dot{\omega }(\phi )\ne 0\), one has \(\epsilon _1\simeq -\epsilon _3-\dot{\omega }(\phi )\dot{\phi }^2/(6H^2\dot{F}(R,\phi ))\).

- 4.
Such a problem is not present in the original formulation of these models for the dark energy issue, where the history of the universe belongs to \(R_0<R\).

- 5.

## Notes

### Acknowledgments

We would like to thank Sergei Odintsov and Sergio Zerbini for comments and suggestions. S. V. would like to thank Amel Duraković for useful discussions, and the hospitality of the Niels Bohr International Academy and in particular of Poul Henrik Damgaard while this work was being completed.

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