# What if gravity becomes really repulsive in the future?

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## Abstract

The current acceleration of the Universe is one of the most puzzling issues in theoretical physics nowadays. We are far from giving an answer in this letter to the question of its nature. Yet, with the observations we have at hand, we analyse the different patterns that the gravitational potential can show in the future. Surprisingly, gravity not only can get weaker in the near future, it can even become repulsive; or equivalently, the gravitational potential may become negative. We show this remark by using one of the simplest phenomenological model we can imagine for dark energy. We have also reviewed the statefinder approach of these models. For completeness, we have also showed the behaviour of the density contrast of dark matter and dark energy for these simple (yet illustrative models). Our results are displayed and we see how they shall evolve in the future.

## 1 Introduction

Hubble’s discovery was crucial for our understanding of the Universe. He showed that the Universe was evolving and not static as it was believed at that time [1]. His discovery was based on observing that the spectrum of far away galaxies was red-shifted which implied that those galaxies were moving away from us. He even measured the galaxies radial outward velocities and realised that it followed a rule: (1) the velocities were proportional to the distances at which the galaxies were located from us and (2) the proportionality factor was a constant, the Hubble constant. About 70 years later, two independent teams [2, 3] realised that by measuring further objects, SNeIa, the Hubble constant was not quite constant, as was already expected. The issue was that the deviation from the constancy was not in the anticipated direction. It was no longer enough to invoke only matter to explain those observations. A new dark component had to be invoked, interacting as far as we know only gravitationally, and named dark energy. This component started recently fuelling a second inflationary era of the visible Universe. Of course, all these observations, and subsequent ones, are telling us how gravity behaves at cosmological scales through the kinematic expansion of our Universe [4, 5, 6, 7, 8, 9].

This kinematic description is linked to the dynamical expansion through the gravitational laws of Einstein theory. To a very good approximation, we may assume that our Universe is homogeneous and isotropic on large scales and that it is filled with matter (standard and dark) and dark energy, where their relative fractional energy densities are \(\varOmega _{\text {m}}=0.309\) and \(\varOmega _{\text {d}}=0.691\), respectively, at present. In addition, the current Hubble parameter is of the order of \(H_0=67.74\) km s\(^{-1}\) Mpc\(^{-1}\). We have fixed those values by using the latest Planck data [7] but please notice that our conclusions in this paper are unaltered by choosing other values for these physical quantities. As regards dark energy, we will assume its energy density to be evolving (or not) in time and its equation of state (EoS) parameter, *w*, to be constant; i.e. we will consider the *w*CDM model as a natural candidate describing our Universe. As is well known: (1) for \(w<-\,1\) the Universe would face a big rip singularity [10, 11, 12], i.e., the Universe would unzip itself in a finite time from now, (2) for \(w=-\,1\) the Universe would be asymptotically de Sitter, and finally (3) if \(w>-\,1\) the Universe would be asymptotically flat locally; i.e. the scalar curvature and the Ricci tensor would vanish for large scale factors. As we next show this pattern is shown also by the behaviour of the gravitational potential.

The paper is organised as follows: in Sect. 2, we review briefly the models to be considered and compare them using a cosmographic/statefinder analysis. In Sect. 3, we present the cosmological perturbations of the models focussing on the asymptotic behaviour of the gravitational potential. Finally, in Sect. 4, we conclude. In Appendix A, we include some formulae useful in Sect. 2.

## 2 Background approach

*t*is the cosmic time,

*a*(

*t*) is the scale factor and \(\delta _{ij}\) is the flat spatial metric. On the other hand, the matter content of the Universe can be separated in three main components: radiation, nonrelativistic matter (baryons and dark matter (DM)) and dark energy (DE). For simplicity, we model these three components using a perfect fluid description where each fluid has energy density \(\rho _\mathrm {i}\) and pressure \(p_\mathrm {i}=w_\mathrm {i}\rho _\mathrm {i}\). Here, i stands for radiation (r) with \(w_\mathrm {r}=1/3\), for nonrelativistic matter (m) with \(w_\mathrm {m}=0\), and for DE (d) with \(w_\mathrm {d}=w\). The Friedmann equation for such model can be written as

*w*: \(\{-\,0.99,\, -\,1,-\,1.01\}\), in order to obtain three qualitatively different types of late-time behaviour for DE: quintessence (\(w\gtrsim -\,1\)), cosmological constant (\(w=-\,1\)) and phantom behaviour (\(w\lesssim -\,1\)).

*n*th derivative of the scale factor with respect to the cosmic time.

^{1}Based on the cosmographic expansion (3), the statefinder hierarchy was developed as a tool to distinguish different DE models [17, 18, 19, 20, 21]. In fact, the statefinder parameters are defined as specific combinations of the cosmographic parameters:

*s*defined in [17, 18] as

*w*CDM model with a radiation component, such as the models considered in this paper, we present in Appendix A the full expressions of the statefinder parameters as functions of the scale factor \(a/a_0\) and the cosmological parameters \(\{\varOmega _\text {i,0},\,w\}\). In the limit \(a\rightarrow +\infty \) the expressions found reduce to

*w*deviates from the nominal value \(-\,1\) the asymptotic values of the statefinder parameters \(S_i^{(1)}\) run away from unity. In fact, for small deviations \(\varDelta w :={| w+1|}\ll 1\) the statefinder parameters depend linearly on \(\varDelta w\) and we find that \(S_{n}^{(1)}<1\) for quintessence models and \(S_{n}^{(1)}>1\) in the case of phantom behaviour. On the other hand, it can be shown that asymptotically

*s*vanishes for \(\varLambda \)CDM, and it gets negative for \(w<-\,1\) and positive for \(-\,1<w\). We have assumed in all our conclusions the presence of radiation no matter how tiny its contribution.

On Fig. 1, we present the evolution of the statefinder hierarchy \(\{S_3^{(1)},\,s\}\) (top panel), \(\{S_3^{(1)},\,S_4^{(1)}\}\) (middle panel) and \(\{S_3^{(1)},\,S_5^{(1)}\}\) (bottom panel) for the three models considered: \(w=-\,0.99\) (blue), \(w=-\,1\) (green) and \(w=-\,1.01\) (red). When the Universe is dominated by radiation and matter the three models are indistinguishable and can be seen to follow the same straight line trajectory in the \(\{S_3^{(1)},\,s\}\), \(\{S_3^{(1)},\,S_4^{(1)}\}\) and \(\{S_3^{(1)},\,S_5^{(1)}\}\) planes. However, as DE starts to dominate at late time the differences between the three models become apparent. The trajectory \(\{S_3^{(1)},\,s\}\) evolves towards the point (1, 0) for the \(\varLambda \)CDM model, then for a quintessence model that trajectory evolves towards the second quadrant in the plane \(\{S_3^{(1)},\,s\}\), i.e. \(S_3^{(1)}<1\) and \(0<s\), and, finally, for a phantom scenario the trajectory \(\{S_3^{(1)},\,s\}\) heads towards the fourth quadrant, i.e. \(1<S_3^{(1)}\) and \(s<0\). For the second group of trajectories, \((\{S_3^{(1)},\,S_4^{(1)}\}\) and \(\{S_3^{(1)},\,S_5^{(1)}\})\), the trajectories of the model with \(w=-\,1\) evolve towards the point (1, 1) that characterises \(\varLambda \)CDM, in the quintessence model the trajectories evolve towards the third quadrant in both panels (\(S_n^{(1)}<1\) for \(n=3,4,5\)). In contrast, for the model with phantom behaviour the trajectories evolve towards the first quadrant in the \(\{S_3^{(1)},\,S_4^{(1)}\}\) and \(\{S_3^{(1)},\,S_5^{(1)}\}\) planes characterised by \(S_n^{(1)}>1\) for \(n=3,4,5\). Finally, by looking at Fig. 1, it seems that the pair \(\{S_3^{(1)},\,S_5^{(1)}\}\) are suitable to distinguish the model with \(w<-\,1\) from \(-\,1<w\).

## 3 Cosmological perturbations: from gravity to DM and DE

^{2}In addition, we will solve the gravitational equations describing the cosmological perturbations at first order using the same methodology we presented in [22]. We remind the reader that the temporal and spatial components of the conservation equation of each fluid imply [22]

*i*) components of the Einstein equations lead to [22]

The behaviour of the gravitational potential and the perturbations is shown in the top panel of Fig. 2 for a given scale. We choose as an example \(k=10^{-3}\) Mpc\(^{-1 }\). As it must, the gravitational potential is constant during the matter era and starts decreasing as soon as dark energy *goes on stage*. This behaviour is independent of the dark energy model considered. However, shortly afterwards, i.e., in our *near future*, the gravitational potential will depend on the specifically chosen EoS for dark energy. In fact: (1) it will decrease until reaching a positive non-vanishing value at infinity for \(w>-\,1\), (2) it will vanish asymptotically for \(w=-\,1\), and amazingly (3) it will vanish and become negative for \(w<-\,1\)! This is in full agreement with the fact that close to the big rip the different structures in our Universe will be destroyed no matter their sizes or bounding energies. When could the gravitational potential vanish and flip its sign? Of course, the answer is model and scale dependent [22]. For the model we have considered, the gravitational potential for the mode \(k=10^{-3}\) Mpc\(^{-1 }\) will vanish in \(8.84\times 10^{10}\) years from the present time or equivalently when the Universe is roughly 213 times its current size. Furthermore, numerical results show that the smaller the scale that is considered (larger *k*), the later the gravitational potential will flip sign [22].

Finally, and most importantly, all these models are in full agreement with observations. In Fig. 3, we show the evolution of the observable \(f\sigma _8\) for the three models mentioned above. This combination of *f*, the relative growth of the linear matter perturbations, and \(\sigma _8\), the root-mean-square mass fluctuation in spheres with radius 8 h\(^{-1}\) Mpc, was proposed in [27] as a discriminant for different models of late-time acceleration that is independent of local galaxy density bias. On the top panel of Fig. 3, we contrast the \(f\sigma _8\) curves of the three models with the available observational data (cf. Table I of [22]). All the three curves, which are practically indistinguishable for the naked eye, are within the error bars of nearly all the points. On the bottom panel of Fig. 3, we present the relative difference, \(\varDelta f\sigma _8\), of the results of each model with regards to \(\varLambda \)CDM.^{3} Despite the small values found in terms of amplitudes, the behaviour observed suggests that the sign of \(\varDelta f\sigma _8\) can distinguish between a phantom (positive \(\varDelta f\sigma _8\)) and a quintessence model (negative \(\varDelta f\sigma _8\)). As a consequence of this difference in sign, the growth of the linear matter perturbations is stronger in a phantom scenario as opposed to \(\varLambda \)CDM and quintessence. This is in full agreement with the results presented in [10] where the decay of the growth suppression factor of the linear matter perturbations is found to be faster in quintessence models and slower in phantom models.

## 4 Concluding remarks

Summarising, what we have shown is that after all gravity might behave the other way around in the future and, rather than the apple falling from the tree, the apple may fly from the earth surface to the branches of the tree, if dark energy is repulsive enough, as could already be indicated by current observations.^{4}

To illustrate these observations, we have considered three models where DE is characterised by a constant parameter of EoS *w* with values \(w=-\,0.99,-\,1,-\,1.01\). After comparing the present and future behaviour at the background level by using a statefinder approach, as illustrated in Fig. 1, we have considered the cosmological perturbations of these models. We have shown that for models with \(w<-\,1\) the gravitational potential changes sign in the future (cf. Fig. 2). We have as well analysed the behaviour of the DM and DE perturbations as shown for example in Fig. 2. Finally, we have proven that no matter the future behaviour of the gravitational potential depicted in Fig. 2, the three models discussed above are in full agreement with the latest observations of \(f\sigma _8\) (cf. Fig. 3).

Before concluding, we would like to recall that in this work, we have considered the existence of phantom matter; however, it might be possible that Nature presents rather a phantom-like behaviour as happens in brane world-models [29, 30] where no big rip takes place and where the perturbations can be stable. In addition, even the presence of phantom matter might not be a problem at a cosmological quantum level where the big rip or other kind of singularities can be washed away [31, 32, 33].

## Footnotes

- 1.
The parameters \(A_{2},\, A_{3},\, A_{4},\, A_{5}\) are also known as the deceleration parameter \(q=-A_{2}\), the jerk \(j=A_{3}\), the snap \(s=A_{4}\) and the lerk \(l=A_{5}\), respectively [13].

- 2.
As long as the speed of sound \(c_s^2\) is not too close to zero and \(w\simeq -\,1\), the value of \(c_s^2\) will not much affect the perturbations of dark matter. A full discussion on the effect of the speed of sound of DE on the perturbations of the late Universe can be found in [24, 25, 26]. Therefore, our choice \(c_s^2=1\) is not crucial in our study, it was taken just for simplicity and because it is common to use it in codes like CAMB and CLASS, though there is no fundamental reason for such a choice.

- 3.
\(\varDelta f\sigma _8^\text {(model)} (\%):= 100[(f\sigma _8^\text {(model)})/(f\sigma _8^{\varLambda \text {CDM}})-\,1]\).

- 4.
Repulsive gravity could occur as well if the effective gravitational constant changes sign. This could happen, for example, in scalar–tensor theories, in particular, for a non-minimally coupled scalar field [28]. However, an anisotropic curvature singularity arises generically at the moment of this transition.

## Notes

### Acknowledgements

The work of IA was supported by a Santander-Totta fellowship “Bolsas de Investigação Faculdade de Ciências (UBI) Santander Totta”. The work of MBL is supported by the Basque Foundation of Science IKERBASQUE. JM is thankful to UPV/EHU for a PhD fellowship. MBL and JM acknowledge financial support from project FIS2017-85076-P (MINECO/AEI/FEDER, UE), and Basque Government Grant no. IT956-16. This research work is supported by the Grant UID/MAT/00212/2013. This paper is based upon work from COST action CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology).

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