# Note on Tsallis holographic dark energy

## Abstract

We explore the effects of considering various infrared (IR) cutoffs, including the particle horizon, the Ricci horizon and the Granda–Oliveros (GO) cutoffs, on the properties of Tsallis holographic dark energy (THDE) model, proposed inspired by Tsallis generalized entropy formalism (Tavayef et al. in Phys Lett B 781:195, 2018). Interestingly enough, we find that for the particle horizon as IR cutoff, the obtained THDE model can describe the late time accelerated universe. This is in contrast to the usual HDE model which cannot lead to an accelerated universe, if one considers the particle horizon as the IR cutoff. We also investigate the cosmological consequences of THDE under the assumption of a mutual interaction between the dark sectors of the Universe. It is shown that the evolution history of the Universe can be described by these IR cutoffs and thus the current cosmic acceleration can also be realized. The sound instability of the THDE models for each cutoff are also explored, separately.

## 1 Introduction

There are various cosmological observations which indicate that our Universe is now experiencing an accelerated expansion phase [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The origin of this accelerated phase is attributed to a mysterious matter which is called dark energy (DE). For reviews on the DE problem and the modified gravity theories, which is called geometric DE, to account for the late-time cosmic acceleration, see, e.g., [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In this regard, the holographic dark energy (HDE) is an interesting attempt which can address this bizarre problem in the framework of quantum gravity by using the holographic hypothesis [27, 28, 29]. This model is in agreement with various astronomical observations [30, 31, 32, 33, 34], and various scenarios of it can be found in [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. Horizon entropy is the backbone of the HDE models, and hence, any change of the horizon entropy affects the HDE model. Another important player in these models is the IR cutoff, and indeed, the various IR cutoffs lead to different HDE models [35, 36, 40, 41].

Since gravity is a long-range interaction, one can also use the generalized statistical mechanics to study the gravitational systems [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. In this regard, due to the fact that the black hole entropy can be obtained by applying the Tsallis statistics to the system [61, 62, 63], three new HDE models with titles THDE, SMHDE and RHDE have recently been proposed [1, 64, 65]. Among these three models, in the absence of an interaction between the cosmos sectors, RHDE, based on the Renyi entropy and the first law of thermodynamics, shows more stability by itself [65]. In fact, in a noninteracting universe, while SMHDE is classically stable whenever SMHDE is dominant in the universe, THDE, built using the Tsallis generalized entropy [78], is never stable at the classical level [1, 64]. It is also worth mentioning that a THDE model whose IR cutoff is the future event horizon has been studied in a noninteracting universe showing satisfactory results [79].

On the other side, the cosmological observations admit an interaction between the two dark sectors of cosmos including DE and DM [80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90]. The existence of such mutual interaction may provide a solution for the coincidence problem [88, 89, 90, 91, 92, 93, 94, 95, 96]. In the present work, we are interested in studying the dynamics of a flat FRW universe filled with a pressureless source and THDE in both interacting and non-interacting cases. In order to build THDE, we shall employ various IR cutoffs, including the apparent and the particle horizons together with the GO and the Ricci cutoffs.

The organization of this paper is as follows. In the next section, we study the evolution of the Universe by considering an interaction between DM and THDE whose IR cutoff is the apparent horizon. Thereinafter, a new THDE is built by employing the particle horizon as the IR cutoff, and then, the cosmic evolution are investigated for both interacting and non-interacting universes in Sect. 3. The cases of the GO and Ricci cutoffs are studied in Sects. 4 and 5, respectively. The last section is devoted to a summary and concluding remarks.

## 2 Interacting THDE with Hubble cutoff

*B*is an unknown parameter. We consider a homogeneous and isotropic flat Friedmann–Robertson–Walker (FRW) universe which is described by the line element

*a*(

*t*) is the scale factor. The first Friedmann equation takes the form

*Q*denotes the interaction term between DE and DM. Throughout this paper, \(Q=3b^2 H(\rho _m+\rho _D)\), where \(b^2\) is a coupling constant, is considered as the mutual interaction between the cosmos sectors [89, 90]. The ratio of the energy densities is also evaluated as

*q*versus \((1+z)\) have been plotted in Figs. 1, 2 and 3. From these figures, one can see that \(\omega _D\) can cross the phantom line, and moreover, the value of the transition redshift is increased as a function of \(b^2\). Finally, we explore the stability of the THDE model as

## 3 THDE with particle horizon cutoff

### 3.1 Non-interacting

*q*and the squared speed of sound (defined in Eq. (15)), are also founded out as

*q*and \(\omega _D\), the model is not stable.

### 3.2 Interacting

*q*and stability for the model are calculated as

*z*. In addition, comparing Figs. 13 and 6 with each other, we observe that the changes in the density parameter of interacting case is slower than the noninteracting case. Moreover, Figs. 14 and 15 indicate that the model behaves as the phantom source, and thus, the model eventually enters the accelerated phase with the EoS for the universe being less than \(-1\) (or equally \(q<-1\)).

## 4 THDE with GO horizon cutoff

### 4.1 Non-interacting

In Figs. 18, 19, 20, 21, the system parameters including \(\omega _D\), *q*, \(\Omega _D\) and \({v}^{2}_{s}\) are plotted for some values of \(\alpha \), \(\beta \) and \(\delta \) by considering the initial conditions \(\Omega ^{0}_D=0.73\) and \(H(a=1)=67\). It is interesting to note here that the model begin to show stability from itself whenever \(q\rightarrow \frac{1}{2}\). Moreover, the depicted curves are some of those which do not cross the phantom line for \(z\ge -1\).

### 4.2 Interacting

*q*and \(\omega _D\) have the same form as those of the non-interacting case meaning that the mutual interaction does not affect them. Thus, we only need the \({\Omega }^\prime _D\) and \(v_{s}^{2}\) parameters evaluated as

## 5 HDE with Ricci horizon cutoff

### 5.1 Non-interacting

*q*, we rewrite Eq. (42) as

It can be seen from Figs. 24, 25, 26, 27 that the current accelerated universe can be achieved. The \(\lambda =1\) case is interesting, because unlike SMHDE [64], this model is stable (unstable) for \(q>0\) (\(q<0\)). Hence, since the Ricci horizon is a special case of the GO cutoff, the GO cutoff can also produce the same results if proper values for the system unknown constants have been chosen.

### 5.2 Interacting

*z*for some values of the unknown constants. It is obvious that the system parameters affected by the mutual interaction. It is also interesting to note that while \(v_{s}^{2}\) was not negative for \(\lambda =1.5\) in the non-interacting case, here, we always have it is not stable for all values of \(v_{s}^{2}<0\).

## 6 Closing remarks

In the shadow of the holographic principle and based on the non-additive generalized Tsallis entropy expression [78], a new holographic dark energy model called THDE has recently been proposed [1]. In this paper, by considering various IR cutoffs, including the particle horizon, the Ricci horizon and the GO cutoff in the background of the FRW universe, we investigated the evolution of the THDE models and studied their cosmological consequences. We found out that when the particle horizon is considered as IR cutoff, then the THDE model can explain the current acceleration of the universe expansion. This is in contrast to the usual HDE model which cannot lead to an accelerated universe, if one consider the particle horizon as IR cutoff [35, 36]. We also explored the sound stability of the THDE models with various cutoffs. In this manner, the assumed mutual interaction between the cosmos sectors makes the model to be stable for some values of the redshift parameter *z*. For the GO and the Ricci horizon cutoffs, we found out that although acceptable behavior for some parameters of the system, including *q*, the density parameter and \(\omega _D\), are achievable, the model is not always stable. Finally, we have explored the effects of considering a mutual interaction between the two dark sectors of the universe on the behavior of the solutions.

## Notes

### Acknowledgements

We thank Shiraz University Research Council. This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Iran. The work of KB was supported in part by the JSPS KAKENHI Grant Number JP 25800136 and Competitive Research Funds for Fukushima University Faculty (18RI009).

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