A Generalized Interacting Tsallis Holographic Dark Energy Model and its thermodynamic implications

The paper deals with a theoretical model for interacting Tsallis holographic dark energy (THDE) whose infrared (IR) cut-off scale is set by the Hubble length. The interaction Q between the dark sectors (dark energy and pressureless dark matter) of the universe has been assumed to be non-gravitational in nature. The functional form of Q is chosen in such a way that it reproduces well known and most used interactions as special cases. We then study the nature of the THDE density parameter, the equation of state parameter, the deceleration parameter and the jerk parameter for this interacting THDE model. Our study shows that the universe undergoes a smooth transition from a decelerated to an accelerated phase of expansion in the recent past and also this transition occurs within the redshift interval [0.637,0.962]. This is well consistent with the present observations. It is shown the evolution of the the normalized Hubble parameter for our model and compared that with the latest Hubble parameter data. Finally, we also investigate both the stability and thermodynamic nature of this model in the present context.


I. INTRODUCTION
Many cosmological observations indicate that our Universe is now experiencing an accelerated expansion phase [1][2][3][4][5]. A possible candidate to explain this cosmic acceleration is to consider some exotic matter, dubbed as dark energy (DE) which consists of approximately 68% of the total energy budget of our universe. However, the origin and nature of this DE are absolutely unknown. On the other hand, the second largest component of our universe is the dark matter (DM) which takes around 28% of the total energy density of the universe. Like the DE sector, DM sector is also not very well understood. Till now, a large number of theoretical models are taken into account to accommodate the present phase of acceleration and some excellent reviews on this topic can be found in [6][7][8]. However, the problem of the onset and nature of cosmic acceleration remains an open challenge of modern cosmology at present.
It is important to mention that observations admit an interaction between the dark sectors (DM and DE) of cosmos which can solve the coincidence problem and the tension in current observational values of the Hubble parameter . The scenario of interaction between DM and DE is one of such alternative models, which is the main subject interest of the present work. Recently, Zadeh et al. [34] investigated the evolution of the THDE models with different IR cutoffs and studied their cosmological consequences under the assumption of a mutual interaction between the dark sectors of the universe. Following [34], in this work, we are also interested in studying the dynamics of a flat FRW universe filled with a pressureless matter and THDE in an interacting scenario. In particular, we explore consequences of interacting THDE model in a more general scenario. In our setups, we study the evolution of our universe by considering an interaction between DM and THDE whose IR cutoff is the Hubble horizon. The nature of the THDE density parameter, the deceleration parameter, the jerk parameter and the THDE equation of state parameter have also been studied for the present model. Furthermore, we also investigate the stability and thermodynamic nature of this particular model in the present scenario. However, the present work is more general and also different from the similar work by Zhai et al. [34] in different ways. Firstly, in this paper, the functional form of the interaction term is chosen in such a way that it can reproduce some well known and most used interactions (including [34]) in the literature for some special cases (for details, see section II). Secondly, we study the evolution of jerk parameter for this general interaction term. We also plot the normalized Hubble parameter for our model and compared that with the latest Hubble parameter data. Additionally, we go one step further by investigating this scenario taking into account the thermodynamic considerations. In particular, we study the nature of the total entropy of the universe surrounded by a cosmological horizon.
The paper is organized as follows. In the next section, we present a THDE model with Hubble scale as IR cutoff. Additionally, the results of considering a mutual interaction between the dark sectors of the universe are also investigated. In section III, we also explore the thermodynamical properties of the present model. Finally, in section IV, we summarize the conclusions of this work.

II. INTERACTING THDE WITH HUBBLE CUTOFF
The THDE model is based on the modified entropyarea relation [32] and the holographic dark energy hypothesis, was proposed in [33] by introducing the following energy density where B is an unknown parameter. By considering the Hubble horizon as the IR cutoff, i.e., L = H −1 , the energy density corresponding to THDE is obtained as In the large scale, our universe is homogeneous and isotropic and its geometry is best described by the spatially flat Friedmann-Robertson-Walker (FRW) metric where a(t) is the scale factor of the universe. Now, in such a spacetime, one can write down Friedmann equations as [6] where, m p denotes the reduced Planck mass, H =ȧ a is the Hubble parameter and an overhead dot represents derivative with respect to the cosmic time t. Also, ρ m and ρ D represent the energy density of pressureless matter and the THDE density, respectively. The energy density parameter of THDE and pressureless matter can be expressed as where, ρ c = 3m 2 p H 2 denotes the critical energy density. Now, equation (4) can be written as Also, the ratio of the energy densities is obtained as Moreover, we assume that DM and DE interact with each other. Accordingly, the energy conservation equations becomeρ where ω D ≡ pD ρD denotes the equation of state (EoS) parameter of THDE, p D is the pressure of THDE and Q indicates the rate of energy exchange between the dark sectors (DM and DE). Positive value of Q indicates that there is an energy transfer from the THDE to the DM, while for Q < 0, the reverse scenario happens. On the other hand, if Q = 0 (i.e., non-intracting case), then the DM evolve as, ρ m ∝ a −3 . Hence, the interaction between the dark sectors is indeed a more general scenario to unveil the dynamics of the universe. In fact, there are many proposed interactions in the literature to study the dynamics of the universe (for details, one can look into [60][61][62][63][64][65][66][67][68][69] and references therein), however, the exact functional form of Q is still unknown to us. From the continuity equations (9) and (10), one can see that the interaction Q could be any arbitrary function of the parameters ρ m , ρ D and H. So, naturally, one can construct various interacting models to understand the dynamics of the universe in this framework. For mathematical simplicity, in the present work, we assume that the interaction is a linear combination of the dark sector densities given as where, the parameters b 1 and b 2 are dimensionless constants. This type of functional forms of Q has been studied recently by several authors [60][61][62][63][64] and the particular [34] and b 2 = 0, b 1 = b in Ref. [39]. Therefore, the general form of Q, given by equation (11), covers a wide range of other popular theoretical models for different choices of b 1 and b 2 . The simplicity of the functional form of Q (as given in equation (11)), however, makes it very attractive and simple to study. Indeed, as DE and DM have not the same energy density (and hence contribution) within the universe dynamics and as we do not yet know their nature, it is reasonable to consider different contributions (b 1 = b 2 ) for these dark components within the interaction term. Now, taking the time derivative of equation (4), and by using equations (8), (9) and (10), we geṫ Similarly, taking the time derivative of equation (2) along with combining the result with equations (10) and (12), we obtain Taking the time derivative of equation (5) and by using equations (8), (12) and (13), we arrive at the following equation for THDE density parameter, as where Ω ′ D = dΩD d(ln a) . Now, for simplicity, we re-expressed equation (5) as which implies, where, h is the normalized Hubble parameter, Ω 0 D is the present THDE density parameter and H 0 = , denotes the present value of H. Later, using equation (14) along with the above equation, we try to show the evolution of h for this model and will compare it with that of observational Hubble parameter data.
The deceleration parameter is defined as which is an important cosmological parameter to investigate the expansion history of the universe. In particular, q < 0 indicates accelerated (ä > 0) expansion phase of our universe, whereas q > 0 indicates a decelerated phase (ä < 0). In our model, q evolves as . (18) It is well known that the jerk parameter, a dimensionless third derivative of the scale factor with respect to cosmic time, provides a comparison between different DE models and the ΛCDM (j = 1) model. It is given by [70][71][72] Finally, in order to estimate the stability of the model we consider the square of sound speed given as Using then equation (10) along with equations (13) and (14), the above equation can be re-expressed as The sign of v 2 s is important to specify the stability of background evolution. v 2 s > 0 (v 2 s < 0) indicates a stable (unstable) model. It is important to note here that the expressions of q, ω D , Ω D and v 2 s are similar to the results of [34] for the special choice, b 1 = b 2 = b. On the otherhand, if b 1 = b 2 = 0, then the equations (13), (18), (14) and (21) match to the relations derived in [33]. As discussed earlier, thus the present work is more general in the literature. The evolution of q(z), as a function of z, has been plotted in figure 3. From this figure, it is clear that our model can describe the current accelerated universe, and the transition redshift z t (i.e., q(z t ) = 0) from the deceleration phase to an accelerated phase occurs within the interval [0.637,0.962], which are in good agreement with the results, 0.5 < z t < 1, as reported in [71][72][73][74][75][76][77][78][79][80]. The evolution of j(z) has also been plotted in figure 4. It is observed that j stays positive and lies      (16), is shown by considering δ = 1.4 and Ω 0 D = 0.73. In this plot, the black dots correspond to the H(z) data consisting 41 data points [81,82] with 1σ error bars and the corresponding error in h(z) is given as [76], are errors in H and H0 respectively. Also, the value of H0 is taken from [83].
within (0.52-0.58) at late time, and further it tends to unity (or ΛCDM model) as z → −1. This is an interesting result of the present analysis. In figure  5, we have shown the evolution of h (equation (16)) for the present model and compared it with the data points for h(z) (within 1σ error bars) which have been obtained from the latest compilation of 41 data points of Hubble parameter measurements ( for details, see  and Ω 0 D as given in figure 6. This plot is for δ = 2.001. [81,82]). We observed from figure 5 that the model reproduces the observed values of h(z) quite well for each data point. Furthermore, we also checked that the nature of the evolution of h(z) is hardly affected by a small change in the values of the parameters (b 2 1 , b 2 2 ).
For understanding the classical stability of our model, we also plot the square of sound speed in figures 6 and 7. It has been found from figure 6 that the model is unstable (v 2 s < 0). However, the situation is completely different, i.e., v 2 s > 0, for some higher value of δ (see figure 7). Thus, the stability of the interacting THDE model crucially depends on the choice of the parameter δ.

III. THERMODYNAMICS OF INTERACTING THDE
In this section, we derive the rate of change of the total entropy and then examinethe validity of generalized second law of thermodynamics. It is well known that thermodynamical analysis of the gravity theory is an exciting research topic in the cosmological context and the thermodynamical properties which hold for a black hole are equally valid for a cosmological horizon [84][85][86][87][88][89][90][91][92][93][94][95]. In addition, the first law of thermodynamics which holds in a black hole horizon can also be derived from the first Friedmann equation in the FRW universe when the universe is bounded by an apparent horizon. This provides well motivation to select the apparent horizon as the cosmological horizon in order to examine the thermodynamic properties of any cosmological model. Motivated by the above arguments, here, we have considered the universe as a thermodynamic system that is bounded by the cosmological apparent horizon with the radius [89] For a spatially flat universe (k = 0), the above equation immediately give which is the Hubble horizon.
If we consider S f and S h are the entropy of the fluid and the entropy of the horizon containing the fluid, then the total entropy (S) of the system can be expressed as According to the laws of thermodynamics, like any isolated macroscopic system, then S should satisfy the following relationṡ In this context, it is important to mention that the generalized second law (GSL) of thermodynamics and thermodynamic equilibrium (TE) refer to the inequali-tiesṠ ≥ 0 andS < 0 respectively. Furthermore, the GSL should be true throughout the evolution of the universe, while the TE should hold at least during the final phases of its evolution. We shall now examine the validity of GSL of thermodynamics and also the TE in the present context. Now, the entropy of the horizon S h can be derived as [93], where, A = 4πr 2 h and r h are the surface area and radius of the apparent horizon respectively. Also, the temperture of the apparent horizon is given by the relation [93] T h = 1 2πr h .
As previously mentioned, we considered only the DE and DM as the components in the energy budget, so we can write where, S d and S m represent the entropies of the DE and DM respectively, and T is the temperture of the composite matter inside the horizon. Therefore, the first law of thermodynamics (T dS = dE + pdV ) can be written for the individual matter contents in the following form where V = 4 3 πr 3 h , is the horizon volume. Also, E d =  (26) and (30) and (29) with respect to time, we obtain (31) Using equation (31) along with the asumption that the fluid temperture T should be equal to that of the horizon temperture T h [64], one can arrive at the expressioṅ In fact, the above relation has already been established in the context of interacting DE, where DE, DM and radiation are inteacting with each other [95]. Equation (32) by simple algebra takes the form which is always positive definite irrespective of the functional forms of H. Here, a 'prime' represents derivative with respect to x = ln(a). Differentiating equation (33) once more, we obtain where, we define φ = H ′′ H ′ − 2 H ′ H . It deserves to mention here that for a thermodynamic equilibrium, S ′′ < 0, which further implies φ < 0. We now try to understand the behavior of S ′′ (or φ) with the evolution of the universe for the interacting THDE   Figure 8 shows that the GSL is always satisfies in the present context. It is evident from figure 9 that φ undergoes a transition from positive to negative value in the past, where the THDE starts to dominate over the DM. It is also found φ to be negative at the current epoch and also remains negative at future, i.e, z → −1. This indicates towards a TE of the universe.

IV. CONCLUSIONS
In this paper, we have studied an accelerating cosmological model for the present universe which is filled with DM and THDE. The DM is assumed to be interact with the THDE whose IR cut-off scale is set by the Hubble length. As already discussed in section II, the functional form of Q is chosen in such a way that it reproduces well known and most used interactions in the literature for some specific values of the model parameters b 1 and b 2 [34,39,60,[64][65][66].
In our setups, the behavior of various quantities, e.g., Ω D , ω D , q, j, h and v 2 s have been studied during the cosmic evolution. In figure 2, the plot of ω D with z shows ω D < − 1 3 at the present epoch which indicates an accelerating phase of the universe. The evolution of q shows that the universe is decelerating at early epoch and accelerating at present epoch. This explains both the observed growth of structures at the early times and the late time cosmic acceleration measurements. Also, the transition between the DM era and the THDE era takes place within the redshift interval [0.637,0.962], which are in good compatibility with several recent studies [71][72][73][74][75][76][77][78][79][80]. It is also observed that j stays positive and approaches to the ΛCDM (j = 1) model as z → −1. Further, we studied the thermodynamic nature of the universe for this model. The basic motivation was to verify whether our model fulfills the thermodynamical requirements of the expanding universe. Our study shows that the GSL of thermodynamics is always satisfies and also indicates towards a TE of the universe.
Furthermore, we noticed that the stability of our model crucially depends on the choice of the parameter δ (see figures 6 & 7). Therefore, we conclude that for the deep understanding of behavior of interacting THDE, more investigations should be done. In a follow-up study, we would like to study the model by considering other IR cutoffs and some non-linear interaction between the dark sectors, which may modify the properties of THDE.

V. ACKNOWLEDGMENTS
The work of KB was supported in part by the JSPS KAKENHI Grant Number JP 25800136 and Competitive Research Funds for Fukushima University Faculty (19RI017).