Tsallis Holographic Dark Energy Reconsidered

We consider the interacting Tsallis Holographic Dark Energy (THDE), with the Granda-Oliveros (GO) scale as the infrared (IR) cutoff, as dynamical vacuum. We analytically solved for the Hubble parameter, in a spatially flat FLRW universe with dark energy and matter as components, and the solution traces the evolutionary path from the prior decelerated to the late accelerated epoch. Without interaction, the model predicts a $\Lambda$CDM like behavior with an effective cosmological constant. We used Pantheon Supernovae type Ia, observational Hubble data (OHD), cosmic microwave background (CMB), and baryon acoustic oscillation (BAO) data to constrain the free parameters of the model. The estimated values of the cosmological parameters were consistent with observational results. We analyzed the behavior of the model using the statefinder and $\omega^\prime_{e}-\omega_{e}$ plane where $\omega_{e}$ and $\omega^\prime_{e}$ corresponds to the effective equation of state and its evolution, respectively. The model shows a quintessence behavior in general, and the model trajectory ends in a point that corresponds to the de Sitter phase. We performed a dynamical analysis of the model, concluding that the prior decelerated and late accelerated phases are unstable and stable equilibria, respectively. We also investigated the thermodynamical nature of the model and found that the generalized second law remains valid in the dynamical vacuum treatment of the model.


I. INTRODUCTION
After the discovery of the accelerated expansion of the universe [1,2], the desire to understand the universe spired, resulting in intensive works in recent literature. The nature and origin of the accelerated expansion of the universe are a mystery even now. Postulating dark energy models and modified theories of gravity are the two approaches that try to unwind the mystery of the universe's accelerated expansion. One of the simplest and best dark energy candidates is the cosmological constant, Λ as given in the most successful model, ΛCDM. However, this model suffers mainly from two problems; one is the cosmological constant problem where there is a discrepancy in the observational and the theoretical value (10 121 times larger) of the energy density of the cosmological the quantum field theory to describe a black hole [9,10]. The initial proposal of the holographic principle considers the entropy of the cosmological horizon as the Bekenstein-Hawking entropy and the dark energy density scales as the square of the Hubble parameter. However, it could not explain the current accelerated expansion of the universe [11,12]. HDE model with particle horizon [13,14] as IR cutoff also could not explain the present acceleration. Taking future event horizon [14] as IR cutoff successfully explained the present accelerated expansion, but it suffered from causality problem. Since any known symmetry does not dodge the interaction between the dark sectors [15], models considering such interactions were proposed, and they yield better consistency with the cosmic observations than non interacting models [16][17][18][19].
Holographic Ricci dark energy model with Ricci scalar curvature as IR cutoff introduced in [20] is a phenomenologically viable model which avoids causality and coincidence problems. Inspired by this model and on a pure dimensional basis, Granda and Oliveros proposed a new IR cutoff [12], a combined function of Hubble parameter and its time derivative.
Tsallis and Cirto [36] introduced a generalized non additive entropy, popularly known as Tsallis entropy, to solve the thermodynamic inconsistencies in non standard systems like a black hole. The pioneer works on the analysis of dark energy models with Tsallis non extensive statistical formulation can be found in [37] and further possibilities in cosmology are probed in [38]. This kind of entropy agrees well with the Friedmann equations and Padmanabhan's proposal of the emergence of space time [39]. Like in the conventional HDE model, it is possible to construct dark energy models using Tsallis entropy, and as a result, Tsallis holographic dark energy (THDE) with Hubble horizon as IR cutoff was introduced in [40]. Taking inspiration from the aforementioned study, dynamics of FRW universe having dark matter and THDE with the apparent horizon, the particle horizon, the Ricci scalar curvature scale, and the Granda-Oliveros (GO) scale as IR cutoffs was studied considering non interacting and interacting scenarios [41][42][43][44]. It is found that the THDE model with particle horizon as IR cutoff explains the current accelerated expansion of the universe, unlike the corresponding conventional HDE model. The results from [41] show that the THDE model is not always stable for the GO scale and the Ricci scalar cutoffs in both interacting and non interacting cases. Whereas in [42] THDE model with the GO scale as IR cutoff shows stability in ( + 1) dimensional FRW universe. Thermodynamical stability studies of THDE with the apparent horizon as IR cutoff in [45] shows that the model does not satisfy the stability conditions in both interacting and non interacting cases. The investigations on the evolution of the THDE with Hubble horizon as IR cutoff, by considering time varying deceleration parameter in FRW universe is discussed in [46], in Brans-Dicke cosmology is discussed in [47,48]. Geometrical diagnosis of THDE model of the universe with the apparent horizon as IR cutoff, considering the interaction between dark sectors of the universe, was made in [49]. Cosmological model in higher dimensional Kaluza-Klien theory, having THDE with Hubble horizon as IR cutoff, and with Generalized Chaplygin Gas (GCG) as cosmic components are studied in [50]. THDE with Hubble horizon as IR cutoff in Rastall framework and on Randall-Sundrum brane has been considered in [51,52]. Dynamical system studies on interacting and non interacting THDE in a fractal universe with Hubble radius and apparent horizon as IR cutoff can be found in [53][54][55][56]. The equivalence between Tsallis entropic dark energy and generalized HDE with cutoffs in terms of particle horizon, future horizon, and its derivatives are established in [57]. Cosmological analysis of the THDE with Hubble horizon as IR cutoff in the axially symmetric Bianchi-I universe within the framework of general relativity has been explained in [58,59]. Sign changeable mutual interactions between dark sectors are also considered to study the effects of anisotropy in the Bianchi universe [60]. Similar analysis of THDE with Hubble horizon and GO scale as IR cutoff in Bianchi-III universe has been discussed in [61,62]. Investigations on dynamics of THDE with Hubble horizon as IR cutoff,   by assuming power law-exponential form for the scale factor have been studied in [63]. Geometrical evolutionary studies in THDE models with Hubble horizon, future event horizon, and GO scale as IR cutoffs corresponding to different interactions have been explored in [64][65][66][67][68][69][70]. Investigations on THDE with GO scale as IR cutoff [71], presuming that this energy density is responsible for inflation, show the potentiality of the model in explaining the early universe. Comparison of THDE model with other HDE models using statefinder analysis was reviewed in [72]. The evolution of cosmological perturbations in THDE models with Hubble horizon and future event horizon as IR cutoff and Bayesian model comparison with ΛCDM as reference model has been scrutinized in [73][74][75]. In [76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93], the THDE model with different IR cutoffs within various modified gravity theories and scalar field theories has also been explored. The growth rate of clustering for different IR cutoffs for the THDE model in the FRW universe can be found in [94].
Cosmological implications through non linear interactions between THDE with Hubble horizon as IR cutoff and cold dark matter in the framework of loop quantum cosmology has been discussed in [95]. Investigations on the THDE model with Hubble horizon as IR cutoff in the higher derivative theory of gravity in [96] show that it is not compatible with late time acceleration as it could not acquire the required value of the equation of state parameter. Due to the quantified non extensivity in Tsallis entropy, studies on modification of Friedmann equation and gravity theory, including emergence proposal of gravity [97][98][99][100][101][102], also flourish in this field. All these works were carried out considering the equation of state parameter of dark energy as varying with the expansion of the universe.
It is to be noted that, in formulating the HDE density, one has to compare the UV cutoff, corresponding to the vacuum energy, with the IR cutoff, representing the large length scale of the universe. There is a broad consensus that the cosmological constant in the standard ΛCDM model can be the vacuum energy, having an equation of state parameter, −1. Hence, the UV cutoff involved in the HDE models indicates that the cor-responding dark energy density is of dynamical vacuum. As a result, reconsidering the THDE as the dynamical vacuum in analyzing the evolutional history of the universe is of practical significance. We are considering the THDE as the dynamical vacuum in the present work. We study the evolution of the universe by considering the interaction, between dark matter and dark energy in conformity with the total conservation of energy, in the THDE model with the GO scale as IR cutoff. Our analysis shows that the model predicts a transition to the late accelerating universe. This work also involves the geometrical and dynamical analysis and thermodynamical study of the model to check the feasibility of explaining the accelerating universe.
The structure of this paper is as follows. In the next section, we present an interacting THDE as a dynamical vacuum.
We analytically solve for the Hubble parameter and investigate its evolutionary behavior. In Sect. III, we constrain the parameters with observational data and discuss its cosmological implications. Along with that, we also analyze the evolutionary trajectory of the model in geometrical plane − and phase plane − plane. In Sect. IV, we perform the dynamical analysis on the interacting THDE model. In Sect. V, we study the thermodynamical properties of the model. In the last section, we summarize the conclusions of the work.

II. INTERACTING THDE MODEL AS DYNAMICAL VACUUM
A generalization of the Boltzmann-Gibbs (BG) theory, now known as the non extensive statistical mechanics, was proposed [32] to address the complexities in non standard systems. For large scale systems, the thermodynamical entropy must be modified to non additive entropy [36]. According to Tsallis and Cirto [36], the quantum correction modified the entropy area relation as, where is the horizon area of the black hole, is a positive [97] constant, and is the positive non additive parameter [36].
This will reduces to the Bekenstein entropy for = 1 4 2 and = 1, with 2 as the Planck length. Following the holographic principle, Cohen et al. [9] have found a relation between the entropy, IR cutoff (L), and the UV cutoff (Λ) as, Following the Tsallis entropy in (1) and substituting for area,  In the present study we adopt the GO scale as IR cutoff, which was originally proposed in reference [12] to study the conventional HDE model and is given by, where and are unknown dimensionless constants and , is the derivative of Hubble parameter with respect to the cosmic time. Using (4) in (3), THDE density can be written as The Friedmann equation for the flat FRW universe is given by where and is the dark matter and dark energy density respectively. The conservation equations including the interaction between THDE and dark matter are given by where and are the derivatives of dark energy and dark matter densities with respect to the cosmic time, and are the pressure of dark energy and matter respectively and represents the interaction, which determines the rate of exchange of energy between the dark sectors. From equation (7) it is clear that has to be a function of energy density and inverse of time. We are adopting a simple function, = 3 where is the coupling constant. None of the previous works has considered this form of interaction in combination with the GO scale in studying the evolution of the FLRW universe.
Since the THDE is considered as a dynamical vacuum, its equation of state is = − and the matter is considered as pressureless. Considering the above assumptions, the equations in (7) reduces to The above equations (8) can be rewritten in terms of density where 0 denotes the present value of Hubble parameter and = ln as The solution of equation (10) is is the present matter density parameter. In sim- is the present dark energy density parameter. The equation (VI) reduces to, 1 = Ω 0 + Ω 0 , for = 0 . Considering this result and using equations (10), (9) and (6), a second order differential equation can be formulated as below where ℎ 2 = 2 / 2 0 . The solution of the above differential equation in terms of scale factor is [103], The constants are then obtained as In the asymptotic limit → 0, the constant can be neglected due to domination of the first two terms in the Hubble parameter equation (12) for 1, consequently the resulting solution represents the decelerated expansion. In the future limit, → ∞, the constant term in the Hubble parameter will dominate over the rest of the terms, indicating an end de Sitter phase. Hence the model predicts a transition into a late accelerating epoch in the evolution of the universe. For = 0 equation (12) will reduce to whereΩ 0 andΩ 0 are the mass density parameters [104] which have the form This shows that, in the absence of interaction, the present model is similar to that of the standard ΛCDM, with an effective cosmological constant corresponding to the mass density parameterΩ 0 . Even though the dark energy density in equation (5)  to the value of model parameters.

A. The model parameter estimation
In this section, we estimate the constant model parameters, , , , , 0 and Ω 0 by contrasting the model with cosmological observational data. The dataset consists of the type Ia supernovae data [108], observational Hubble data (OHD) [109,110], cosmic microwave background (CMB) data [111], and baryon acoustic oscillation (BAO) data [112]. We have applied the Markov chain Monte Carlo (MCMC) method by employing the emcee python package [113] using the lmfit python library [114] to constrain the model parameters.
The theoretical distance modulus of SN Ia is given by The 2 function of SN Ia data can be expressed as where is the observational Hubble parameter measurement, Combining the type Ia supernovae data and the OHD, the total 2 function takes the form The estimated values of the model parameters using the SNIa data and OHD are given in the third row of where * is the redshift at the photon decoupling epoch. We adopt the distance prior measurement value = 1.7502 ± 0.0046 at the redshift * = 1089.92 from the Planck 2018 observations [111]. The corresponding 2 [117] from the CMB data is where is the variance of the measurement.
The estimated values of the model parameters using the SNIa+OHD+CMB are given in the fourth row of The distance-redshift relation determined by a BAO measurement is given by the acoustic peak parameter, , which is defined in terms of THDE model parameters as follows where is the redshift of the acoustic peak parameter. We adopt the value of = 0.484 ± 0.016 at the redshift = 0.35 from the SDSS-BAO distance data [118]. The corresponding 2 from the BAO data is where is the variance of the measurement.
The estimated values of the model parameters using the SNIa+OHD+BAO are given in the fifth row of Table I The estimated values of the model parameters using the type Ia supernovae data, the OHD, the CMB, and the BAO data are given in the sixth row of Table I Chatelier-Braun principle [55,119]. The uncertainty on is relatively high compared to the value at which the maximum likelihood function peaks for all datasets. Evidence for such plausibility using SNIa and OHD can be observed in the work of C. P. Singh [120] on Holographic dark energy. The best fit value of obtained using all the datasets is similar to the previous results from THDE models with future event horizon as IR cutoff [105,115], which are stable against the background perturbations. Concurrently, it is smaller than the best fit value ( > 2) obtained from the studies on the THDE model with Hubble horizon as IR cutoff [115], which is unstable against the background perturbation. Furthermore, it is greater than the best fit value (less than 1) obtained from the study on the growth of matter fluctuations of the THDE model [73] with future event horizon as IR cutoff using the Gold-2017 dataset of 18 uncorrelated 8 measurements, SNIa data, and OHD.

B. Evolution of cosmological parameters
Estimating luminosity and redshift from the cosmological observables contributes to determining the present Hubble pa- The deceleration parameter which measures the rate of cosmic expansion is obtained in the following form In the limit → −1, the deceleration parameter → −1 and in the limit → ∞, the deceleration parameter will tend to a positive value since second term in equation (28)  Planck data [122] and the derived value Ω = 0.29 ± 0.07 from WMAP results [128]. Our estimation of the current values of the matter density parameter are similar to the current value obtained for past THDE models [105,115] and slightly higher than the value for the THDE model with future event horizon as IR cutoff [73].

C. The age of the universe
The age of the universe can be estimated using the present cosmological observational data, even though systematic and statistical uncertainties will arise during observation and estimation. Theoretically, considering the interacting THDE model, the age of the universe can be calculated by slightly rearranging the equation (12) and the resultant equation takes the form Using the best estimated values of the parameters, the age of the universe is evaluated as 14 [107,115]. Our results are closer to the standard value of age 13.8 ± 0.02 Gyrs obtained from Planck mission and 13.72 ± 0.12 Gyrs from WMAP + BAO + SN data assuming ΛCDM model [121,122] and 13.5 +0. 16 −0.14 (stat.) ±0.23(0.33)(sys.) Gyrs from the oldest globular cluster [129].

D. Evolution in − plane and − plane
To check further, the reliability of THDE model being a generalized model of dark energy in contrast to the present observational data we studied the evolution using the geometrical pair called the statefinder pair { , } constructed from the scale factor and its derivatives, first defined by Sahni et al. [130,131]. The relationship connecting the statefinder pair and the scale factor of the universe are given by It is lucid from (30), is third order derivative of ' ' and is related to ' ' and ' ' linearly. The statefinder pair { , } can also be expressed in terms of Hubble parameter and its derivatives as Using equation (12) in the above equations will results in the following two expressions in terms of model parameters The parametric plot of { , } is obtained as shown in Fig. 6. The statefinder evolutionary trajectory of the non interacting THDE model with the Hubble horizon as IR cutoff in the flat FRW universe in the past works [65,66,106] shows that for 1 < < 2, the model is quintessenc like and approaches ΛCDM in the future. The interacting THDE model with the Hubble horizon as IR cutoff in flat FRW universe in past works [49,65] shows Chaplygin gas like behavior for 1 < < 2 and the distance to the ΛCDM point from the present time will be larger or smaller depending on the corresponding larger or smaller value of coupling constant. In contrast to these re- In addition, we have obtained the effective equation of state parameter using the relation, where and are the effective pressure and energy density respectively. Using the equation (12) in (33) the effective equation of state parameter for this model takes the following The asymptotic limit of this is as follows. As → −∞, The evolutionary trajectory of the model in the phase plane of − , the effective equation of state parameter and its derivative, is depicted in Fig. 7 The autonomous equations in terms of phase space variables can be obtained using the Friedmann equation as, in which and are small compared to˜and˜. Linearizing the set of equations (37) and (38) with respect to and , will result in a matrix equation where 2 × 2 matrix on the right hand side of the above equation is the Jacobian at the critical points corresponding to the autonomous system and the partial derivatives are calculated about the critical points (˜,˜). Universe undergoes a de Sitter expansion at this point. Our analysis shows that the system emerges from a decelerated expansion and ends on a de Sitter epoch. The previous works [64,107] in the dynamical system analysis of interacting and where is the speed of light and is the Boltzmann constant.
The evolution of with respect to the scale factor is shown in Fig. 10. The rate of change of horizon entropy with respect to scale factor is given by The second derivative of horizon entropy with respect to the The numerical simulations and analysis of the observational data on Hubble parameter shows that < 0 and > 0 [134][135][136][137]. Fig. 11 clearly shows that the horizon entropy satisfies ≥ 0, always, which in turn implies that the horizon entropy is always increasing. Finally, in the asymptotic limit → ∞, the entropy approaches zero, consequently horizon entropy attains a constant value. That is, the horizon entropy asymptotically reaches constant at the end de Sitter epoch. The behavior of with scale factor in Fig. 12 guarantees the convexity, thereby ensuring that entropy does not grow unboundedly.
The GSL of thermodynamics stipulates that the entropy of the horizon together with the entropy of the matter inside the horizon must always increase with time [138]. Since it is clear the matter entropy is much less than the horizon entropy (smaller in the order of 35) [139,140], the total entropy of the universe can approximately be taken as the horizon entropy [141]. Since the horizon entropy is an increasing function as shown in Fig. 10 and Fig. 11, the GSL is considered to be satisfied. Certain past studies [55] of interacting THDE model with Hubble horizon as IR cutoff with the form of interaction , where 2 1,2 are the coupling constants, and with the varying equation of state of DE, shows the plausibility of violation of GSL depending on the evolution of the universe. Unlike those studies, the present analysis by considering DE as dynamical vacuum guarantees the validity of GSL throughout the evolution of the universe.
As evident from the equation (16), when the interaction parameter = 0, the present model reduces to a model, like ΛCDM with an effective cosmological constantΩ 0 , and the value of which is around 0.735 and 0.729 for the best estimated parameter values using the fourth and sixth datasets, respectively. Results from various studies, like in [142,143], support that a ΛCDM like model would satisfy the GSL. Following this, it can be concluded that the present model with = 0 will satisfy the GSL throughout the evolution.

VI. CONCLUSION
Recently much interest has been arisen in the new holo-