Cosmic Consequences of Kaniadakis and Generalized Tsallis Holographic Dark Energy Models in the Fractal Universe

We investigate the recently proposed holographic dark energy models with the apparent horizon as the IR cutoff by assuming Kaniadakis and generalized Tsallis entropies in the fractal universe. The implications of these models are discussed for both the interacting (Γ = 3Hbρm) and noninteracting (b2 = 0) cases through different cosmological parameters. Accelerated expansion of the universe is justified for both models through deceleration parameter q. In this way, the equation of state parameter ωd describes the phantom and quintessence phases of the universe. However, the coincidence parameter ~r =Ωm/Ωd shows the dark energyand dark matter-dominated eras for different values of parameters. It is also mentioned here that the squared speed of sound gives the stability of the model except for the interacting case of the generalized Tsallis holographic dark energy model. It is mentioned here that the current dark energy models at the apparent horizon give consistent results with recent observations.


Introduction
To acquire a unified understanding of various entropy measures and how they connect to each other in a generalized form, it is required to recall characteristics of "classical" entropies. In information theory, information can be collected through the probability distribution of some events that belong to the sample space of all possible events which are called entropies. Gibbs was the first who stated a hypothesis [1] which was the source of inspiration for people to define numerous entropies [2][3][4]. We hear and read frequently as it is claimed due to Gibbs that the black hole (BH) entropy is proportional to the area of the BH boundary in spite of having proportionality to the volume of BH. In recent times, these entropies have been the source for the modeling of cosmic evolution in different setups [5][6][7]. It is a matter of fact that to retrieve thermodynamical extensivity for nonstandard systems, the entropies generalizing that of Boltzman-Gibbs (BG) become necessary. For the study of BH, generalized entropies have been employed [8][9][10][11], also for the construction of new holographic dark energy (HDE) models [12,13]. Besides this, it has been revealed that such kind of entropies can affect the Jeans mass [14], can provide a theoretical basis for the modified Newtonian dynamics (MOND) theory [15], may be inspired by the quantum features of gravity [16], and even may illustrate inflation without assuming inflation [17]. The foundation stone of primary HDE [18] is a holographic principle, and it is proposed on Bakenstein entropy [19][20][21]. It is observed that the Hubble horizon is a proper casual boundary for the universe meeting thermodynamics and conservation laws [22][23][24][25]. However, HDE suffers from some problems when the Hubble horizon is considered the IR cutoff [18,26]. Some other HDE models based on generalized entropy can give a considerable description of accelerated expansion of the universe even when the apparent horizon is used as the IR cutoff [12,13,27]. Consequently, more suitable models of HDE may be found using generalized entropies.
The Tsallis entropy is a generalized form of BG entropy which was presented in 1988 by Constantino Tsallis [28] as a fundamental to generalize the standard statistical mechanics. In the literature, there is a wide debate regarding the physical relevance of Tsallis entropy [29,30]. However, at the start of the 21st century, there is an identified increasing wide spectrum of an artificial, natural, and social complex system which certifies the consequences and predictions that resulted from this nonadditive entropy (i.e., nonextensive statistical mechanics). In this regard, one of the most precise investigated frameworks is developed by Kaniadakis which is the Tsallis nonextensive statistical mechanics and the generalized power law statistics [28].
A lot of work has been done by Dubey et al. [31], Sharma et al. [32][33][34], Srivastava et al. [35], and Ghaffari et al. [17,36] on cosmic expansion in various theories of gravity by using recently obtained DE models such as new Tsallis HDE (NTHDE), Réyni HDE (RHDE), and Barrow HDE (BHDE). They have made versatile studies on accelerated expansion of the universe through various cosmological parameters and planes and found consistent results with recent Planck's data [37]. In recent times, the Kaniadakis statistics have been studied as generalized entropy measures [2,4] with some gravitational and cosmological consequences [7,38]. In view of these generalized entropies, HDE models have been developed by Moradpour et al. [11]. They have examined the deceleration parameter, EoS parameter, and coincidence parameter for these models and found consistent results with recent Planck's data. It is suggested that generalized entropies must obey fundamental laws of thermodynamics such as the zeroth law [39][40][41][42][43]. The above arguments and work done are the sources of motivation due to which we are going to examine BH entropy in the different well-known generalized entropy formalisms and study their capability in representing the current accelerated expansion of the universe by formulating their corresponding HDE models.
In the next section, Tsallis entropy of BH will be calculated using the relation of Boltzmann and Tsallis entropies. Additionally, Kaniadakis entropy of BH will be computed using the relationship of Kaniadakis statistics with Tsallis entropy. Moreover, Sharma-Mittal and Rényi entropies of BH will also be discussed by applying their relation with Tsallis entropy. In Section 3, the Kaniadakis holographic dark energy (KHDE) will be discussed along with some of its cosmological consequences such as deceleration parameter q, EoS parameter ω K d , and the dimensionless ratio called coincidence parameterr = Ω m /Ω K d for both the interacting and noninteracting cases. The stability of the achieved model for both the interacting and noninteracting cases is also analyzed by the squared speed of sound C 2 s . In Section 4, the NTHDE of BH will be discussed. We will find some cosmological parameters using this model for both the noninteracting and interacting cases such as deceleration parameter q, EoS parameter ω T d , and the dimensionless ratio called coincidence parameterr = Ω m /Ω T d . The squared speed of sound C 2 s will also be evaluated to investigate the stability of the model taking both the interacting and noninteracting cases into account. In Section 5, achieved results are compared with the observational data and some concluding remarks about our work have been discussed.

Tsallis and Kaniadakis Entropies of BH
Both the Gibbs [1] and Shannon [44] entropies of distribution with W states working in the unit k B = 1 leads to the relation explicitly dealt in [11]: where P i represents the probability of occupying the ith state for the classical system. The so-called von Neumann entropy which is a quantum mechanical form of this entropy is represented as The utilization of Equation (2) for the classical system goes back to the proposal of Boltzmann, where ρ represents the state density in the phase space [45]. One can obtain the so-called Bekenstein entropy (S BH = A/4) by applying Equation (2) to a purely gravitational system [19]. Since degrees of freedom are disseminated on the horizon without any preference w.r.t. one another, one may consider that P i is equal for all of them [20,21] permitting us to write P i = 1/W. In such manner, Equations (1) and (2) lead to the Boltzmann entropy (S = ln ðWÞ), and thus, we have [7] for horizon entropy and accordingly WðAÞ. As a unique free parameter generalized entropy, the Tsallis entropy is defined as [4] where Q is an unknown free parameter named as the nonextensive or Tsallis parameter (S T Q → S for Q → 1). When the probability distribution meets the conditions P i = 1/W, Equation (4) yields The quantum features of gravity [7,46] are also a source for the existence of Q parameter. Now, taking δ = 1 − Q and utilizing Equation (3) with Equation (5), one can find easily In the scenario of loop quantum gravity, it is acquired by applying the Tsallis entropy definition to BH [11,47] that 2 Advances in High Energy Physics satisfying S T Q → S whenever Q → 1 and β = ln ð2Þ/π ffiffi ffi 3 p [11,47]. Furthermore, Equation (6) and (7) become accordingly the same when we consider β = ln ð2Þ/π ffiffi ffi 3 p . Another single-free parameter generalized entropy is Kaniadakis entropy (K-entropy) [2,3,11], defined as where K represents an unknown parameter, and the limit K → 0 is a way to obtain Boltzmann-Gibbs entropy [2,3]. Comparing Equation (8) with (4) and (5), it can easily obtain Furthermore, by taking P i = 1/W, Equation (8) assists in getting [2,3,11] Combining Equation (10) with Equation (3) yields It is observed that Sharma-Mittal and Rényi entropies can be obtained as a function of Tsallis entropy as [13,48] which leads to where R is an unknown parameter. It is suggested that β = ln ð2Þ/π ffiffi ffi 3 p ; otherwise, ðln ð2Þ/βπ ffiffi ffi 3 p ÞS BH would occur in mathematical results rather than S BH .

Kaniadakis Holographic Dark Energy
As it was claimed by the HDE hypothesis that if the current accelerated universe is driven by vacuum energy, then its total amount stored in a packet with size L 3 should not go beyond the energy of BH having the same size as it [18]. By keeping in mind this, one can generate the following relation in view of Kaniadakis entropy (11) as for the vacuum energy ρ K d . Now, taking the Hubble horizon of the cosmos as the IR cutoff (i.e., L = 1/H ⇒ A = 4π/H 2 ), we obtain where the constant c 2 is unknown [18], κ belongs to a set of real numbers [3], and H = _ a/a is the Hubble parameter. Now, it is clear that we have ρ K d → 3c 2 H 2 /8 (the wellknown Bekenstein entropy-based HDE) when k → 0 [18]. Considering the pressureless fluid (with energy density ρ m ) and the dark energy candidate (with pressure p d and density ρ K d ), the energy-momentum conservation laws for the fractal universe take the form where the "dot" represents the derivative w.r.t. cosmic time t, and the phenomenal term Γ represents the interaction between dark matter and DE (it also gives the flow of energy between the two fluids) and has different mathematical values for both the linear and nonlinear cases, among which some linear cases are Γ 1 = 3Hb 2 ðρ m + ρ d Þ, Γ 2 = 3Hb 2 ρ m , and Γ 3 = 3Hb 2 ρ d [49][50][51][52][53][54] while some nonlinear cases are Γ 4 = 3 [53,54] with b 2 being the coupling constant. We have chosen Γ = 3Hb 2 ρ m [49] as it is simple and leads to precise results. The fractal profile is either timelike or spacelike. We have chosen the timelike fractal profile in the power law form as υ = a −γ [36,55] with a being the scale factor depending upon cosmic time t and γ being a positive constant. In the fractal universe, the Friedmann equations can be obtained as where □υ = ð1/ ffiffiffiffiffiffi −g p Þ∂ μ ð ffiffiffiffiffiffi −g p ∂ μ υÞ with simplified relation as From simplification of (16), one can find 3 Advances in High Energy Physics where Δ = 3 − γ − 3b 2 , z represents the redshift parameter, H o is the value of the Hubble parameter at t = 0, and Ω m o = 8πρ m o /3H 2 o with ρ m o being a constant of integration. The scope and importance of cosmological parameters is increasing day by day as they are favorable tools to analyze and track the history and evaluation of the universe. The parameter q is one which decides whether the universe is facing accelerated expansion or not (i.e., q < 0 gives the accelerated expansion of the universe while q > 0 when the universe has decelerating expansion behavior) [56,57]. The equation of state (EoS) parameter ω d is one which decides the phases of the cosmos (i.e., ω d < −1 represents the phantom phase of the universe, and −1 < ω d < −1/3 describes the quintessence phase while ω d > −1/3 gives the vacuum phase of the universe) [58,59]. The squared speed of sound C 2 s is another important cosmological parameter which decides whether the model is stable or not (i.e., C 2 s > 0 describes the stable model while C 2 s < 0 only when the model is unstable) [57,58,60]. The ratio of Ω m = ρ m /ρ c and Ω d = ρ d /ρ c called the coincidence parameter given byr = Ω m /Ω d = ρ m /ρ d decides the dark energy-and dark matter-dominated eras of the universe (i.e., 0 < Ω m /Ω d < 1 describes the dark energydominated era, and Ω m /Ω d > 1 gives the dark matterdominated era) [61]. In the upcoming, we will discuss these cosmological parameters.
Differentiating (18) with respect to cosmic time t and substituting all necessary values in it, one can get Using the transformation d/dt = −ð1 + zÞHðd/dzÞ from cosmic time t to the redshift parameter z, simplification of (21) gives the relation for H in terms of the redshift parameter as where the "prime" denotes the derivative with respect to redshift parameter z. The deceleration parameter q which is of great importance to decide the accelerated expansion of the universe can be found as Substituting the corresponding values and simplifying (17), we obtain The mathematical formalism for the EoS parameter is ω K d = p K d /ρ K d , and the relation for ω K d can be obtained by using (15) and (24) in this formula as Some other important mathematical formulas of cosmological consequences which we will use later on are The ratio Ω m /Ω K d which is of great importance to decide about the dark energy-and dark matter-dominated eras of the universe for the Kaniadakis entropy content of BH is given by Advances in High Energy Physics The stability of the system is examined by a perturbed parameter called the squared speed of sound (C 2 s ). The mathematical formalism for this parameter is given by where ρ eff = ρ m + ρ K d and p eff = p K d . Differentiating these relations with respect to the redshift parameter leads to the following mathematical result: Variation of q against z has been plotted for interacting ( Figure 1) and noninteracting (Figure 2) cases, respectively. By considering κ = 1400,2500,3600 with fixed values of other parameters as Ω m o = 0:315, b 2 = 0:5, H o = 67:9, ω = −0:3, c 2 = 0:313, and γ = 0:127, we obtain the cosmic acceleration phase in both cases. Also, the values of the deceleration parameter lie in the range ½−1, 0Þ which is a compatible range with observational data. In Figure 3, EoS parameter ω K d for the KHDE model is plotted versus deceleration parameter q for the interacting case while in Figure 4, the same is plotted for the noninteracting case. The evolved constant parameters are taken, the same as in Figures 1 and 2. The phantom phase of the universe is observed for the interacting case. For the noninteracting case, the quintessence phase is achieved for κ = 1400, 2500 and the phantom phase for κ = 3600. Moreover, the EoS parameter ω K d → −1 as z → −1 which coincides with the ΛCDM model. In Figures 5 and 6, the coincidence parameterr = Ω m /Ω K d is plotted against q for interacting and noninteracting cases, respectively. The dark energydominated era is recovered for the interacting case while for the noninteracting case, when κ = 1400, the interval − 0:95 < z < 0:65 gives the energy-dominated era while z > 0:65 results in the matter-dominated era; for κ = 2500, the matter-dominated era is obtained when z > 0:83, and for κ = 3600, the matter-dominated era is achieved when z > 1:1. The squared speed of sound C 2 s which decides the stability of the model is examined for both cases (interacting and noninteracting) in Figures 7 and 8, respectively. For the

New Tsallis Holographic Dark Energy
Considering the pattern of (14) and (15) and taking Tsallis entropy (6) into account, it is easy to have NTHDE as where T 2 is an unknown real number [18]. Taking D 2 = πT 2 and the apparent horizon as the IR cutoff (H = 1/L), we have in which X = δπ/H 2 , where δ belongs to a set of real numbers [11]. Substituting all required assumed and obtained values in (29), we get where λ = exp ðπδ/2H 2 Þ sinh ðπδ/2H 2 Þ. Considering all assumptions, differentiating (18) w.r.t. cosmic time t, and putting necessary values in it, we reach the following result:   Advances in High Energy Physics The deceleration parameter q for the NTHDE model is obtained as Substituting the concern values from the NTHDE model and simplifying Equation (17) yield The relation for EoS parameter ω T d is obtainable after simplifications of Equations (32) and (35) as The coincidence parameter for the obtained model is

Advances in High Energy Physics
The relation for a perturbed parameter called the squared speed of sound which is given in (28) is obtainable as where β = exp ðπδ/2H 2 Þ cosh ðπδ/2H 2 Þ.
The graph of deceleration parameter q against redshift parameter z is plotted in Figures 9 and 10 for interacting and noninteracting cases, respectively. Constant parameters are δ = 4100,4700,5300, Ω m o = 0:315, b 2 = 0:5, H o = 67:9, ω = −0:3, D 2 = 0:313, and γ = 0:127. Required results (accelerated expansion) of the universe are achieved in both cases. EoS parameter ω T d for the NTHDE model is plotted versus deceleration parameter q for interacting and noninteracting cases in Figures 11 and 12, respectively. The involved parameters are taken, the same as in Figures 9 and 10. The phantom phase of the universe is observed for the interacting case, and the quintessence phase is achieved for the noninteracting case. In Figures 13 and 14, the coincidence parameterr = Ω m /Ω T d is plotted against q for interacting and noninteracting cases, respectively. The dark energy-dominated era is recovered for the interacting case whereas for the noninteracting case, when δ = 4100, the interval −0:95 < z < 0:8 gives the energy-dominated era while z > 0:8 results in the matter-  8 Advances in High Energy Physics dominated era; for δ = 4700, the matter-dominated era is obtained when z > 1, and for δ = 5300, the matterdominated era is achieved when z > 1:23. In Figures 15 and  16, the squared speed of sound C 2 s has been plotted to examine the stability of the model for both the interacting and noninteracting cases, respectively. For the interacting case, C 2 s < 0 while for the noninteracting case, it leads to positive values which describes the stable model.

Deceleration
Parameter. For the KHDE model, the deceleration parameter q provides the accelerated universe in both the interacting and noninteracting cases. The results obtained in both cases for q are compared with Planck's observational data [62] presented in Table 1. It has been found that the results of the KHDE model are consistent with the observational data at the present epoch in both the interacting and noninteracting cases. At z = 0, for the interacting case, we have achieved ðκ, qÞ = ð1400,−0:9937Þ, ð2500,− 0:9939Þ, ð3600,−0:9944Þ, and for the noninteracting case, we obtained ðκ, qÞ = ð1400,−0:9773Þ, ð2500,−0:9781Þ, ð3600, −0:9800Þ. The deceleration parameter q has given that NTHDE can model the accelerated universe in both the interacting and noninteracting cases. The results obtained in both cases for q are compared with Planck's observational data [62] presented in Table 1. It has been found that the results of the NTHDE model are consistent with the observational data at the present epoch in both the interacting and noninteracting cases. At z = 0, for the interacting case, we achieve ðδ, qÞ = ð4100,−0:99976Þ, ð4700,−0:99973Þ, ð5300,− 0:99968Þ, and for the noninteracting case, we obtain ðδ, qÞ = ð4100,−0:99954Þ, ð4700,−0:99948Þ, ð5300,−0:99939Þ.

EoS
Parameter. For the KHDE model, the EoS parameter ω K d has illustrated the phantom phase of the universe at different values of κ for the interacting case. However, it shows the quintessence phase of the universe for κ = 1400, 2500 and the phantom phase for κ = 3600 when the noninteracting case is under consideration. Moreover, we compared these results with Planck's observational data [62] presented in     Planck+lowE+lensing+BAO+TT, TE, EE 9 Advances in High Energy Physics phase of the universe for the noninteracting case. Moreover, we compared these results obtained with Planck's observational data [62] presented in Table 2. The comparison shows that results obtained by the NTHDE model have consistency with the observational data at z = 0 in both cases. For the interacting case, we found the values of ðδ, ω T d Þ at the present epoch as ð4100,−1:032Þ, ð4700,−1:023Þ, ð5300,−1:017Þ, and for the noninteracting case, we achieve ðδ, ω T d Þ = ð4100,− 0:99959Þ, ð4700,−0:99958Þ, ð5300,−0:99955Þ.

Coincidence
Parameter. For the KHDE model, the coincidence parameterr = Ω m /Ω K d is achieved for both the interacting and noninteracting cases. For the interacting case, we obtain the dark energy-dominated era for −0:95 < z < 1. For the noninteracting case, when κ = 1400, the interval −0:95 < z < 0:65 gives the energy-dominated era while z > 0:65 results in the matter-dominated era; for κ = 2500, the matter-dominated era is obtained when z > 0:83, and for κ = 3600, the matter-dominated era is achieved when z > 1:1. The coincidence parameter for the NTHDE modelr = Ω m / Ω K d has been examined for both the interacting and noninteracting cases. For the interacting case, we got the dark energydominated era in the interval −0:95 < z < 1. For the noninteracting case, when δ = 4100, the interval −0:95 < z < 0:8 gives the energy-dominated era while z > 0:8 results in the matterdominated era; for δ = 4700, the matter-dominated era is obtained when z > 1, and for δ = 5300, the matterdominated era is achieved when z > 1:23.

Squared
Speed of Sound. The squared speed of sound C 2 s which decides the stability of the model is examined for both the interacting and noninteracting cases. For the interacting case, C 2 s > 0 in the interval −0:95 < z < 0:63 which is a justification for the stable model in this interval while the KHDE model is unstable when z > 0:63. For the noninteracting case model, behavior is stable when z < −0:1873 while it becomes unstable when z > −0:1873. For the NTHDE model, the squared speed of sound C 2 s is examined for both the interacting and noninteracting cases. For the interacting case, C 2 s < 0 which gives that the achieved model is unstable for the interacting case but it has given positive values of C 2 s for the noninteracting case in the interval −0:95 < z < 2, which is a justification for the stable model. Ghaffari et al. investigated the cosmological consequences of the interacting THDE model with the apparent radius in the fractal universe [36]. They constructed various cosmological parameters such as the EoS parameter, the deceleration parameter, and the evolution equation. They suggested that THDE described the transition that took place from the deceleration phase of the universe to the accelerated phase, eventually in both the noninteracting and interacting scenarios. Also, it is checked that the free parameters of the models are compatible with the latest observational results by using the Pantheon supernovae data, 6df, eBOSS, BOSS DR12, CMB Planck 2015, and Gamma-Ray Burst. They also found unstable behavior of the THDE model in both scenarios. However, in our case, KHDE and NTHDE with the apparent horizon in the fractal universe have provided con-sistent results with recent Planck's data [37] (as mentioned in Tables 1 and 2). It is also found that at the present epoch, the KHDE model is stable for the interacting case but unstable for the noninteracting case. The NTHDE model shows unstable behavior for the interacting case while stable behavior for the noninteracting case.

Data Availability
I have mentioned all the results in the manuscript and references therein.