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

Authors: Abdul Jawad; Abdul Malik Sultan
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

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=3Hb2ρm) 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 energy- and 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.


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## Body

## 1. 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 [2–4]. 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 [5–7]. 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 [8–11], 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 [19–21]. It is observed that the Hubble horizon is a proper casual boundary for the universe meeting thermodynamics and conservation laws [22–25]. 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. [32–34], 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 [39–43]. 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 Section3, the Kaniadakis holographic dark energy (KHDE) will be discussed along with some of its cosmological consequences such as deceleration parameter q, EoS parameter ωdK, and the dimensionless ratio called coincidence parameter r~=Ωm/ΩdK 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 Cs2. 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 ωdT, and the dimensionless ratio called coincidence parameter r~=Ωm/ΩdT. The squared speed of sound Cs2 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.


## 2. Tsallis and Kaniadakis Entropies of BH

Both the Gibbs [1] and Shannon [44] entropies of distribution with W states working in the unit kB=1 leads to the relation explicitly dealt in [11]:
(1)S=−∑i=1WPilnPi,where Pi 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
(2)S=−Trρlnρ.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 (SBH=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 Pi is equal for all of them [20, 21] permitting us to write Pi=1/W. In such manner, Equations (1) and (2) lead to the Boltzmann entropy (S=lnW), and thus, we have [7]
(3)SBH=A4=lnW⇒W=expA4,for horizon entropy and accordingly WA. As a unique free parameter generalized entropy, the Tsallis entropy is defined as [4]
(4)SQT=11−Q∑i=1WPiQ−Pi,where Q is an unknown free parameter named as the nonextensive or Tsallis parameter (SQT→S for Q→1). When the probability distribution meets the conditions Pi=1/W, Equation (4) yields
(5)SQT=W1−Q−11−Q.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
(6)SQT=11−Qexp1−QSBH−1=2expδSBH/2δsinhδSBH2.In the scenario of loop quantum gravity, it is acquired by applying the Tsallis entropy definition to BH [11, 47] that
(7)SQT=11−Qexp1−Qln2πβ3SBH−1,satisfying SQT→S whenever Q→1 and β=ln2/π3 [11, 47]. Furthermore, Equation (6) and (7) become accordingly the same when we consider β=ln2/π3.Another single-free parameter generalized entropy is Kaniadakis entropy (K-entropy) [2, 3, 11], defined as
(8)SK=−∑i=1WPi1+K−Pi1−K2K=12Σi=1WPi1−K−PiK+Σi=1WPi1+K−Pi−K,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
(9)SK=S1+KT+S1−KT2.Furthermore, by takingPi=1/W, Equation (8) assists in getting [2, 3, 11]
(10)SK=WK−W−K2K.Combining Equation (10) with Equation (3) yields
(11)SK=1KsinhKSBH.It is observed that Sharma-Mittal and Rényi entropies can be obtained as a function of Tsallis entropy as [13, 48]
(12)SSM=1R1+1−QSTR/1−Q−1,S=11−Qln1+1−QST,which leads to
(13)SSM=1RexpRSBH−1,S=SBH,where R is an unknown parameter. It is suggested that β=ln2/π3; otherwise, ln2/βπ3SBH would occur in mathematical results rather than SBH.


## 3. 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 sizeL3 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
(14)Λ4=ρdK∝SKL4,for the vacuum energy ρdK. Now, taking the Hubble horizon of the cosmos as the IR cutoff (i.e., L=1/H⇒A=4π/H2), we obtain
(15)ρdK=3c2H48πκsinhπκH2,where the constant c2 is unknown [18], κ belongs to a set of real numbers [3], and H=ȧ/a is the Hubble parameter. Now, it is clear that we have ρdK→3c2H2/8 (the well-known Bekenstein entropy-based HDE) when k→0 [18]. Considering the pressureless fluid (with energy density ρm) and the dark energy candidate (with pressure pd and density ρdK), the energy-momentum conservation laws for the fractal universe take the form
(16)ρ̇m+3H+υ̇υρm=Γ,(17)ρ̇d+3H+υ̇υρd+pd=−Γ,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 Γ1=3Hb2ρm+ρd, Γ2=3Hb2ρm, and Γ3=3Hb2ρd [49–54] while some nonlinear cases are Γ4=3Hb2ρm2/ρm+ρd+ρm, Γ5=3Hb2ρd2/ρm+ρd+ρd, Γ6=3Hb2ρd2/ρm+ρd+ρd+ρm, and Γ7=3Hb2ρm2/ρm+ρd+ρd+ρm [53, 54] with b2 being the coupling constant. We have chosen Γ=3Hb2ρm [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
(18)H2+Hυ̇υ−ω6υ̇2=13ρm+ρd,(19)Ḣ+H2−Hυ̇υ+ω3υ2−□υ2υ=−16ρ+3p,where □υ=1/−g∂μ−g∂μυ with simplified relation as □υ=−υ¨+3Hυ̇. From simplification of (16), one can find
(20)ρm=3ΩmoHo28π1+zΔ,where Δ=3−γ−3b2, z represents the redshift parameter, Ho is the value of the Hubble parameter at t=0, and Ωmo=8πρmo/3Ho2 with ρmo 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 parameterq 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 ωd is one which decides the phases of the cosmos (i.e., ωd<−1 represents the phantom phase of the universe, and −1<ωd<−1/3 describes the quintessence phase while ωd>−1/3 gives the vacuum phase of the universe) [58, 59]. The squared speed of sound Cs2 is another important cosmological parameter which decides whether the model is stable or not (i.e., Cs2>0 describes the stable model while Cs2<0 only when the model is unstable) [57, 58, 60]. The ratio of Ωm=ρm/ρc and Ωd=ρd/ρc called the coincidence parameter given by r~=Ωm/Ωd=ρm/ρd decides the dark energy- and dark matter-dominated eras of the universe (i.e., 0<Ωm/Ωd<1 describes the dark energy-dominated era, and Ωm/Ωd>1 gives the dark matter-dominated 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
(21)Ḣ=−3κΩmoHo2Δa−Δ−8πκωH2γ3a−2γ48πκ−48πκγ−8πκωγ2a−2γ−12c2H2sinhπκ/H2+6πκc2coshπκ/H2.Using the transformationd/dt=−1+zHd/dz from cosmic time t to the redshift parameter z, simplification of (21) gives the relation for H in terms of the redshift parameter as
(22)H′=8πκωH2γ31+z2γ−1+3κΩmoHo2Δ1+zΔ−1H48πκ−48πκγ−8πκωγ21+z2γ−12c2H2sinhπκ/H2+6πκc2coshπκ/H2,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
(23)q=−1−ḢH2=−1+8πκωH2γ31+z2γ+3κΩmoHo2Δ1+zΔH248πκ−48πκγ−8πκωγ21+z2γ−12c2H2sinhπκ/H2+6πκc2coshπκ/H2.Substituting the corresponding values and simplifying (17), we obtain
(24)pdK=38πκγ−32πκc2H1+zcoshπκH2−4c2H31+zsinhπκH2H′−c2H4γ−3sinhπκH2+3κb2ΩmoHo21+zΔ.The mathematical formalism for the EoS parameter isωdK=pdK/ρdK, and the relation for ωdK can be obtained by using (15) and (24) in this formula as
(25)ωdK=−1+2πκ1+zcothπκ/H2H3γ−3−41+zHγ−3H′+3κb2ΩmoHo21+zΔc2H4γ−3sinhπκ/H2.Some other important mathematical formulas of cosmological consequences which we will use later on are(26)Ωm=ρmρc,Ωd=ρdρc,ΩmΩd=ρmρd,ρc=3H28π.The ratioΩm/ΩdK 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
(27)ΩmΩdK=κΩmoHo21+zΔc2H4sinhπκ/H2.The stability of the system is examined by a perturbed parameter called the squared speed of sound (Cs2). The mathematical formalism for this parameter is given by
(28)Cs2=∂peff∂ρeff,where ρeff=ρm+ρdK and peff=pdK. Differentiating these relations with respect to the redshift parameter leads to the following mathematical result:
(29)Cs2=1γ−3κΩmoHo2Δ1+zΔ−1+c24H3H′sinhπκ/H2−2πκHH′coshπκ/H22πκc21+zcoshπκH2H′2+HH′×1+z−1+HH′′+2πκH−2H′2tanhπκH2−41+zsinhπκH2c23H2H′2+H3H′1+z−1+H3H′′−2πκ×H′2cothπκH2+3κb2ΩmoHo2Δ1+zΔ−1−c2γ−3HH′4H2sinhπκH2−2πκcoshπκH2.Variation ofq 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 Ωmo=0.315, b2=0.5, Ho=67.9, ω=−0.3, c2=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 −1,0 which is a compatible range with observational data. In Figure 3, EoS parameter ωdK 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 ωdK→−1 as z→−1 which coincides with the ΛCDM model. In Figures 5 and 6, the coincidence parameter r~=Ωm/ΩdK is plotted against q for interacting and noninteracting cases, respectively. The dark energy-dominated 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 Cs2 which decides the stability of the model is examined for both cases (interacting and noninteracting) in Figures 7 and 8, respectively. For the interacting case, Cs2>0 in the interval −0.95<z<0.63 which leads to the stable model in this interval. The noninteracting case model is stable when z<−0.18 while it becomes unstable when z>−0.18.Figure 1
Behavior of deceleration parameterq against redshift parameter z considering different values of κ when Ωmo=0.315, b2=0.5, Ho=67.9, ω=−0.3, c2=0.313, and γ=0.127.Figure 2
Behavior ofq against z.Figure 3
Graph of EoSωdK versus deceleration parameter q.Figure 4
Variation of EoS parameterωdK versus deceleration parameter q.Figure 5
Plot of coincidence parameterr~=Ωm/ΩdK against deceleration parameter q.Figure 6
Plot ofr~=Ωm/ΩdK against q.Figure 7
Variation of squared speed of soundCs2 versus redshift parameter z.Figure 8
Variation of squared speed of soundCs2 versus redshift parameter z.


## 4. 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
(30)ρdT∝SQTL4⇒ρdT=3T28πSQTL4,where T2 is an unknown real number [18]. Taking D2=πT2 and the apparent horizon as the IR cutoff (H=1/L), we have
(31)ρdT=2D2ρcXexpX2sinhX2,in which X=δπ/H2, where δ belongs to a set of real numbers [11]. Substituting all required assumed and obtained values in (29), we get
(32)ρdT=3H4D24π2δλ,where λ=expπδ/2H2sinhπδ/2H2. Considering all assumptions, differentiating (18) w.r.t. cosmic time t, and putting necessary values in it, we reach the following result:
(33)H′=16π2δHγ3ω1+z2γ−1+6πδΩmoHo2Δ1+zΔ−1H−116π2δ6−6γ−ωγ21+z2γ−9D2λ16H2−πδcothπγ/2H2+1.The deceleration parameterq for the NTHDE model is obtained as
(34)q=−1−ḢH2=−1+16π2δγ3ω1+z2γ+6πδΩmoHo2H−2Δ1+zΔ96π2δ1−γ−16π2δωγ21+z2γ−9D2λ16H2−πδcothπγ/2H2+1.Substituting the concern values from the NTHDE model and simplifying Equation (17) yield
(35)pdT=9b2ΩmoHo21+zΔ8πγ−3−3D2H4H′λ4π2δ+3D2H1+zλH′4πγ−3cothπδ2H2+1−4H2πδ.The relation for EoS parameterωdT is obtainable after simplifications of Equations (32) and (35) as
(36)ωdT=pdTρdT=πδb2ΩmoHo21+zΔ2H4D2γ−3λ−1+1+zγ−3πδ4H3cothπδ2H2+πδ4H3−4HH′.The coincidence parameter for the obtained model is(37)r~=ΩmΩdK=πδΩmoHo21+zΔ2D2H4λ.The relation for a perturbed parameter called the squared speed of sound which is given in (28) is obtainable as
(38)Cs2=1γ−33πδΩmoHo2Δ1+zΔ−1+6D2HH′λ4H2−πδ−πδcothπδ/2H29πδb2ΩmoHo2Δ1+zΔ−1+6D2×γ−3HH′λπδcothπδ2H2−4H2+πδ+6D2πδβ1+zH′2+HH′1+z−1+HH′′−πδH−2H′2+πδ×H−2H′2tanhπδ2H2+6D2πδ1+zλH′2+HH′1+z−1+HH′′−πδH−2H′2−πδH−2H′2cothπδ2H2−24D21+zλ3H2H′2+H3H′1+z−1+H3H′′−πδH′2−πδH′2cothπδ2H2,where β=expπδ/2H2coshπδ/2H2.The graph of deceleration parameterq against redshift parameter z is plotted in Figures 9 and 10 for interacting and noninteracting cases, respectively. Constant parameters are δ=4100,4700,5300, Ωmo=0.315, b2=0.5, Ho=67.9, ω=−0.3, D2=0.313, and γ=0.127. Required results (accelerated expansion) of the universe are achieved in both cases. EoS parameter ωdT 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 parameter r~=Ωm/ΩdT 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-dominated era; for δ=4700, the matter-dominated era is obtained when z>1, and for δ=5300, the matter-dominated era is achieved when z>1.23. In Figures 15 and 16, the squared speed of sound Cs2 has been plotted to examine the stability of the model for both the interacting and noninteracting cases, respectively. For the interacting case, Cs2<0 while for the noninteracting case, it leads to positive values which describes the stable model.Figure 9
Variation of deceleration parameterq against redshift parameter z considering different values of δ when Ωmo=0.315, b2=0.5, ω=−0.3, Ho=67.9, D2=0.3136, and γ=0.127.Figure 10
Graph ofq against z.Figure 11
Plot of EoS parameterωdT versus deceleration parameter q.Figure 12
Plot of EoS parameterωdT versus deceleration parameter q.Figure 13
Behavior of coincidence parameterr~=Ωm/ΩdT against deceleration parameter q.Figure 14
Behavior of coincidence parameterr~=Ωm/ΩdT against q.Figure 15
Plot of squared speed of soundCs2 versus redshift parameter z.Figure 16
Graph ofCs2 against z.


## 5. Conclusions

We have investigated the interacting and noninteracting behaviors of KHDE and NTHDE with the apparent horizon and extracted various cosmological parameters by varying the value ofκ for KHDE and δ for NTHDE and keeping all other parameters fixed as Ωmo=0.315, b2=0.5, Ho=67.9, ω=−0.3, c2=0.313, D2=0.313, and γ=0.127. The cosmological consequences resulted as follows.


### 5.1. Deceleration Parameter

For the KHDE model, the deceleration parameterq 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 κ,q=1400,−0.9937,2500,−0.9939,3600,−0.9944, and for the noninteracting case, we obtained κ,q=1400,−0.9773,2500,−0.9781,3600,−0.9800. 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 δ,q=4100,−0.99976,4700,−0.99973,5300,−0.99968, and for the noninteracting case, we obtain δ,q=4100,−0.99954,4700,−0.99948,5300,−0.99939.Table 1qObservational schemes−0.644−0.223+0.223BAO+TDSL+Masers+Pantheon−0.6401−0.187+0.187BAO+TDSL+Masers+Pantheon+Ho−0.930−0.218+0.218BAO+TDSL+Masers+Pantheon+Hz−1.2037−0.175+0.175BAO+TDSL+Masers+Pantheon+Ho+Hz


### 5.2. EoS Parameter

For the KHDE model, the EoS parameterωdK 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 Table 2. The comparison shows that results obtained by the KHDE model have consistency with the observational data at z=0 in both cases. For the interacting case, we found the values of κ,ωdK at the present epoch as 1400,−1.16,2500,−1.12,3600,−1.08 while for the noninteracting case, we achieve κ,ωdK=1400,−0.988,2500,−0.997,3600,−1.007. For the NTHDE model, the EoS parameter ωdT tells about the phantom phase of the universe at different values of δ for the interacting case while it shows the quintessence 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 δ,ωdT at the present epoch as 4100,−1.032,4700,−1.023,5300,−1.017, and for the noninteracting case, we achieve δ,ωdT=4100,−0.99959,4700,−0.99958,5300,−0.99955.Table 2ωdObservational schemes−1.56−0.48+0.60Planck+lowE+TT−1.58−0.41+0.52Planck+lowE+TT, TE, EE−1.57−0.40+0.50Planck+lowE+lensing+TT, TE, EE−1.40−0.10+0.10Planck+lowE+lensing+BAO+TT, TE, EE


### 5.3. Coincidence Parameter

For the KHDE model, the coincidence parameterr~=Ωm/ΩdK 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 model r~=Ωm/ΩdK has been examined for both the interacting and noninteracting cases. For the interacting case, we got the dark energy-dominated 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 matter-dominated era; for δ=4700, the matter-dominated era is obtained when z>1, and for δ=5300, the matter-dominated era is achieved when z>1.23.


### 5.4. Squared Speed of Sound

The squared speed of soundCs2 which decides the stability of the model is examined for both the interacting and noninteracting cases. For the interacting case, Cs2>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 Cs2 is examined for both the interacting and noninteracting cases. For the interacting case, Cs2<0 which gives that the achieved model is unstable for the interacting case but it has given positive values of Cs2 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 consistent 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.


## 5.1. Deceleration Parameter

For the KHDE model, the deceleration parameterq 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 κ,q=1400,−0.9937,2500,−0.9939,3600,−0.9944, and for the noninteracting case, we obtained κ,q=1400,−0.9773,2500,−0.9781,3600,−0.9800. 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 δ,q=4100,−0.99976,4700,−0.99973,5300,−0.99968, and for the noninteracting case, we obtain δ,q=4100,−0.99954,4700,−0.99948,5300,−0.99939.Table 1qObservational schemes−0.644−0.223+0.223BAO+TDSL+Masers+Pantheon−0.6401−0.187+0.187BAO+TDSL+Masers+Pantheon+Ho−0.930−0.218+0.218BAO+TDSL+Masers+Pantheon+Hz−1.2037−0.175+0.175BAO+TDSL+Masers+Pantheon+Ho+Hz


## 5.2. EoS Parameter

For the KHDE model, the EoS parameterωdK 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 Table 2. The comparison shows that results obtained by the KHDE model have consistency with the observational data at z=0 in both cases. For the interacting case, we found the values of κ,ωdK at the present epoch as 1400,−1.16,2500,−1.12,3600,−1.08 while for the noninteracting case, we achieve κ,ωdK=1400,−0.988,2500,−0.997,3600,−1.007. For the NTHDE model, the EoS parameter ωdT tells about the phantom phase of the universe at different values of δ for the interacting case while it shows the quintessence 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 δ,ωdT at the present epoch as 4100,−1.032,4700,−1.023,5300,−1.017, and for the noninteracting case, we achieve δ,ωdT=4100,−0.99959,4700,−0.99958,5300,−0.99955.Table 2ωdObservational schemes−1.56−0.48+0.60Planck+lowE+TT−1.58−0.41+0.52Planck+lowE+TT, TE, EE−1.57−0.40+0.50Planck+lowE+lensing+TT, TE, EE−1.40−0.10+0.10Planck+lowE+lensing+BAO+TT, TE, EE


## 5.3. Coincidence Parameter

For the KHDE model, the coincidence parameterr~=Ωm/ΩdK 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 model r~=Ωm/ΩdK has been examined for both the interacting and noninteracting cases. For the interacting case, we got the dark energy-dominated 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 matter-dominated era; for δ=4700, the matter-dominated era is obtained when z>1, and for δ=5300, the matter-dominated era is achieved when z>1.23.


## 5.4. Squared Speed of Sound

The squared speed of soundCs2 which decides the stability of the model is examined for both the interacting and noninteracting cases. For the interacting case, Cs2>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 Cs2 is examined for both the interacting and noninteracting cases. For the interacting case, Cs2<0 which gives that the achieved model is unstable for the interacting case but it has given positive values of Cs2 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 consistent 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.



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*Source: 5519028-2021-04-20.xml*