Barrow holographic dark energy with Hubble horizon as IR cutoff
BBarrow holographic dark energy with Hubble horizon as IR cutoff
Shikha Srivastava , Umesh Kumar Sharma , Department of Mathematics, Institute of Applied Sciences and Humanities, GLA UniversityMathura-281406, Uttar Pradesh, India E-mail: [email protected] E-mail: [email protected]
Abstract
In this work, we propose a non-interacting model of Barrow holographic dark energy(BHDE) using Barrow entropy in a spatially flat FLRW Universe considering the IR cut-off as the Hubble horizon. We study the evolutionary history of important cosmologicalparameters, in particular, EoS ( ω B ), deceleration parameter and, the BHDE and matterdensity parameter and also observe satisfactory behaviours in the BHDE the model. Inaddition, to describe the accelerated expansion of the Universe the correspondence of theBHDE model with the quintessence scalar field has been reconstructed. Keywords : BHDE, FLRW, quintessencePACS: 98.80.-k
According to a sequence of past and latest observational data [1–6], the dark sector of our Uni-verse is filled with two dark fluids, namely the dark energy (DE) and dark matter (DM). Theformer fluid drives the current accelerating stage of the Universe while the later is responsiblefor the structure formation of the Universe. It is also observed from the observational datathat about ninety-six per cent of the total energy density of the Universe is coming from thiscombined dark component, where the contribution of DE is around sixty-eight per cent of thetotal energy allocation of the Universe while the DM contributes approximately twenty-eightper cent of the total energy allocation of the Universe. Although, the origin, evolution and thecharacteristics of these dark sector of the DE are not distinctly recognized yet [7–12]. However,the nature of DM appears to be partially known by indirect gravitational effects, although, theDE has endured being exceptionally mysterious. As a consequence, in the last couple of years,various cosmological models have been proposed and explored. The non-interacting models ofthe Universe are considered to be the simplest model leading to two independent evolution ofthese dark components where DE and DM are conserved separately. In a more general form ofavailable cosmological models interaction between DE and DM are permitted. [13].The quantitative description of dark energy can suitably be substituted by holographicdark energy (HDE) [14, 15], deriving from holographic principle (HP) [16–20]. Notably one can1 a r X i v : . [ phy s i c s . g e n - ph ] O c t ecieve the consequence of vacume energy of holographic source at astrophysical scales structureDE [14, 15] due to the relationship between the longest length of QF T (quantum field theory)with its UV cutoff [21]. Interestly both the simple [22–31] and extended [32–55] versions ofHDE represents to lead to a fascinating cosmological behaviour and they also constrained withobservations [56–64]. The Universe horizon entropy remains proportional to its area, whichbecomes the most pevetel step in the application of HP at cosmological context, similar to ablack hole BH (Bekenstein-Hawking) entropy. Although, for the black-hole structure, recentlyBarrow propoesd that the QG (quantum-gravitational) impacts can lead to introduce fractal,intricate behaviours. He emphasized that this complex structure gives finite volume but withthe finite (or infinite) area and hence a deformed expression of black-hole entropy [65] S B = (cid:18) AA (cid:19) , (1)where, A and A are Planck area and standard horizon area, respectively. A new exponent∆, contributing to the quantum-gravitational deformation was in ∆ = 1 corresponding to itsthe most fractal and intricate structure and corresponds to the usual BH entropy when ∆ =0. It is important to mention that the standard “quantum-corrected” entropy with correctionslogarithmic [66,67] is different from the above quantum-gravitationally corrected entropy. How-ever, it resembles non-extensive Tsallis entropy [68–70], but the physical principles and involvedfoundations are totally different.Recently, Saridakis [71], constructed the BHDE, by using the usual HP, however applyingthe Barrow entropy instead of the BH entropy. Also, for the limiting case as ∆ = 0, theBHDE possesses standard HDE, although The BHDE, in general, is a new scenario with nicecosmological behavior and richer structure. While standard HDE is given by the inequality ρ B L ≤ S , where L denotes horizon length, and under the imposition S ∝ A ∝ L [14], the Eq.(1) will give ρ B = CL ∆ − , (2)where C is a parameter having [ L ] − − ∆ dimensions [71]. As expected, in the case ∆ = 0,the above expression gives the standard HDE ρ B = 3 c M p L − , where C = 3 c M p with c themodel parameter and M p is the Planck mass. Depending on the parameter ∆, the BHDE willdeviate by the standard one, which may lead to distinct cosmological behaviour. If we take intoconsideration the IR cut off L as the Hubble horizon ( H − ), then the energy density of BHDEis obtained as ρ B = CH − ∆ . (3)Saridakis [72], using Barrow entropy presented a modified cosmological scenario besides theBekenstein-Hawking one. For the evolution of the effective DE density parameter, the analyticalexpression was obtained and shown the DM to DE era of the Universe. Using the Barrowentropy on the horizon in place of the standard Bekenstein-Hawking one, the potency of thegeneralized second law of thermodynamics has also been examined [73]. Mamon et al. [74]studied interacting BHDE model and also the validity of the generalized second law by assumingdynamical apparent horizon as the thermodynamic boundary. More recently, Anagnostopouloset al. [75] have shown that the BHDE is in concurrence with observational information, andit can serve as a decent contender for the depiction of DE. The authors examined the Tsallis2onextensive form of the logarithmically amended Barrow entropy [76]. Barrow’s new idea ofentropy have some significant examinations [77,78]. Now, we discuss the similarity and differencewith other work in literature. Recently, an interacting model of the BHDE has been proposed byusing Barrow entropy. In particular, the evolution of a spatially flat FLRW Universe filled withBHDE and pressureless DM that interact with each other through a well-motivated interactionterm has been investigated by taking the Hubble horizon as IR cutoff [74]. While in this workwe focus the BHDE model without interaction in a flat FLRW Universe with Hubble horizonas IR cutoff. We organize the present work in the following way. The cosmological parametersof BHDE model are discussed in Sect. 2. In Sect. 3, we investigate the stability of the BHDEmodel. The correspondence between the BHDE and quintessence scalar field model and alsopotentials for scalar field models are discussed in Sect. 4. In Sect. 5, we draw our conclusions. The first Friedmann equation for a flat FRW Universe which is composed by pressureless DM( ρ m ) and BHDE ( ρ B ) is H = 13 m p ( ρ B + ρ m ) , (4)expounding the dimensionless density parameter as Ω i = ρ i /ρ c , where ρ c = 3 m p H is known asthe critical energy density, we can obtainΩ B = ρ B m p H = C m p H − ∆ , Ω m = ρ m m p H . (5)The conservation law corresponding to dust and BHDE are˙ ρ m + 3 Hρ m = 0 , (6)˙ ρ B + 3 Hρ B (1 + ω B ) = 0 . (7)Here ω B = p B /ρ B is EoS parameter and p B is pressure of BHDE. Differentiating Eq. (4)with respect to time and solving Eqs. (6), (7) and Eq. (5), we can obtain˙ HH = −
32 (1 + ω B Ω B ) . (8)Substituting Eq. (3) in Eq. (7), we will find˙ HH = − ω B )2 − ∆ , (9)by the help of Eq. (7) and Eq. (8), we get ω B = − ∆2 − (2 − ∆)Ω B . (10)From Eq. (8), we can find that Ω (cid:48) B = d Ω B d (ln a ) = (1 − ∆)Ω B ˙ HH . (11)3 = Δ = Δ = - - - - - - q Figure 1: Expansion of deceleration parameter verses redshift z distinct estimations of ∆.Here, we take Ω B = 0 . (cid:48) B = − − ∆) (cid:16) − Ω B − (2 − ∆)Ω B (cid:17) Ω B . (12)As we know that deceleration parameter is given as q = − (cid:2) HH (cid:3) , (13)putting Eq. (9) in Eq. (13) and Using Eq. (10), we get q = (cid:20) − (1 + ∆)Ω B − (2 − ∆)Ω B (cid:21) . (14)The evolutionary behaviour of the deceleration parameter is plotted for the BHDE modelversus redshift z by finding its numerical solution using the initial values of Ω B as Ω B = 0.70.The value of deceleration parameter q decides the nature of the Universe such as if q <
0, itis accelerating and if q >
0, it is in decelerating phase. It is proposed by various observationsthat the Universe is in an accelerated expansion phase and the value of deceleration parameterlies in the range − ≤ q <
0. The evolutionary behaviour of the deceleration parameter versusredsfit z is plotted in Fig. 1. We observe from this figure that the deceleration parameter ofthe BHDE model transits from an early decelerated phase to the current accelerated phase forthe different values of the parameter ∆. The evolutionary behaviour of the EoS parameter ω B = Δ = Δ = - - - - - - - ω B Figure 2: The behaviour of the EoS parameter of BHDE ω B against z for different ∆. Here,we take Ω B = 0 . z for the BHDE model is plotted in Fig. 2 for the different values of the parameter ∆. Wecan observe that the EoS parameter of the BHDE remains in quintessence era, and approachesto the cosmological constant ( ω B = −
1) at future for ∆ =1.7, 1.8 and 1.9. It is significant thatEoS parameter of the BHDE gives pleasant conduct and it tends to be quintessence-like for thevarious estimations of ∆.The modified cosmological scenario has been constructed by applying the FLT (first law ofthermodynamics) and using the Barrow entropy [72]. Moreover, Saridakis [72], proposed thiscosmological modified scenario with varying parameter ∆ by applying the non-extensive ther-modynamics. In [71–73], the researcers proposed that this cosmological modified scenario givesa description of both inflation and late-time acceleration of the Universe with varying parameterfrom the FLT. In Fig. 3(a) and 3(b) we have plotted the energy density for the BHDE Ω B andenergy density of matter Ω m as a function of redshift. The thermal history of the Universe, inparticular, the successive sequence of matter and DE era can be observed from these figures.In the papers [71–73], it has also been proposed by some of the researchers, relating to the latetimes Universe evolution, a new term that appear due to the varying non-extensive exponentconstituted an effective DE sector. Futher, it has been shown that Universe shows usual thermalhistory, with the successive sequence of matter and De epochs, and the transition to accelerationis in agreement with the observed behavior. 5a) Δ = Δ = Δ = - Ω B (b) Δ = Δ = Δ = - Ω m Figure 3: (a) Expansion of BHDE density parameter Ω B with z redshift for distinct value of∆.Here, we take Ω D = 0 .
70 (b) The evolution of matter energy density parameter Ω m with z (redshift) for distinct value of ∆. Here, we take Ω B = 0 . . Squared sound speed lies in a range 0 ≤ v s ≤ ω B (cid:54) = − v s from 1 to 0 and when ω B > − ω B < − ω B > − v s = dp T dρ B , [81]. Generally, we give the squared speed of sound in the form: v s = dp B dρ B = 2∆(1 − ∆)Ω B −
2∆ + ∆Ω B [2 − (2 − ∆)Ω B ] . (15)6 = . Δ = . Δ = . - - - - - - - - v s Figure 4: The behaviour of squared sound speed v s verses redshift z for distinct value of ∆.Here, we take Ω B = 0 . . The squared speed of sound v s represented by Eq. (15), is plotted in Fig. 4 as a function ofredshift ( z ) for different values of the parameter ∆, and it can be observed by the figure thatfor ∆ <
2, The BHDE model is unstable during the cosmic evolution.
In view of this segment, we will deliberate the correspondence of the BHDE with quintessencescalar field model. The potentials and dynamics for quintessence scalar field are also recon-structed. By comparing energy densities of the scalar field models and the BHDE model givenin Eq. (3), we obtain the correspondence. Here we also compare the EoS parameter for scalarfields (quintessence ) with EoS of BHDE model stated by Eq. (10). Both the dark energy,canonical and non-canonical are well described by using scalar fields. In the present article, wehave chosen the quintessence field as canonical. As we know that for quintessence scalar field ω B > − p = ˙ φ − V ( φ ) , ρ = ˙ φ V ( φ ) , (16)where V ( φ ) stands for scalar field potential and φ stands for scalar field. V ( φ ) = 1 − ω B ρ B , (17)˙ φ = (1 + ω B ) ρ B . (18)7 (cid:61) (cid:68) (cid:61) (cid:68) (cid:61) Φ V (cid:72) Φ (cid:76) Figure 5: The potential reconstruction for the Barrow holograhpic quintessence for different ∆,here V ( φ ) and φ are taken in unit of ρ c and M p respectively and Ω B = 0 .
70 . Δ = Δ = Δ = ϕ Figure 6: The evolution of the scalar-field φ ( z ) for quintessence of Barrow holograhpic, here φ is taken in unit of M p and Ω B = 0 .
70. 8oreover, from the equation of a flat FLRW 3 M p H = ρ m + ρ B , we can obtain E ( z ) ≡ H ( z ) H = (cid:18) Ω m (1 + z ) (1 − Ω B ) (cid:19) / , (19)here Ω m represents the present fractional energy density of pressureless matter. By the help ofEq. (19) we can express Eqs. (17) and (18) respectively [87] V ( φ ) ρ c = 12 (1 − ω B ) Ω B E , (20)˙ φ ρ c = (1 + ω B ) Ω B E . (21)Where ρ c = 3 M P H gives present critical density of the Universe. For the propose correspon-dence between the BHDE and the quintessence scalar field, namely, in particular, we recognize ρ with p . Futhermore, quintessence field takes over Barrow nature such as E, Ω B and ω B arerepresented by Eqs. (10), (12) and (19). According to [87], we assume ˙ φ > φ (scalar field) with respect to z (redshift), given as dφdz M p = (cid:112) ω B ) Ω B z . (22)By integrating Eq. (22), we can without much of a stretch acquire the evolutionary structureof the field as : φ ( z ) = (cid:90) z dφdz dz, (23)here, the field amplitude ( φ ( z )), at the present epoch ( z = 0) is constant to be zero such as φ (0) = 0. In this manner, this can be constituted as Barrow Holographic quintessence DEmodel and recreate the potential of the BHDE.In Fig. 5 , V ( φ ), the reconstructed quintessence potential is graphed, and φ ( z ) which is alsorecreated by Eqs. (22) and (23) , plotted in Fig. 6. Assuming present fractional energy densityis Ω B0 = 0 . . , chosen curves are portrayed for the different cases of ∆. The dynamics of thescalar field explicitly observed from Figs. 5 and 6. Clearly, the scalar field φ moves down thepotential with kinetic energy ˙ φ slowly diminishing. In this paper, we have constructed the BHDE model without interaction which depends onBarrow entropy proposed by Barrow recently, it begins with the modification proposal of theblack-hole structure because of some QG effects. We have studied the evolution of a spatiallyflat FLRW Universe composed of the BHDE and pressureless DM by considering the IR cutoffas Hubble horizon. We have explored the behaviour of cosmological parameters such as DP(deceleration parameter), the EoS parameter, the BHDE and matter-energy density parameterduring the cosmic evolution. The correspondence between BHDE model and the quintessence9calar field models have also been done to clarify the late-time cosmic accelerated expansion ofthe Universe. We finish up our outcomes as follows : • It has been observed that the BHDE model exhibits a smooth transformation from anearly deceleration era to present acceleration era of the Universe and in a good agreement withrecent cosmic observations. • The behaviour of EoS parameter can be observed by varying the exponent ∆. The EoSparameter of the BHDE model varies in the quintessence region for different values of the pa-rameter ∆. • We extricated a straightforward DE (Differential Equation) for the evolution of the DEdensity parameter, and we presented the behaviour of BHDE and matter density parameter.As we appeared, the situation of the BHDE can portray the Cosmos usual history, with thearrangement of matter and DE era. • We observe that for ∆ <
2, the BHDE model is not stable for all values of z during thecosmic evolution. It can be fixed by taking other IR cutoffs, probable interactions between theUniverse sectors, different entropy corrections or even a mix of these scenarios. Indeed, thiscontemplation can likewise modify and increase predictions and behaviour of BHDE. These aresubjects concentrated in future to turn out to be all the more near the various properties ofBHDE, and thus the cause of the DE. • We proposed a correspondence between BHDE model and quintessence scalar-field model.We have a look at that the BHDE with ∆ < The author S. Srivastava thankfully recognize the utilization of administrations and office gaveby GLA University, Mathura, India to lead this research work. The author U. K. Sharma thanksthe IUCAA, Pune, India for awarding the visiting associateship.
References [1] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], “Measurements of Ωand Λ from 42 high redshift supernovae,”
Astrophys. J. (1999) 565-586.[2] A. G. Riess et al. [Supernova Search Team], “Observational evidence from supernovae foran accelerating universe and a cosmological constant,”
Astron. J. (1998), 1009-1038.[3] P. Ade et al. [Planck], “Planck 2015 results. XIII. Cosmological parameters,”
Astron.Astrophys. (2016), A13. 104] T. Abbott et al. [DES], “First Cosmology Results using Type Ia Supernovae from theDark Energy Survey: Constraints on Cosmological Parameters,”
Astrophys. J. Lett. (2019) no.2, L30.[5] N. Aghanim et al. [Planck Collaboration], “Planck 2018 results. VI. Cosmological param-eters,” arXiv:1807.06209 [astro-ph.CO].[6] L. Amendola et al. , “Cosmology and fundamental physics with the Euclid satellite,”
LivingRev. Rel. (2018) no.1, 2.[7] S. Capozziello, “Dark Energy Models toward observational tests and data,” Int. J. Geom.Meth. Mod. Phys. (01) (2007), 53.[8] S. Capozziello, V. Cardone and A. Troisi, “Dark energy and dark matter as curvatureeffects,” JCAP (2006) 001.[9] C. Escamilla-Rivera, M. A. C. Quintero and S. Capozziello, “A deep learning approachto cosmological dark energy models,” JCAP (2020) 008.[10] S. Capozziello, Ruchika and A. A. Sen, “Model independent constraints on dark energyevolution from low-redshift observations,” Mon. Not. Roy. Astron. Soc. (2019), 4484[11] S. Capozziello, V. F. Cardone, A. Troisi, “ Reconciling dark energy models with f ( R )theories,” Phys. Rev. D (4), (2005) 043503.[12] S. Capozziello, V. F. Cardone, E. Elizalde, S. Nojiri and S. D. Odintsov, “Observationalconstraints on dark energy with generalized equations of state,” Phys. Rev. D (2006),043512.[13] L. Amendola, “Coupled quintessence,” Phys. Rev. D (2000), 043511.[14] M. Li, “A Model of holographic dark energy,” Phys. Lett. B (2004), 1.[15] S. Wang, Y. Wang and M. Li, “Holographic Dark Energy,”
Phys. Rept. (2017), 1-57.[16] L. Susskind, “The World as a hologram,”
J. Math. Phys. (1995), 6377-6396. [arXiv:hep-th/9409089 [hep-th]][17] W. Fischler and L. Susskind, “Holography and cosmology,” [arXiv:hep-th/9806039 [hep-th]].[18] G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conf. Proc. C (1993),284-296 [arXiv:gr-qc/9310026 [gr-qc]]..[19] P. Horava and D. Minic, “Probable values of the cosmological constant in a holographictheory,”
Phys. Rev. Lett. (2000), 1610-1613 [hep-th/0001145].[20] R. Bousso, “The Holographic principle,” Rev. Mod. Phys. (2002), 825-874.[21] A. G. Cohen, D. B. Kaplan and A. E. Nelson, “Effective field theory, black holes, and thecosmological constant,” Phys. Rev. Lett. (1999), 4971-4974.1122] D. Pavon and W. Zimdahl, “Holographic dark energy and cosmic coincidence,” Phys. Lett.B (2005), 206-210.[23] B. Wang, C. Y. Lin and E. Abdalla, “Constraints on the interacting holographic darkenergy model,”
Phys. Lett. B (2006), 357-361.[24] R. Horvat, “Holography and variable cosmological constant,”
Phys. Rev. D (2004),087301.[25] M. R. Setare and E. N. Saridakis, “Correspondence between Holographic and Gauss-Bonnet dark energy models,” Phys. Lett. B (2008), 1-4.[26] Q. G. Huang and M. Li, “The Holographic dark energy in a non-flat universe,”
JCAP (2004), 013.[27] S. Nojiri and S. D. Odintsov, “Unifying phantom inflation with late-time acceleration:Scalar phantom-non-phantom transition model and generalized holographic dark energy,” Gen. Rel. Grav. (2006), 1285-1304.[28] B. Wang, Y. g. Gong and E. Abdalla, “Transition of the dark energy equation of state inan interacting holographic dark energy model,” Phys. Lett. B (2005), 141-146.[29] M. R. Setare, “Interacting holographic dark energy model in non-flat universe,”
Phys.Lett. B (2006), 1-4.[30] M. R. Setare and E. N. Saridakis, “Non-minimally coupled canonical, phantom and quin-tom models of holographic dark energy,”
Phys. Lett. B (2009), 331-338.[31] H. Kim, H. W. Lee and Y. S. Myung, “Equation of state for an interacting holographicdark energy model,”
Phys. Lett. B (2006), 605-609.[32] E. N. Saridakis, “Holographic Dark Energy in Braneworld Models with a Gauss-BonnetTerm in the Bulk. Interacting Behavior and the w =-1 Crossing,”
Phys. Lett. B (2008), 335-341.[33] M. Tavayef, A. Sheykhi, K. Bamba and H. Moradpour, “Tsallis Holographic Dark Energy,”
Phys. Lett. B (2018), 195.[34] L. P. Chimento and M. G. Richarte, “Dark radiation and dark matter coupled to holo-graphic Ricci dark energy,”
Eur. Phys. J. C (2013) no.4, 2352.[35] A. Jawad, S. Husain, S. Rani and S. Qummer, “Generalized ghost Tsallis holographicdark energy model in RS-II braneworld and dynamical Chern-Simons modified gravity,” Int. J. Geom. Methods Mod. Phys. (2020), doi: 10.1142/S0219887820501248.[36] H. Moradpour, S. A. Moosavi, I. P. Lobo, J. P. Morais Gra¸ca, A. Jawad and I. G. Salako,“Thermodynamic approach to holographic dark energy and the R´enyi entropy,”
Eur. Phys.J. C (2018) no.10, 829.[37] S. Nojiri and S. D. Odintsov, “Covariant Generalized Holographic Dark Energy and Ac-celerating Universe,” Eur. Phys. J. C (2017) no.8, 528.1238] V. Srivastava and U. K. Sharma. “Tsallis holographic dark energy withhybrid expansion law,” Int. J. Geom. Methods Mod. Phys. (2020) 2050144,doi:10.1142/S02198878205014443.[39] A. S. Jahromi, S. A. Moosavi, H. Moradpour, J. P. Morais Gra¸ca, I. P. Lobo, I. G. Salakoand A. Jawad, “Generalized entropy formalism and a new holographic dark energy model”,
Phys. Lett. B (2018), 21.[40] B. Pourhassan, A. Bonilla, M. Faizal and E. M. C. Abreu, “Holographic Dark Energyfrom Fluid/Gravity Duality Constraint by Cosmological Observations,”
Phys. Dark Univ. (2018), 41-48.[41] E. N. Saridakis, “Restoring holographic dark energy in brane cosmology,” Phys. Lett. B (2008), 138-143.[42] G. Varshney, U. K. Sharma, and A. Pradhan, “Reconstructing the k -essence and thedilation field models of the THDE in f ( R, T ) gravity”.
Eur. Phys. J. Plus (2020),541.[43] S. Nojiri, S. D. Odintsov and E. N. Saridakis, “Holographic inflation,”
Phys. Lett. B (2019), 134829.[44] M. Bouhmadi-Lopez, A. Errahmani and T. Ouali, “The cosmology of an holographicinduced gravity model with curvature effects,”
Phys. Rev. D (2011), 083508.[45] U. K. Sharma, “Reconstruction of quintessence field for the THDE with swampland cor-respondence in f ( R, T ) gravity,” arXiv:2005.03979 [physics.gen-ph].[46] Y. Gong and T. Li, “A Modified Holographic Dark Energy Model with Infrared InfiniteExtra Dimension(s),”
Phys. Lett. B (2010), 241-247.[47] M. Jamil, E. N. Saridakis and M. R. Setare, “Holographic dark energy with varyinggravitational constant,”
Phys. Lett. B (2009), 172-176.[48] R. G. Cai, “A Dark Energy Model Characterized by the Age of the Universe,”
Phys. Lett.B (2007), 228-231.[49] M. R. Setare and E. C. Vagenas, “Thermodynamical Interpretation of the InteractingHolographic Dark Energy Model in a non-flat Universe,”
Phys. Lett. B (2008), 111-115.[50] E. N. Saridakis, “Ricci-Gauss-Bonnet holographic dark energy,”
Phys. Rev. D (2018)no.6, 064035.[51] E. N. Saridakis, “Holographic Dark Energy in Braneworld Models with Moving Branesand the w=-1 Crossing,” JCAP (2008), 020.[52] U. K. Sharma and V. C. Dubey, “Exploring the Sharma-Mittal HDE models with differentdiagnostic tools,” Eur. Phys. J. Plus (2020), 391.[53] R. C. G. Landim, “Holographic dark energy from minimal supergravity,”
Int. J. Mod.Phys. D (2016) no.04, 1650050. 1354] C. Q. Geng, Y. T. Hsu, J. R. Lu and L. Yin, “Modified Cosmology Models from Thermo-dynamical Approach,” Eur. Phys. J. C (2020) no.1, 21.[55] E. N. Saridakis, K. Bamba, R. Myrzakulov and F. K. Anagnostopoulos, “Holographicdark energy through Tsallis entropy,” JCAP (2018), 012.[56] S. M. R. Micheletti, “Observational constraints on holographic tachyonic dark energy ininteraction with dark matter,” JCAP (2010), 009.[57] M. Li, X. D. Li, S. Wang and X. Zhang, “Holographic dark energy models: A comparisonfrom the latest observational data,” JCAP (2009), 036 [0904.0928 [astro-ph.CO]].[58] X. Zhang, “Holographic Ricci dark energy: Current observational constraints, quintomfeature, and the reconstruction of scalar-field dark energy,” Phys. Rev. D (2009),103509 [0901.2262 [astro-ph.CO]].[59] R. D’Agostino, “Holographic dark energy from nonadditive entropy: cosmological pertur-bations and observational constraints,” Phys. Rev. D (2019) no.10 103524[1903.03836[gr-qc]]].[60] X. Zhang and F. Q. Wu, “Constraints on holographic dark energy from Type Ia supernovaobservations,” Phys. Rev. D (2005), 043524 [astro-ph/0506310].[61] C. Feng, B. Wang, Y. Gong and R. K. Su, “Testing the viability of the interactingholographic dark energy model by using combined observational constraints,” JCAP (2007), 005 [0706.4033 [astro-ph]].[62] E. Sadri, “Observational constraints on interacting Tsallis holographic dark energymodel,” Eur. Phys. J. C (2019) no.9, 762 [1905.11210[astro-ph.CO]].[63] Z. Molavi and A. Khodam-Mohammadi, “Observational tests of Gauss-Bonnet like darkenergy model,” Eur. Phys. J. Plus (2019) no.6, 254 [1906.05668 [gr-qc]].[64] J. Lu, E. N. Saridakis, M. R. Setare and L. Xu, “Observational constraints on holographicdark energy with varying gravitational constant,”
JCAP (2010), 031 [0912.0923 [astro-ph.CO]].[65] J. D. Barrow, “The Area of a Rough Black Hole,” [arXiv:2004.09444 [gr-qc]].[66] S. Carlip, “Logarithmic corrections to black hole entropy from the Cardy formula,” Class.Quant. Grav. (2000), 4175-4186 [gr-qc/0005017].[67] R. K. Kaul and P. Majumdar, “Logarithmic correction to the Bekenstein-Hawking en-tropy,” Phys. Rev. Lett. (2000), 5255-5257[68] G. Wilk and Z. Wlodarczyk, “On the interpretation of nonextensive parameter q in Tsallisstatistics and Levy distributions,” Phys. Rev. Lett. (2000), 2770[69] C. Tsallis and L. J. L. Cirto, “Black hole thermodynamical entropy,” Eur. Phys. J. C (2013), 2487 1470] C. Tsallis, “Possible Generalization of Boltzmann-Gibbs Statistics,” J. Statist. Phys. (1988), 479-487[71] E. N. Saridakis, “Barrow holographic dark energy,” [arXiv:2005.04115 [gr-qc]].[72] E. N. Saridakis, “Modified cosmology through spacetime thermodynamics and Barrowhorizon entropy,” [arXiv:2006.01105 [gr-qc]].[73] E. N. Saridakis and S. Basilakos, “The generalized second law of thermodynamics withBarrow entropy,” [arXiv:2005.08258 [gr-qc]].[74] A. A. Mamon, A. Paliathanasis and S. Saha, “Dynamics of an Interacting Barrow Holo-graphic Dark Energy Model and its Thermodynamic Implications,” [arXiv:2007.16020[gr-qc]].[75] F. K. Anagnostopoulos, S. Basilakos and E. N. Saridakis, “Observational constraints onBarrow holographic dark energy,” Eur. Phys. J. C (2020) 826.[76] E. M. C. Abreu and J. A. Neto, “Barrow black hole corrected-entropy model and Tsallisnonextensivity,” [arXiv:2009.10133 [gr-qc]].[77] E. M. C. Abreu and J. Ananias Neto, “Barrow fractal entropy and the black hole quasi-normal modes,” Phys. Lett. B (2020) 135602.[78] E. M. C. Abreu and J. A. Neto, “Thermal features of Barrow corrected-entropy black holeformulation,”
Eur. Phys. J. C (2020) 776.[79] S. Vagnozzi, L. Visinelli, O. Mena and D. F. Mota, “Do we have any hope of detectingscattering between dark energy and baryons through cosmology?,” Mon. Not. Roy. Astron.Soc. , (2020) 1139.[80] E. Calabrese, R. Putter, D. Huterer, E. V. Linder and A. Melchiorri, “Future CMBconstraints on early, cold, or stressed dark energy,”
Phys. Rev. D (2011) 023011.[81] P. J. E. Peebles and B. Ratra, “The cosmological constant and dark energy,” Rev. Mod.Phys. (2003) 559.[82] S. Capozziello, “Curvature quintessence,” Int. J. Mod. Phys. D (2002) 483.[83] E. Piedipalumbo, M. De Laurentis and S. Capozziello, “Noether symmetries in InteractingQuintessence Cosmology,” Phys. Dark Univ. (2020) 100444.[84] B. A. Bassett, P. S. Corasaniti and M. Kunz, “The Essence of quintessence and the costof compression,” Astrophys. J. Lett. (2004) L1.[85] I. Zlatev, L. M. Wang and P. J. Steinhardt, “Quintessence, cosmic coincidence, and thecosmological constant,”
Phys. Rev. Lett. (1999) 896. [astro-ph/9807002].[86] R. R. Caldwell and E. V. Linder, “The limits of quintessence,” Phys. Rev. Lett. (2005)141301; [astro-ph/0505494].[87] X. Zhang, “Reconstructing holographic quintessence,” Phys. Lett. B648