Barrow HDE model for Statefinder diagnostic in FLRW Universe
aa r X i v : . [ g r- q c ] J a n Barrow HDE model for Statefinder diagnostic in FLRW Universe
Anirudh Pradhan , Archana Dixit , Vinod Kumar Bhardwaj , , Department of Mathematics, Institute of Applied Sciences and Humanities, GLAUniversityMathura-281 406, Uttar Pradesh, India E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
Abstract
We have analyzed the Barrow holographic dark energy (BHDE) in the frameworkof the flat FLRW Universe by considering the various estimations of Barrow exponent △ . Here we define BHDE, by applying the usual holographic principle at a cosmologicalsystem, for utilizing the Barrow entropy rather than the standard Bekenstein-Hawking.To understand the recent accelerated expansion of the universe, considering the Hubblehorizon as the IR cut-off. The cosmological parameters, especially the density parameter(Ω D ), the equation of the state parameter ( ω D ), energy density ( ρ D ) and the decelera-tion parameter( q ) are studied in this manuscript and found the satisfactory behaviors.Moreover, we additionally focus on the two geometric diagnostics, the statefinder ( r, s )and O m ( z ) to discriminant BHDE model from the Λ CDM model. Here we determinedand plotted the trajectories of evolution for statefinder ( r, s ), ( r, q ) and O m ( z ) diagnosticplane to understand the geometrical behavior of the BHDE model by utilizing Planck 2018observational information. Finally, we have explored the new Barrow exponent △ , whichstrongly affect the dark energy equation of state that can lead it to lie in the quintessenceregime, phantom regime, and exhibits the phantom-divide line during the cosmologicalevolution. Keywords : FLRW universe, Barrow HDE, Hubble horizon, Statefinder diagnosticPACS: 98.80.-k, 98.80.Jk,
The well-proven accelerated expansion of the universe is the greatest achievement of 20 th cen-tury [1, 2]. The dark energy (DE) with immense negative pressure is considered one of themysterious reasons behind the accelerated expansion of the universe. The WMAP experimentalso suggests that the Universe is made up of 4% of the Baryonic matter, 23% DM and 73%DE [3]. In the path of expansion, the universe passes through different phases of DE/matter.The DE is typically defined by the EoS parameter ( ω ) and the ranges include − / < ω < − ω < − ω = 1) for the cosmological constant.1ecently, researchers show a great interest in HDE models, since these HDE models devel-oped as applications of DE by following holographic principle [4]. The holographic principlederives from the thermodynamics of the black hole. String theory provides a relation betweenthe IR cutoff of quantum field theory linked to vacuum energy [4]- [7]. This concept has beenutilized widely in cosmological contemplations, especially in the late-time period of the Universe,at present known as, holographic dark energy models [8]- [22]. During this phase, we would liketo mention that Nojiri-Odintsov cut-off [8] gave the most general holographic dark energy andit is intriguing that it might be applied to covariant hypotheses [23]. So for solving the darkenergy puzzle, (HDE) speculation is a promising approach [16]- [18]. The new HDE models canbe proposed by utilizing holographic speculation and a generalized entropy. In addition to thedark energy model, it is also found that the HDE is important to analyze the early evolutionof the Universe, such as the inflationary evolution [24]- [29]. It is worth mentioned here somelatest papers [30]- [35] and their references on HDE in various scenarios.In this present work, we consider a spatially flat, homogeneous, and isotropic spacetime asthe underlying geometry. Here we study the behavior of different cosmological parameters (thedeceleration parameter, the energy density parameter, and the equation of state parameter )during the cosmic evolution by assuming the Hubble horizon as the infrared (IR) cut-off. TheHubble horizon as an IR cut-off is suitable to clarify the ongoing accelerated expansion of theDE models.In this direction, many cosmologists have presented mathematical diagnostics r, s , knownas statefinder parameters. For observing the nature of DE models, statefinder parameters arethe most important parameter [36]- [37]. In order to discriminate the various DE models, thetrajectories can be represented graphically in r − s and r − q planes. The state finder parametersare also analyzed [38, 39]. DE models like Chaplygin gas models, quintessence model, cosmo-logical constant and braneworld model as explored in [40] - [44].Barrow holographic dark energy (BHDE) is also a fascinating alternative scenario for thequantitative description of DE which is based on the holographic hypothesis [45]- [49] andapplying the recently proposed Barrow entropy [50] instead of the normal Bekenstein-Hawking[51,52]. Saridakis [53] have shown that the BHDE includes basic HDE as a sub-case in the limitwhere Barrow entropy becomes the usual Bekenstein-Hawking. Anagnostopoulos et al. [54]have shown that the BHDE is an agreement with observational data, and it can serve as a goodcandidate for the description of DE. Barrow holographic dark energy models have been studiedby several authors [55] - [60] in different contexts.On the other hand, concerning various cosmological theories, where DE interacts with DMhas extended much attention in the literature [61].The essential aspect in holographic principle in the cosmological level, is that the universeHorizon entropy is proportional to its area, as similar to the Bekenstein-Hawking entropy, witha black hole. The entropy of the black hole shown by Barrow can be modified as [50] S B = (cid:18) BB (cid:19) △ ≤ △ ≤ , (1)where B is the Planck area and B is the normal horizon area.2here is a quantum-gravitational deformation which enumerated by the parameter △ , for △ = 0 related to the Bekenstein-Hawking standard entropy and △ = 1 corresponding tothe most complex and fractal structure. The aim of the present manuscript is to examinethe BHDE model by taking the Hubble radius as an IR cutoff and analyzing the behavior ofcosmological parameters for a flat FRW universe. We extend our analysis to BHDE, inspired bythe works [53] with a similar IR cut-off which gives the ongoing stage progress of the Universe.We get the statefinder parameters for BHDE which accomplish the worth of Λ CDM modeland show consistency with the quintessence model for appropriate estimation of parameters.The plan of this manuscript is as follow: In section 2, we introduce the BHDE model proposedin [53] with a general interaction term between the dark components (BHDE and DM) of theuniverse and also study its cosmological evolution by considering the basic field equations. Thebehavior of state finder pair for BHDE has been discussed in section 3, we explore the O m diagnostic in section 4. Finally, section 5 is devoted to conclusions. In this section we develop the scenario of Barrow holographic dark energy, where the inequality ρ D L S , is given by the standard HDE. Here L is the horizon length under the assumption S ∝ A ∝ L [9] by using the Barrow entropy (1) obtain as lead to ρ D = CL △− , (2)where C is a parameter with dimensions [ L ] − −△ and L denotes the IR cutoff. In the casewhere △ = 0 as expected, the above expression provides the standard holographic dark energy ρ D = 3 c M p L − (here M p is the Plank mass), where C = 3 c M p and with the model parameter c . The above relation leads to some interesting results in the holographic and cosmological se-tups [53,54]. In [56,62], Barrow entropy was added in the structure of “gravity-thermodynamics”conjecture, according to which the first law of thermodynamics can be applied on the universeapparent horizon. As a result, one obtains a modified cosmology, with extra terms in the Fried-mann equations depending on the new exponent △ , which disappear in the case △ = 0, i.e whenBarrow entropy becomes the standard Bekenstein-Hawking one. Although this framework isdefined in a very effective way in the universe of late time. It should be noted here that thevalue △ = 1 corresponds to the maximal deformation, while the value △ = 0 corresponds tothe simplest horizon structure, and the normal Bekenstein entropy [51, 52] can be recoveredin this case. It is essential to note here that the entropy in equation (1), is close to Tsallis’non-extensive entropy [63, 64]. In the case where the deformation effects are quantified with △ , Barrow holographic dark energy will leave the regular one, leading to numerous cosmolog-ical variations. Recently, Barrow et al. [62] have used Big Bang Nucleosynthesis (BBN) datain order to impose constraints on the exponent of Barrow entropy. They have shown that theBarrow exponent should be inside the bound △ . . × − in order not to spoil the BBN epoch.Therefore, the BHDE is surely a more general structure than the standard HDE scenario.Here we concentrate on the general case of ( △ > H − as the IR cutoff ( L ), we can write the energy density of BHDE as ρ D = CH −△ (3)3et us consider a spatially flat, homogeneous and isotropic, FLRW universe the standardmetric is given by ds = − dt + a ( t )( dr + r d Ω ) (4)In a flat FLRW Universe, the field equations for BHDE are given as : H = 13 8 πG ( ρ D + ρ m ) (5)where ρ D is the energy density of BHDE and ρ m is the energydensity of matter respectively. The energy density parameter of BHDE and matter can begiven as Ω m = πρ m G H and Ω D = πρ D G H .We know that the relation Ω BD + Ω m = 1 (6)The conservation law BHDE and matter are defined as :˙ ρ m + 3 Hρ m = 0 (7)˙ ρ D + 3 H ( p D + ρ D ) = 0 (8)From Eq. (3), we get ˙ ρ D = 3 C − △ ) H −△ (cid:18) ∆Ω D (∆ − D + 2 − (cid:19) (9)Now, Eqs. (5), (7) and (8) and combining the outcome with the Eq. (6), we obtained˙ HH = 32 (cid:18) ∆Ω D (∆ − D + 2 − (cid:19) (10)The deceleration parameter q is written as q = − − ˙ HH (11)By using Eq. (10), the deceleration parameter q is also written as q = 1 − (∆ + 1)Ω D (∆ − D + 2 . (12)By utilizing the Eq. (8) with Eqs. (9) and (10), we get the expression for the EoS parameterderived as: ω D = − ∆(∆ − (cid:16) − ( z +1) Ω m0 − Ω m0 +( z +1) Ω m0 +1 (cid:17) + 2 , (13)where dash is the derivative, here we differentiate the EoS parameter ω D with respect to lna then we get ω ′ D . By using the Eqs. (9) and (10), we find ω ′ D = − (Ω D −
1) Ω D ((∆ − D + 2) . (14)4imilarly by using the Eqs. (9) and (10), we obtained Ω D as:Ω ′ D = − D (Ω D − − D + 2 (15) Δ = Δ = Δ = Δ = - - - - q · . Figure 1: Plot of deceleration parameter ( q ) with redshift z The evolution of q has been plotted in Fig. 1. As we observed from Fig. 1, the BHDE modelcan explain the universe’s history very well, with the sequence of an early matter-dominatedera. Here we plot the q versus z for a various choice of Barrow exponent △ . Which explainsthat the model is stable in the era of matter dominance. Moreover, we analyzed that in thehigh redshift phase, we have q → −
1, while at z → −
1. It is worth mentioning that, cosmosmay cross the phantom line ( q < −
1) for z < − △ . The deceleratingparameter approaches positive to negative values when the universe is overcome by dark energy.However, our findings based on the different values of the △ . If we take △ = 0 . , . , . , . q is deceleration to accelerating for the present time. Additionally,the transition redshift z t = 0 occurs within the interval − . < z t < .
25, which are in goodcompatibility with different recent studies (see Refs. [65]- [71] for more details about the modelsand cosmological datasets used). It has also been observed that the parameter z t depends onthe values of △ in such a way that, as △ increases, the parameter z t also increases. Accordingto the Planck measurement of Ω D , the value of r is 0 . ± . = Δ = Δ = Δ = - - - - - - ω D · . Figure 2: Plot of EoS parameter ( ω D ) with redshift z Next, we have shown the evolution of the EoS parameter ω D in Fig. 2 by considering differ-ent values of △ with respect to redshift z . The expression for equation of state parameter ω D represents in Eq. (13). One of the main efforts in observational cosmology is the measurementof EoS for dark energy (DE). Interestingly, we observed that for different values of △ , the EoSparameter ω D lies in the quintessence regime ( ω D > −
1) at the present epoch, however it entersin the phantom regime ( ω D < −
1) in the far future (i.e., z → − − ve value in between the region − − / − △ ,Barrow holographic dark energy can lie in the quintessence or in the the phantom regime, orexhibit the phantom-divide crossing during the cosmological evolution.6 Ω m Ω D
30 40 z D e n s it y P a r a m e t e r · . Figure 3: Plot of density parameter (Ω D ) with redshift z In this segment, we discuss cosmological development in the scenario of Barrow holographicdark energy. Figure 3 shows the evolution of the BHDE density parameter Ω D as a function ofthe redshift parameter z . From Fig. 2, it is evident that Ω approaches unity as the universeevolves to high redshift, and Where Ω D is the density parameter for BHDE, and the Ω m represents the density parameter of matter. By the assumption, [72], it has been seen that thecurrent universe is near a spatially at geometry (Ω ≈ z →
0, Ω > < z → ∞ , Ω = 1. In order to get a vigorous investigation to separate among DE models, many authors [36, 37]have presented a new mathematical diagnostic pair ( r, s ), known as statefinder parameter, whichis developed from the scale factor. These parameters ( r, s ) is geometrical in the behavior andit is developed from the space-time metric directly.The dynamics of the universe are comprehensively described by statefinder ( r, s ). These aredetermined as r = ... aaH (16) s = ( r − q − ) (17)The relation between the statefinder parameters r and s in terms of energy density can be7xpressed as r = (∆(∆(∆ + 12) − − D + 3(∆(4 − D + 3(∆ − D + 8((∆ − D + 2) (18) s = − D −
1) (2(∆ − D − ∆ + 2)((∆ − D + 2) (19)(a) Δ = Δ = Δ = Δ = - - r · . (b) Δ = Δ = Δ = Δ = - - s · . Figure 4: (a) Plot of r with redshift z (b) Plot of s with redshift z The evolution of r and s with redshift z for FLRW universe has been analyzed in Figs.4 a and 4 b [74]. The primary parameter r of Oscillating dark energy (ODE), at high redshift,approaches standard Λ CDM behavior while at low redshift it goes deviates significantly fromthe standard behavior and the second parameter s shows opposite in behavior [75]. Figures 4 a and 4 b portray evaluation of r and s for different values of Barrow parameter △ and approachesto the Λ CDM , by taking the value (for Ω m = 0 .
27 and H = 69 .
5) are in good arrangementwith recent observations. As expected, for △ = 0 the above modified Friedmann equationsreduce to Λ CDM scenario. The study of the statefinder provides a very useful method to splitthe conceivable depravity of different cosmological models by determining the parameters r ,and s for the higher order of the scale factor. 8a) Δ = Δ = Δ = Δ = Q(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15) Λ C(cid:16)(cid:17)
Chaplygingas - - (cid:18)(cid:19)(cid:20) (cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26) r (b) Δ = Δ = Δ = Δ = (cid:27)(cid:28)(cid:29)(cid:30) S(cid:31) ! Λ CDM - - - - q r · . Figure 5: (a) Plot of ( r − s ) with z (b) Plot of ( r − q ) with z In Fig. 5 a , we plot r − s trajectories which divided into two regions. The region r > s < r − s plane, shows a behaviour similar to a Chaplygin gas (CG) model [76] whereas theregion r < s > △ = 0 . , . , . , .
85 andapproaches Λ
CDM at late times. The trajectories in both region coincide for all differentvalues of △ . The statefinder r, s of BHDE model approaches to the Λ CDM . In addition, wealso plot the evolution trajectory in the r − q plane in figure 5 b . The ( r − q ) trajectories aredivided into two regions through the point ( r, q ) = (1 , − r > q < − r − q plane shows a behaviour similar to the phantom model, while the region r < q > − r, q = 1 , − O m ( z ) Diagnostics
The Om diagnostic analysis is also a very useful geometrical diagnosis that can be used for suchanalysis. In the investigation of the statefinder parameter ( r, s ) the higher order derivative of a ( t ) are utilized. The Ist order derivative are used in O m diagnostic analysis because it containsonly the Hubble parameter.The O m diagnostic can be considered as an easier diagnostic [77]. It might be noticed thatthe O m diagnostic has been additionally applied to [78]- [80]. This arrangement of parameterscan be written as: Om ( z ) = H ( z ) H − z ) − .0 0.5 1.0 1.5 2.01234 zO" ( ) · . Figure 6: Plot of O m ( z ) with redshift z In figure 6, we plot the O m ( z ) evolution with redshift. The positive curve of the O m ( z )trajectories shows the phantom behaviour ( ω < −
1) whereas the negative curve implies that DEbehaves like quintessence ( ω > − < z < .
0, the O m ( z ) is diminishing monotonically and the curve lies inphantom region. The new diagnostic of dark energy O m is acquainted with separate Λ CDM from other DE models. Where H is the current estimation of the Hubble parameter. Herewe demonstrated that the slope of O m ( z ) can recognize dynamical DE from the cosmologicalconsistent in a robust way. In this model, we have discussed the BHDE, by considering the typical holographic principle ata cosmological system, by utilizing the Barrow entropy, rather than the standard Bekenstein-Hawking. Here we have also discussed the evolution of a spatially flat FLRW universe composedof pressure less dark matter and Barrow holographic dark energy. By considering the Hubblehorizon as the infrared cut-off, we have found the exact solution and the calculated cosmologi-cal parameters like the behavior of the density parameter, the EoS parameter, the decelerationparameter, statefinder, and O m diagnostic parameters, etc. We also plotted the trajectories in( r − s ) , ( r − q ), and O m to discriminate the various DE model from the existing BHDE modelsduring the cosmic evolution.The main highlights of the models are as per the following: • It has been found that the BHDE model exhibits a smooth transition from earlydeceleration era ( q >
0) to the present acceleration ( q <
0) era of the universe in Fig.10. Also, the value of this transition redshift is in good accordance with the currentcosmological observations and obtained for the different values of the △ . • It has been observed in Fig. 2 that the new Barrow exponent △ essentially influencesthe dark energy equation of state and as per its worth it lies in the quintessence regime( ω D > − ω D < −
1) inthe far future (i.e., z → −
1) by using different values of △ . • The energy density parameter is also discussed and shown in Fig. 3. We found that inthe cosmic evaluation of BHDE Ω, approaches unity. which is a good agreement withrecent observations. • We have also discussed the statefinder ( r, s ) in terms of the dimensionless densityparameters and Barrow exponent △ . We have plotted r verses z in Fig. 4 a . The r ( z ) parameter of oscillating dark energy (ODE) depicts in high red shift region andapproaches to standard Λ CDM . Similarly we have obtained s ( z ) in Fig. 4 b , where s ( z )parameter shows opposite behaviour to the primary parameter r . • The excellent diagnostics of DE is shown in Figs. 5 a and 5 b which are ( r, s ) and ( r, q ).Here we take the value Ω m = 0 . H = 69 . △ = 0 . , . , . , . r, s ) and( r, q ) obtained the Chaplygin gas (CG) model, steady state (SS) model, quintessences(Q-model) etc. Now we observe that the statefinders play a very important role in theFLRW universe with BHDE. • The O m -diagnostic technique is used to check the stability of the model and variousperiods of the Universe. We plot the trajectory in Om(z)plane to separate the conductof the DE models in Fig. 6. The positive inclination of the curve shows the phantom-likebehavior of the modelIn summary, in the manuscript, the physical behavior of the cosmological parameters arestudied through their graphical representation. This BHDE model is in a good agreement withcosmological informations, and it can fill in as a decent possibility for the graphical representa-tion. The author (AP) thanks the IUCAA, Pune, India for providing the facility under visitingassociateship. The authors are also thankful to the anonymous referee for his/her constructivecomments which helped to improve the quality of paper in present form.
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