Exploring the Sharma-Mittal HDE models with different diagnostic tools
EExploring the Sharma-Mittal HDE models with different diagnostic tools
Umesh Kumar Sharma ∗ , Vipin Chandra Dubey † , Department of Mathematics,Institute of Applied Sciences and Humanities, GLA UniversityMathura-281406, Uttar Pradesh, India
In this paper, we have examined the Sharma-Mittal holographic dark energy model (SMHDE) inthe framework of an isotropic and spatially homogeneous flat Friedmann-Robertson-Walker(FRW)Universe by considering different values of parameter δ and R , where the infrared cut-off is takencare by the Hubble horizon. We examined the SMHDE model through the analysis of Statefinderhierarchy and the growth rate of perturbation. The evolutionary trajectories of the statefinderhierarchy S , S S , S versus redshift z, show satisfactory behavior throughout the Universeevaluation. One promising tool for investigating the dark energy models is the composite nulldiagnostic(CND) { S − (cid:15) } , where the evolutionary trajectories of the S − (cid:15) pair present differentproperty and the departure from ΛCDM could be well evaluated. Additionally, we investigated thedynamical analysis of the model by ω D − ω (cid:48) D pair analysis. I. INTRODUCTION
Various cosmological observations show that ourUniverse is going through accelerated expansion phaseat present [1–6]. The concept of dark energy (DE)was used to explain this accelerated expansion of thecosmos, where DE has negative pressure [7–10]. Thereare basically two methods to explain the late-timeacceleration of the Universe. Firstly, the dynamical darkenergy model in which the matter part of the Einsteinfield equation can be changed. In all the theories andmodels, the cosmological constant model is the simplestone and was elucidated by Einstein [11–14], which givesresultant in the face of the equation of state parameter(EoS) ω = −
1, the most basic applicant for dark energyis the cosmological constant and it is consistent fromthe prospective of observations, except the coincidenceand fine-tuning problem [12, 15]. As an answer to theproblem various dynamical dark energy models are givenas an alternative like quintessence [16, 17], phantom[18], k -essence [19–21], tachyon [22] and Chaplygin gas[23]. Secondly, by modified gravity theories, which areachieved by modifying the geometric part of Einsteinfield equation [24–29].Many dark energy models have been proposed sofar to explain the accelerated expansion phase of theUniverse inspired by the holographic principle, which inlast propagates the theory that the degree of freedomis dependent on the bounding area and not on volume[30–33]. M.Li, in 2004, proposed Holographic darkenergy (HDE) taking future event horizon as IR cutoffto describe the accelerated expansion scenario of theUniverse [34]. The holographic dark energy modelhas been taken into account broadly and studied in ∗ [email protected] † [email protected] the literature [35–39], as ρ D ∝ Λ , and the relationbetween the UV cutoff Λ, entropy S and IR cutoff L isΛ L ≤ ( S ) , that shows that the combination of theentropy with the IR cut-offs gives energy density of HDEmodel. The declaration of ρ D is the focal point and isachieved by consideration of the dimensional analysis& the holographic principle instead of the inclusion ofthe expression of the dark energy into the Lagrangian.This is the basis for the importance of the HDE andthe original Holographic DE model is dependent onBekenstein-Hawking entropy S = A G , where A = 4 πL ,so the density is ρ D = c πG L − , here c is numericalconstant. Three years after the HDE i. e. in 2007,Cai proposed the Agegrapic dark energy (ADE) modeltaking length measure as the age of the Universe [40].Due to some confusion in the original ADE modelproposed by Cai, Wei and Cai in 2008, proposed theNew agegraphic dark energy (NADE) model consideringconformal time as time scale [41]. The Ricci dark energywas proposed by Gao et al. [42] replacing future eventhorizon with Ricci scalar curvature inspired by theholographic principle.Recently, Different entropies [43–46] have also beenused to propose some new forms of dark energy modelin the investigation of gravitational and the cosmo-logical incidences. Inspired by hologrphic principleand using various system entropies, some new formof dark energy modolels were proposed, for example,the R´ e nyi holograpic dark energy (RHDE) model [47],Tsallis holographic dark energy (THDE) model [48],Tsallis agegraphic dark energy (TADE) model [49] andSharma-Mittal holograpic dark energy (SMHDE) model[50]. These newly proposed dark energy models wereinvestigated by various researchers in different scenario[51–63].As the number of dark energy models is increasingday by day, the diagnostic tools which can discrim- a r X i v : . [ phy s i c s . g e n - ph ] M a r inate them are required. The statefinder hierarchyand the growth rate of linear perturbations, as nulldiagnostics for the ΛCDM model, was introduced byArabsalmani and Sahni [64] to discriminate the differentdark energy models from the ΛCDM model. Thestatefinder hierarchy contains high derivatives of scalefactor a ( t ), model-independent and is a geometricaldiagnostic [65]. Previously to check scale-independentconsistency between the structure growth and theexpansion history, the growth rate of the structurewas used in [66–68]. It can be combined with thestatefinder hierarchy or act as a cosmic growth historydiagnostic to serve on a composite diagnostic. Fourholographic DE models were discriminated against bythese two diagnostics in [65]. In [68–76] these diagnosticswere considered. Recently, the discrimination betweenTHDE models ΛCDM model investigated by one of theauthors through statefinder hierarchy in the nonflatUniverse considering apparent horizon as IR cutoff[77]. The ω D − ω (cid:48) D analysis [78] can also be utilized torecognize the difference in dark energy models, which isbased on the behavior of EoS for the dark energy models.In this work, we have explored the newly proposedSharma-Mittal Holographic Dark Energy (SMHDE)model through the diagnostic tools described above inthe flat FRW Universe by taking the Hubble horizon asan infrared cutoff, which has not been explored earlier.Also, we have examined the deviation of the SMHDEmodel from ΛCDM using these diagnostic tools. Thispaper is structured as follows; In Section II, we brieflyvisit the Sharma-Mittal holographic dark energy. SectionIII is dedicated to discussing the flat FRW cosmologicalmodel. Section IV is divided into three subsections A, Band C for the methods of the statefinder hierarchy diag-nostic and growth rate of perturbations and diagnosticby the ω D − ω (cid:48) D analysis. Finally, in the last section, wehave given inferences. II. SHARMA-MITTAL HDE MODEL
Recently, inspired by holographic principle and usinggeneralized entropy measure, proposed by Sharma-Mittal[46], a new form of holographic dark energy model isproposed in [50], called Sharma-Mittal holographic darkenergy.By combining the Tsallis and R´ e nyi entropies [44, 45],which are two well-known generalized one-parametric en-tropy measures, with each other, a two-parametric en-tropy, which was introduced by Sharma-Mittal, is definedin [46] S SM = (cid:0) Aδ + 1 (cid:1) R/δ − R , (1)where A = 4 πL and the IR cutoff is L. Where two free parameters are R and δ . By considering proper limitsof R, R´ e nyi and Tsallis entropies can be recovered fromit. The Sharma-Mittal entropy becomes R´ e nyi entropyin the limit R →
0, and in limit R → − δ , it becomesTsallis entropy. The energy density is obtained when theUV cutoff and IR cutoff are taken into consideration aswas suggested by Cohen et al.[79]. ρ D ∝ S SM L = ⇒ ρ D = 3 c S SM πL , (2)Considering Hubble horizon cut-off L = H , when we takethe aforementioned equation into consideration then theenergy density of Sharma-Mittal HDE model [50] is ρ D = (cid:0) c H (cid:1) (cid:16)(cid:0) πδH + 1 (cid:1) R/δ − (cid:17) πR , (3)where c is a numerical constant as usual. III. THE COSMOLOGICAL MODEL
For the flat FRW Universe, the metric is given as : ds = − dt + a ( t ) (cid:16) dr + r d Ω (cid:17) (4)In a flat FRW Universe, the first Friedmann equation,involving dark matter and SMHDE is defined as : H = 13 (8 πG ) ( ρ D + ρ m ) , (5)where ρ D and ρ m represent the energy density ofSMHDE and matter, respectively. The energy densityparameter of SMHDE and pressureless matter using thefractional energy densities, can be given asΩ m = 8 πρ m G H , Ω D = 8 πρ D G H , (6)Now Eq. (5) with help of Eq. (6) can be written as:1 = Ω D + Ω m (7)The conservation law for matter and SMHDE are givenas : ˙ ρ m + 3 Hρ m = 0 , (8)˙ ρ D + 3 H ( ρ D + p D ) = 0 . (9)in which ω D = p D /ρ D represents the SMHDE EoSparameter. Now, using differential with time of Eq. (5)in Eq. (8), and Eqs. (9) combined the result with theEq. (7), we get ˙ HH = − (cid:0) − Ω D ) (cid:0) πδ + H (cid:1)(cid:1) × (cid:16) πc H (cid:0) πδH + 1 (cid:1) R/δ + πδ − πδ Ω D − H Ω D + H (cid:17) (10)By Eq. (10), The deceleration parameter q is found as q = − − ˙ HH = − D − (cid:0) πδ + H (cid:1) D −
1) ( πδ + H ) − πc H (cid:0) πδH + 1 (cid:1) R/δ (11)Now, taking the differential with respect to time of Eq.(3), we get˙ ρ D = 4 ρ D ˙ HH − c H ˙ H (cid:18) πδH + 1 (cid:19) Rδ − (12)Now by using the Eqs. (12) with Eqs. (9) and (10),we gets expression for EoS parameter as: ω D = Ω − D (1 − (Ω D − (cid:0) πδ + H (cid:1) × πc H (cid:0) πδH + 1 (cid:1) R/δ − (2Ω D −
1) ( πδ + H ) ) (13)Also, taking the time differential of the energy densityparameter Ω D with Eqs. (10) and (12), we findΩ (cid:48) D = − (3 (Ω D −
1) ( π (cid:0) − c (cid:1) H (cid:0) πδH + 1 (cid:1) R/δ + πδ Ω D + H Ω D ) × ( π (cid:0) − c (cid:1) H (cid:0) πδH + 1 (cid:1) R/δ − πδ + 2 πδ Ω D + 2 H Ω D − H ) − (14)where the dot is the derivative while taking time intoconsideration and prime lets us obtain the derivativewith respect to ln a. IV. THE METHODS OF DIAGNOSTIC
In this work, we used three diagnostic tools, statefinderhierarchy, the growth rate of perturbations and ω D − ω (cid:48) D pair. We shall explore the SMHDE model to discriminatefrom the ΛCDM model with the help of three diagnostictools in this section. A. The Statefinder Hierarchy diagnostic
Here, statefinder hierarchy diagnostic will be reviewedand then the growth rate of structure of the SMHDEmodel will be described. The Taylor expansion of the δ = - δ = - δ = -
800 LCDM - - S ( ) R = = = - - S ( ) FIG. 1: Graph of S (1)3 versus redshift z, for non- interactingSMHDE with Hbbble radius as the IR cutoff. Here, H ( z =0) = 67, Ω m ( z = 0) = 0 . R = 10000 and different valuesof δ (upper panel) and H ( z = 0) = 67, Ω m ( z = 0) = 0 . δ = −
600 and different values of R (below panel). scale factor a ( t ) a = z +1 , around the present epoch t isgiven as: a ( t ) a = ∞ (cid:88) n =1 A n ( t ) n ! [ H ( t − t )] n (15)Where A n = a n aH n , a n is the n th derivative of the scalefactor a verses cosmic time t and n ∈ N. The statefinder δ = - δ = - δ = -
800 LCDM - - - - - - - - - S ( ) R = = = - - - - - S ( ) FIG. 2: Graph of S (2)3 versus redshift z, for non- interactingSMHDE with Hbbble radius as the IR cutoff. Here, H ( z =0) = 67, Ω m ( z = 0) = 0 . R = 10000 and different valuesof δ (upper panel) and H ( z = 0) = 67, Ω m ( z = 0) = 0 . δ = −
600 and different values of R (below panel). hierarchy S n is defined as follows [64]: S = A + 3Ω m , S = A and S = A + 9Ω m , (16)Aforementioned gives the diagnostics for the model(ΛCDM) with n ≥
3, i.e., S n | ΛCDM = 1. Hence bythe use of Ω m = q +1)3 the statefinder hierarchy S (1)3 , S (1)4 can be written as: S (1)3 = A , and S (1)4 = A + 3( q + 1) , (17)For ΛCDM model, S (1) n = 1. In [80] it gives a path for δ = - δ = - δ = -
800 LCDM - - S ( ) R = = = - - S ( ) FIG. 3: Graph of S (1)4 versus redshift z, for non- interactingSMHDE with Hbbble radius as the IR cutoff. Here, H ( z =0) = 67, Ω m ( z = 0) = 0 . R = 10000 and different valuesof δ (upper panel) and H ( z = 0) = 67, Ω m ( z = 0) = 0 . δ = −
600 and different values of R (below panel). construction of second Statefinder S (1)3 = S namely S (2)3 = S (1)3 − (cid:0) q − (cid:1) (18)In concordance cosmology S (1)3 = 1 while S (2)=03 .Hence, (cid:110) S (1)3 , S (2)3 (cid:111) = { , } gives a model independentmeans for forming a distinction between the dark energymodels from the cosmological constant [80]. Eq. (18)gives the second member of the Statefinder hierarchy S (2) n = S (1) n − α (cid:0) q − (cid:1) , (19) δ = - δ = - δ = -
800 LCDM - - - - - - - S ( ) R = = = - - - - - - - S ( ) FIG. 4: Graph of S (2)4 versus redshift z, for non- interactingSMHDE with Hbbble radius as the IR cutoff. Here, H ( z =0) = 67, Ω m ( z = 0) = 0 . R = 10000 and different valuesof δ (upper panel) and H ( z = 0) = 67, Ω m ( z = 0) = 0 . δ = −
600 and different values of R (below panel). where α is an arbitrary constant. In concordance cos-mology S (2) n = 0 and (cid:110) S (1) n , S (2) n (cid:111) = { , } , (20)Some of degeneracies in S (1) n can be removed by usingthe second statefinder S (2) n . For the dark energy model,we have S (1)3 = 12 (9 ω D ) ( ω D + 1) Ω D + 1 (21) S (2)3 = ω D + 1 (22) S (1)4 = − (cid:0) ω D (cid:1) ( ω D + 1) Ω D −
12 (27 ω D ) ( ω D + 1) (cid:18) ω D + 76 (cid:19) Ω D + 1 (23) S (2)4 = − ω D ( ω D + 1) Ω D − ( ω D + 1) (cid:18) ω D + 76 (cid:19) (24)where S (2)4 = S (1)4 − ( q − ) and q − = (3 ω D ) Ω D . As wedemonstrate in figures 1, 2, 3, 4 the Statefinder hier-archy (cid:110) S (1) n , S (2) n (cid:111) give us a nice way to differenciatingdynamical dark energy models from ΛCDM model.Fig. 1, shows the evolutionary trajectories of S (1)3 ( z )for the SMHDE model by considering different valuesof δ (upper panel) and R (below panel). Hence, weinvestigate two cases. The first is varying δ with a fixed R (upper panels), the second is varying R and a fixed δ (below panel). For the evolution of S (1)3 ( z ) in theSMHDE with varying R , the separation of curvilinearshape is not distinct from the SMHDE with varying δ . In the case of varying δ or R in SMHDE (upperpanel or below panel), the curves which are of S (1)3 ( z )have the trajectories on the line of being similar andthe trend, which is being followed by of curves S (1)3 ( z )is monotonically decreasing at the high-redshift regionand then followed by close degeneration together intoΛCDM S (1)3 = 1, at low-redshift region. This showsthat different values of δ or R have quantitative impactson the S (1)3 ( z ). Although, in both panels, the curvesdiscriminate well from ΛCDM in the high-redshift regionbut highly degenerate in the low-redshift region.Fig. 2, shows the evolutionary trajectories of S (2)3 ( z )for the SMHDE model by considering different valuesof δ (upper panel) and R (below panel). For theevolution of S (2)3 ( z ) in the SMHDE with varying R,the differentiation of curvilinear shape is not distinctfrom the SMHDE with varying δ . In the case of varying δ or R in the SMHDE (upper panel or below panel),the curves which are of S (2)3 ( z ) have the trajectorieson the line of being similar and the trend, which isbeing followed by of curves S (2)3 ( z ) is monotonicallyincreasing at the high-redshift region and then followedby close degeneracy together with ΛCDM S (2)3 = 0, atlow-redshift region. This shows that different values of δ or R have quantitative impacts on the S (2)3 ( z ). In Fig. 3,we give the graph for S (1)4 evolution versus z i.e. redshiftfor the SMHDE model by considering different values of δ (upper panel) and R (below panel). We can say thatthe evolutionary trajectories of S (1)4 ( z ) are like that of S (1)3 ( z ). Quantitative impacts the SMHDE model arefound by adopting different values of δ and R .In Fig. 4, we give the graph for S (2)4 evolution versus z i.e. redshift for the SMHDE model by consideringdifferent values of δ (upper panel) and R (below panel).We can say that the evolutionary trajectories of S (2)4 are like that of S (2)3 ( z ). Quantitative impacts on theSMHDE model are found by adopting different values of δ and R and this is endorsed by the figures.Therefore, in all the plots i.e. Fig. 1-4, there is adrawback that the curves are highly degenerate in thehigh-redshift region and superposing that of ΛCDMin the low-redshift region. It means that the singlegeometric diagnostic is not sufficient. It will be better tocombine with the growth rate of perturbations, as CNDfor getting more clear discrimination. B. Growth rate of perturbations
The fractional growth parameter (cid:15) ( z ) [66, 67] is deter-mined as (cid:15) ( z ) = f ( z ) f ΛCDM ( z ) (25)Here f ( z ) = d log δd log a is the growth rate of structure. Here, δ = δρ m ρ m , with δρ m and ρ m being the the density pertur-bation and energy density of matter (including CDM andbaryons), respectively. If the perturbation is in the lin-ear fashion and without any interaction between DM andDE, then we can say that the equation of perturbationat late times can be:¨ δ + 2 ˙ δH = 4 πδGρ m (26)Here, Newton’s gravitational constant is representedby G . So, the approx growth rate of linear density per-turbation can be reflected by [81]: f ( z ) (cid:39) Ω m ( z ) γ (27) γ ( z ) = ( − ω D ) ( − ωD )) (1 − Ω m ( z ))125 ( − ωD ) +35 − ω D − ω D (28)where Ω m ( z ) = ρ m ( z )3 H ( z ) M p , the fractional density ofmatter, Ω is constant or varies slowly with time. (cid:15) ( z )= 1 and γ (cid:39) .
55 are the values for the ΛCDM model[81, 82]. For other models (cid:15) ( z ) exhibits differences fromΛCDM which would be the possible reason for its use asa diagnostic. By applying the composite null diagnostic CN D ≡ { S n, (cid:15) } where { S n, (cid:15) } = { , } for ΛCDM, wecan make use of both matter perturbational as well asgeometrical information of cosmic evolution. While, we δ = - δ = - δ = - - ϵ ( z ) R = = = - ϵ ( z ) FIG. 5: Graph of (cid:15) ( z ) versus redshift z, for non- interactingSMHDE with Hbbble radius as the IR cutoff. Here, H ( z =0) = 67, Ω m ( z = 0) = 0 . R = 10000 and different valuesof δ (upper panel) and H ( z = 0) = 67, Ω m ( z = 0) = 0 . δ = −
600 and different values of R (below panel). can analyze and present only one-side information ofcosmic evolution by using one single diagnostic tool.For the diagnose of diverse theoretical DE models,having CND pairs, (cid:110) S (1)3 , (cid:15) (cid:111) and (cid:110) S (1)4 , (cid:15) (cid:111) , the evolutionof the fractional growth parameter (cid:15) ( z ) is analysed.Fig. 5 is the evolutionary trajectories of (cid:15) ( z ) versusredshift z for a spatially homogeneous and an isotropicflat FRW Universe of SMHDE model by consideringdifferent values of δ (upper panel) and R (below panel).For the evolution of (cid:15) ( z ) in the SMHDE with varying R , the differentiation of curvilinear shape is not distinctfrom the SMHDE with varying δ . We can say thatthe evolutionary trajectories of (cid:15) ( z ) have similar evo-lutionary trajectories. It is clear from Fig. 5, that the δ = - δ = - δ = - ★ LCDM0.85 0.90 0.95 1.000.960.981.001.021.041.061.081.10 ϵ ( z ) S ( ) R = = = ★ LCDM0.85 0.90 0.95 1.000.960.981.001.021.041.061.081.10 ϵ ( z ) S ( ) FIG. 6: Graph of Graph of S (1)3 versus (cid:15) ( z ), for non- inter-acting SMHDE with Hbbble radius as the IR cutoff. Here, H ( z = 0) = 67, Ω m ( z = 0) = 0 . R = 10000 anddifferent values of δ (upper panel) and H ( z = 0) = 67,Ω m ( z = 0) = 0 . δ = −
600 and different values of R (belowpanel). evolutionary trajectories of (cid:15) ( z ) comes closer to 1 frompast to future.The evolutionary trajectories of (cid:110) S (1)3 , (cid:15) (cid:111) of SMHDEmodel are plotted in Fig. 6 for a spatially homogeneousand an isotropic flat FRW Universe of SMHDE modelby considering different values of δ (upper panel) and R (below panel). The fixed point (1 ,
1) in this figurepresented by by star symbol denotes the Λ CDM. Thetrend of curves (cid:110) S (1)3 , (cid:15) (cid:111) is monotonically decreasingfrom the high-redshift region to low red-shift region forthe SMHDE model. This figure clearly detpicts thedeviation from ΛCDM model (cid:110) S (1)3 = 1 , (cid:15) = 1 (cid:111) for all δ = - δ = - δ = - ★ LCDM0.85 0.90 0.95 1.000.960.981.001.021.041.061.081.10 ϵ ( z ) S ( ) R = = = ★ LCDM0.85 0.90 0.95 1.000.960.981.001.021.041.061.081.10 ϵ ( z ) S ( ) FIG. 7: Graph of Graph of S (1)4 versus (cid:15) ( z ), for non- inter-acting SMHDE with Hbbble radius as the IR cutoff. Here, H ( z = 0) = 67, Ω m ( z = 0) = 0 . R = 10000 anddifferent values of δ (upper panel) and H ( z = 0) = 67,Ω m ( z = 0) = 0 . δ = −
600 and different values of R (belowpanel). values of δ and R of the SMHDE model.Fig. 7 is the the evolutionary trajectories of the CNDpair (cid:110) S (1)4 , (cid:15) (cid:111) for the SMHDE model by considering dif-ferent values of δ (upper panel) and R (below panel). Theevolutionary trajectories of (cid:110) S (1)4 , (cid:15) (cid:111) shows similar char-acteristic as the curves of (cid:110) S (1)3 , (cid:15) (cid:111) . These results showsthat adopting different values of δ and R has quantita-tive impacts and the deviation from ΛCDM can be seenin this figure. ★ LCDM δ = - δ = - δ = - - - - - - - ω D ω D ' R = = = ★ LCDM - - - - - - - - ω D ω D ' FIG. 8: The evolution trajectories in the ω D − ω (cid:48) D plane ofthe SMHDE model, for non- interacting SMHDE with Hbbbleradius as the IR cutoff. Here, H ( z = 0) = 67, Ω m ( z = 0) =0 . R = 10000 and different values of δ (upper panel) and H ( z = 0) = 67, Ω m ( z = 0) = 0 . δ = −
600 and differentvalues of R (below panel).
C. The ω D − ω (cid:48) D analysis The sign of ω (cid:48) D can be used in the thawing andfreezing models [78] and ω D is the equation of stateparameter characterizing the dark energy model. Hence ω D − ω (cid:48) D pair analysis has been used to differentiate thesimilar model behaviours [83–89]. where ω (cid:48) D = dω D d log a .We investigated the dynamical diagnosis ω D − ω (cid:48) D forSMHDE model which is also utilized widely in theliterature. In this dynamical analysis, the fixed point ω D = − ω (cid:48) D = 0 represents to the standard ΛCDM in the ω D − ω (cid:48) D diagram.The evolutionary trajectories of ω (cid:48) D and ω D plane areshown in Fig. 8, for an isotropic and spatially homoge-neous flat FRW Universe of SMHDE model by consid-ering different values of δ (upper panel) and R (belowpanel). It is clear from ω D − ω (cid:48) D trajectories by consider-ing different values of δ (upper panel) that for all values of δ ω D ≥ − ω D = −
1. It also depicts that currently SMHDE modellies in the thawing region ( ω D < , ω (cid:48) D >
0) as well as infreezing region ( ω D ≥ −
1) which means presently, cosmicexpansion is accelerating.
V. CONCLUSIONS
The paper uses the Sharma-Mittal Holographic DarkEnergy (SMHDE) model in flat FRW Universe byconsidering different Sharma-Mittal parameter δ . Thiscan be summarized as • We studied the deviation of SMHDE model fromΛCDM regarding different values of Sharma-Mittalparameter δ by the use of the diagnostics ofstatefinder hierarchy and growth rate of structure.The statefinder hierarchy gives analytical expres-sions of S (1)3 , S (2)3 , S (1)4 and S (2)4 , for SMHDE in ascosmological parameters. To check the growth rateof structure (cid:15) ( z ) has been calculated analyticallyfor the SMHDE model. We tested the SMHDEmodel using (cid:110) S (1)3 , (cid:15) (cid:111) diagnostics. We plottedthe evolution curves of S (1)3 , S (2)3 , S (1)4 and S (2)4 with respect to cosmic time z and (cid:15) ( z ). Theseevolutionary trajectories shows that the SMHDEmodel shows ΛCDM behaviour at late time.We have plotted the evolutionary trajectories of (cid:110) S (1)3 , (cid:15) (cid:111) plane which depicts that SMHDE modelfor all values of δ shows same deviation fromΛCDM model. • The various diagnostic methods for dark energyhave been discussed. we have also examined the ω D − ω (cid:48) D pair analysis for our SMHDE modelin subsection C . These analysis are used todifferentiate among various dark energy models.In the subsection 5 . ω D − ω (cid:48) D for SMHDE model where thederivative with respect to log a is denoted byprime notation. The evolutionary trajectories of ω D − ω (cid:48) D shows that presently cosmic expansionis accelerating since our SMHDE model lies in thethawing region ( ω D < , ω (cid:48) D > ω D crosses the phantom divide line ω D = − ω D ≥ − ω D − ω (cid:48) D pair, the growth rate of per-turbations and statefinder hierarchy to diagnosethe SMHDE model. Some other diagnostic toollike statefinder diagnostic can also be used todiscriminate the SMHDE from ΛCDM model.We hope that in future high precision observa-tions, for example, SNAP-type investigation canbe equipped for deciding the cosmological param- eters exactly and consequently identify the correctcosmological model and closer to understand theproperties of the SMHDE model. Acknowledgments
The authors are thankful for valuable suggestions givenby Dr. Prateek Pandey, GLA University, Mathura, India,in this research work. [1] A. G. Riess et al. [Supernova Search Team], “Observa-tional evidence from supernovae for an accelerating uni-verse and a cosmological constant,” Astron. J. (1998)1009. doi:10.1086/300499.[2] S. Perlmutter et al. [Supernova Cosmology Project Col-laboration], “Measurements of Ω and Λ from 42 highredshift supernovae,” Astrophys. J. (1999) 565.doi:10.1086/307221 [astro-ph/9812133].[3] N. Aghanim et al. , Planck 2018 results. VI. Cosmologi-cal parameters. preprint (2018), arXiv:1807.06209 [astro-ph.CO].[4] M. Colless et al. [2DFGRS Collaboration], “The2dF Galaxy Redshift Survey: Spectra and red-shifts,” Mon. Not. Roy. Astron. Soc. (2001) 1039doi:10.1046/j.1365-8711.2001.04902.x[5] M. Tegmark et al. [SDSS Collaboration], “Cosmologicalparameters from SDSS and WMAP,” Phys. Rev. D (2004) 103501 doi:10.1103/PhysRevD.69.103501[6] D. N. Spergel et al. [WMAP Collaboration], “First yearWilkinson Microwave Anisotropy Probe (WMAP) obser-vations: Determination of cosmological parameters,” As-trophys. J. Suppl. (2003) 175 doi:10.1086/377226[7] E. J. Copeland, M. Sami and S. Tsujikawa, “Dynamicsof dark energy,” Int. J. Mod. Phys. D (2006) 1753doi:10.1142/S021827180600942X [hep-th/0603057].[8] K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov,“Dark energy cosmology: the equivalent description viadifferent theoretical models and cosmography tests,” As-trophys. Space Sci. (2012) 155 doi:10.1007/s10509-012-1181-8[9] W. Yang, S. Pan, E. Di Valentino, R. C. Nunes,S. Vagnozzi and D. F. Mota, “Tale of stable inter-acting dark energy, observational signatures, and the H tension,” JCAP (2018) 019. doi:10.1088/1475-7516/2018/09/019 [arXiv:1805.08252 [astro-ph.CO]].[10] L. Amendola, S. Tsujikawa, “Dark energy: theory andobservations.” Cambridge University Press (2010).[11] V. Sahni, A. Starobinsky, ”The case for a positive cos-mological Λ-term,” Int. J. Mod. Phys. D (04) (2000)373-443.[12] P. J. E. Peebles and B. Ratra, “The Cosmological con-stant and dark energy,” Rev. Mod. Phys. (2003) 559.doi:10.1103/RevModPhys.75.559[13] T. Padmanabhan, “Dark energy and gravity,” Gen. Rel.Grav. (2008) 529. doi:10.1007/s10714-007-0555-7[14] S. M. Carroll, “The Cosmological constant,” Living Rev.Rel. (2001) 1. doi:10.12942/lrr-2001-1[15] S. Weinberg, “The Cosmological Constant Problem,” Rev. Mod. Phys. (1989) 1.doi:10.1103/RevModPhys.61.1[16] P. J. E. Peebles and B. Ratra, “Cosmology with a time-variable cosmological’constant,” Int. J. Mod. Phys. A (1988) L17-L20.[17] M. S.Turner, “Making sense of the new cosmology“. Int.J. Mod. Phys. A, (2002) 180.[18] R. R. Caldwell and M. Kamionkowski et al. , “Phan-tom energy and cosmic doomsday,” Phys. Rev. Lett. (2003) 071301. doi:10.1103/PhysRevLett.91.071301[19] T. Chiba, “Tracking K-essence,” Phys. Rev. D (2002)063514. doi:10.1103/PhysRevD.66.063514[20] C. Armendariz-Picon, V. F. Mukhanov and P. J. Stein-hardt, “Essentials of k essence,” Phys. Rev. D (2001) 103510 doi:10.1103/PhysRevD.63.103510 [astro-ph/0006373].[21] M. Malquarti, E. J. Copeland, A. R. Liddle and M. Trod-den, “A New view of k -essence,” Phys. Rev. D (2003) 123503 doi:10.1103/PhysRevD.67.123503 [astro-ph/0302279].[22] A. Sen, “Universality of the tachyon potential,” J. HighEnergy Phys. (12)(2000) 027.[23] A. Y. Kamenshchik, U. Moschella et al. , “An Alterna-tive to quintessence,” Phys. Lett. B (2001) 265.doi:10.1016/S0370-2693(01)00571-8[24] C.H. Brans and R.H. Dicke, “Mach’s principle and arelativistic theory of gravitation,” Phys. Rev. D (3)(1961) 925.[25] A. De Felice and S. Tsujikawa, “f(R) theories,” LivingRev. Rel. (2010) 3 doi:10.12942/lrr-2010-3[26] S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Modi-fied Gravity Theories on a Nutshell: Inflation, Bounceand Late-time Evolution,” Phys. Rept. (2017) 1doi:10.1016/j.physrep.2017.06.001[27] S. Nojiri and S. D. Odintsov, “Unified cosmic his-tory in modified gravity: from F(R) theory to Lorentznon-invariant models,” Phys. Rept. (2011) 59doi:10.1016/j.physrep.2011.04.001[28] S.Maity and P. Rudra, (2018). Gravitational Baryo-genesis in Ho ˇ r ava-Lifshitz gravity. arXiv preprintarXiv:1802.00313.[29] T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov,“ f ( R, T ) gravity,” Phys. Rev. D (2011) 024020doi:10.1103/PhysRevD.84.024020[30] L. Susskind, “The World as a hologram,” J. Math. Phys. (1995) 6377 doi:10.1063/1.531249[31] P. Horava and D. Minic, “Probable values of the cos-mological constant in a holographic theory,” Phys. Rev. Lett. (2000) 1610 doi:10.1103/PhysRevLett.85.1610[32] S. D. Thomas, “Holography stabilizes the vac-uum energy,” Phys. Rev. Lett. (2002) 081301.doi:10.1103/PhysRevLett.89.081301[33] S. D. H. Hsu, “Entropy bounds and dark energy,” Phys.Lett. B (2004) 13 doi:10.1016/j.physletb.2004.05.020[34] M. Li, “A Model of holographic dark energy,” Phys. Lett.B (2004) 1 doi:10.1016/j.physletb.2004.10.014[35] S. Wang, Y. Wang and M. Li, “HolographicDark Energy,” Phys. Rept. (2017) 1doi:10.1016/j.physrep.2017.06.003[36] S. Nojiri and S. D. Odintsov, “Unifying phantom in-flation with late-time acceleration: Scalar phantom-non-phantom transition model and generalized holo-graphic dark energy,” Gen. Rel. Grav. (2006) 1285doi:10.1007/s10714-006-0301-6[37] A. Sheykhi, “Holographic Scalar Fields Models ofDark Energy,” Phys. Rev. D (2011) 107302doi:10.1103/PhysRevD.84.107302[38] S. Srivastava, U. K. Sharma and A. Pradhan,“New holo-graphic dark energy in bianchi- III
Universe with k -essence,” New Astron. , 57 (2019).[39] Y. Z. Ma, Y. Gong and X. Chen, “Features of holo-graphic dark energy under the combined cosmolog-ical constraints,” Eur. Phys. J. C (2009) 303doi:10.1140/epjc/s10052-009-0876-7[40] R. G. Cai, “A Dark Energy Model Characterized bythe Age of the Universe,” Phys. Lett. B (2007) 228doi:10.1016/j.physletb.2007.09.061[41] H. Wei and R. G. Cai, “A New Model of Age-graphic Dark Energy,” Phys. Lett. B (2008) 113doi:10.1016/j.physletb.2007.12.030[42] C. Gao and F. Wu et al. , “A Holographic Dark EnergyModel from Ricci Scalar Curvature,” Phys. Rev. D (2009) 043511. doi:10.1103/PhysRevD.79.043511[43] C. Tsallis and L. J. L. Cirto, “Black hole thermo-dynamical entropy,” Eur. Phys. J. C (2013) 2487doi:10.1140/epjc/s10052-013-2487-6[44] C. Tsallis, “Possible Generalization of Boltzmann-Gibbs Statistics,” J. Statist. Phys. (1988) 479.doi:10.1007/BF01016429[45] A.R´ e nyi , in Proceedings of the 4th Berkely Symposiumon Mathematics, Statistics and Probability (UniversityCalifornia Press, Berkeley, CA, 1961) pp. 547561.[46] B.D. Sharma and D.P. Mittal, “ New non-additive mea-sures of entropy for discrete probability distributions,” J.Math. Sci. (1975) 28-40; B.D. Sharma, D.P. Mittal, J.Comb. Inf. Syst. Sci. 2 (1977) 122.[47] H. Moradpour, S. A. Moosavi, I. P. Lobo, J. P. MoraisGraa, A. Jawad and I. G. Salako, “Thermodynamicapproach to holographic dark energy and the Rnyientropy,” Eur. Phys. J. C (2018) no.10, 829doi:10.1140/epjc/s10052-018-6309-8[48] M. Tavayef, A. Sheykhi, K. Bamba and H. Moradpour,“Tsallis Holographic Dark Energy,” Phys. Lett. B (2018) 195 doi:10.1016/j.physletb.2018.04.001[49] M. Abdollahi Zadeh, A. Sheykhi and H. Morad-pour, “Tsallis Agegraphic Dark Energy Model,”Mod. Phys. Lett. A (2019) no.11, 1950086doi:10.1142/S021773231950086X[50] A. Sayahian Jahromi, S. A. Moosavi, H. Moradpour,J. P. Morais Graa, I. P. Lobo, I. G. Salako and A. Jawad,“Generalized entropy formalism and a new holographicdark energy model,” Phys. Lett. B (2018) 21 doi:10.1016/j.physletb.2018.02.052[51] A. Jawad, K. Bamba, M. Younas, S. Qummerand S. Rani, “Tsallis, Rnyi and Sharma-MittalHolographic Dark Energy Models in Loop Quan-tum Cosmology,” Symmetry (2018) no.11, 635.doi:10.3390/sym10110635[52] S. Nojiri, S. D. Odintsov and E. N. Saridakis,“Modifiedcosmology from extended entropy with varying expo-nent,” Eur. Phys. J. C (2019) 242.[53] Q. Huang, H. Huang, J. Chen, L. Zhang and F. Tu,“Stability analysis of a Tsallis holographic dark energymodel,” Class. Quant. Grav. , no. 17, (2019) 175001.S. Ghaffari, H. Moradpour, I. P. Lobo, J. P. Morais Graaand V. B. Bezerra, “Tsallis holographic dark energy inthe BransDicke cosmology,” Eur. Phys. J. C (2018)no.9, 706 doi:10.1140/epjc/s10052-018-6198-x[54] E. N. Saridakis, K. Bamba, R. Myrzakulov andF. K. Anagnostopoulos, “Holographic dark energythrough Tsallis entropy,” JCAP (2018) 012doi:10.1088/1475-7516/2018/12/012[55] V. C. Dubey, S. Srivastava, U. K. Sharma and A. Prad-han, “Tsallis holographic dark energy in Bianchi-I Uni-verse using hybrid expansion law with k -essence,” Pra-mana (2019) no.5, 78. doi:10.1007/s12043-019-1843-y[56] E. Sadri, “Observational constraints on interacting Tsal-lis holographic dark energy model,” Eur. Phys. J. C (2019) no.9, 762 doi:10.1140/epjc/s10052-019-7263-9[57] E. M. Barboza, Jr., R. d. C. Nunes, E. M. C. Abreu andJ. Ananias Neto, “Dark energy models through nonex-tensive Tsallis statistics,” Physica A (2015) 301doi:10.1016/j.physa.2015.05.002[58] Golanbari, T., Saaidi, K., Karimi, P. (2020). Renyi en-tropy and the holographic dark energy in flat space time.arXiv preprint arXiv:2002.04097.[59] Sharma, U. K., Dubey, V. C. (2020). Interacting R´enyiholographic dark energy with parametrization on the in-teraction term. arXiv preprint arXiv:2001.02368.[60] S. Ghaffari, A. H. Ziaie, V. B. Bezerra and H. Moradpour,“Inflation in the Rnyi cosmology,” Mod. Phys. Lett. A (2019) no.01, 1950341 doi:10.1142/S0217732319503413[61] V. C. Dubey, et al. “Tsallis holographic dark energy Mod-els in axially symmetric space time.” Int. J. Geom. Meth-ods Mod. Phys. Eur. Phys. J. C
12, 1020 (2019).[63] A. Iqbal and A. Jawad, “Tsallis, Renyi and Shar-maMittal holographic dark energy models in DGPbrane-world,” Phys. Dark Univ. (2019) 100349.doi:10.1016/j.dark.2019.100349[64] M. Arabsalmani and V. Sahni, “The Statefinder hi-erarchy: An extended null diagnostic for concor-dance cosmology,” Phys. Rev. D (2011) 043501.doi:10.1103/PhysRevD.83.043501[65] J. F. Zhang and J. L. Cui et al. , “Diagnosingholographic dark energy models with statefinder hi-erarchy,” Eur. Phys. J. C (2014) no.10, 3100.doi:10.1140/epjc/s10052-014-3100-3[66] V. Acquaviva and A. Hajian et al. , “Next Gen-eration Redshift Surveys and the Origin of Cos-mic Acceleration,” Phys. Rev. D (2008) 043514.doi:10.1103/PhysRevD.78.043514[67] V. Acquaviva and E. Gawiser, “How to Falsify the GR+LambdaCDM Model with Galaxy Red-shift Surveys,” Phys. Rev. D (2010) 082001.doi:10.1103/PhysRevD.82.082001[68] R. Myrzakulov and M. Shahalam, “Statefinder hierar-chy of bimetric and galileon models for concordancecosmology,” JCAP (2013) 047. doi:10.1088/1475-7516/2013/10/047[69] J. Li and R. Yang et al. , “Discriminating dark energymodels by using the statefinder hierarchy and the growthrate of matter perturbations,” JCAP (2014) 043.doi:10.1088/1475-7516/2014/12/043[70] Y. Hu and M. Li et al. , “Impacts of different SNLS3 light-curve fitters on cosmological consequences of interact-ing dark energy models,” Astron. Astrophys. (2016)A101. doi:10.1051/0004-6361/201526946[71] A. Mukherjee, N. Paul and H. K. Jassal, “Constrain-ing the dark energy statefinder hierarchy in a kinematicapproach,” JCAP (2019) 005 doi:10.1088/1475-7516/2019/01/005[72] Cui, J. L., et al., ”A closer look at interacting dark energywith statefinder hierarchy and growth rate of structure”.JCAP (09) (2015) 024.[73] Zhou, L., Wang, S., ”Diagnosing ΛHDE model withstatefinder hierarchy and fractional growth parameter”.China Phys. Mech. Astron. (7) (2016) 670411.[74] Majumdar, A., Chattopadhyay, S., ”A study of modifiedholographic Ricci dark energy in the framework of f (T)modified gravity and its statefinder hierarchy”. Can. J.Phys. (5) ( 2018) 477-486.[75] Zhao, Z., Wang, S., ”Diagnosing holographic type darkenergy models with the Statefinder hierarchy, compositenull diagnostic and ω D − ω (cid:48) D pair”. China Phys. Mech.Astron. (3) (2018) 039811.[76] Yu, F., et al., ”Statefinder hierarchy exploration ofthe extended Ricci dark energy”. Eur. Phys. J. C (6)(2015)274.[77] Srivastava, Vandna, and Umesh Kumar Sharma.”Statefinder hierarchy for Tsallis holographic dark en-ergy.” New Astronomy (2020): 101380.[78] R. R. Caldwell and E. V. Linder, “The Limits ofquintessence,” Phys. Rev. Lett. (2005) 141301. doi:10.1103/PhysRevLett.95.141301[79] A. G. Cohen, D. B. Kaplan and A. E. Nelson,“Effective field theory, black holes, and the cosmo-logical constant,” Phys. Rev. Lett. (1999) 4971doi:10.1103/PhysRevLett.82.4971[80] V. Sahni and A. Shafieloo et al. , “Two new diagnos-tics of dark energy,” Phys. Rev. D (2008) 103502.doi:10.1103/PhysRevD.78.103502[81] L. M. Wang and P. J. Steinhardt, “Cluster abundanceconstraints on quintessence models,” Astrophys. J. (1998) 483. doi:10.1086/306436[82] E. V. Linder, “Cosmic growth history and expan-sion history,” Phys. Rev. D (2005) 043529.doi:10.1103/PhysRevD.72.043529[83] M. Malekjani and A. Khodam-Mohammadi, “AgegraphicDark Energy Model in Non-Flat Universe: StatefinderDiagnostic and w − w (cid:48) Analysis,” Int. J. Mod. Phys. D (2010) 1857. doi:10.1142/S0218271810018086[84] A. Khodam-Mohammadi and M. Malekjani, “Cosmic Be-havior, Statefinder Diagnostic and w − w (cid:48) Analysis for In-teracting NADE model in Non-flat Universe,” Astrophys.Space Sci. (2011) 265. doi:10.1007/s10509-010-0422-y[85] U. K. Sharma and A. Pradhan, “Diagnosing Tsallis holo-graphic dark energy models with statefinder and ω D − ω (cid:48) D pair,” Phys. Lett. A (13) (2019) 1950101.[86] G. Varshney and U. K. Sharma et al. , “Statefinder di-agnosis for interacting Tsallis holographic dark energymodels with ω − ω (cid:48) pair,” New Astron. (2019) 36.doi:10.1016/j.newast.2019.02.004[87] N. Zhang and Y. B. Wu et al. , “Diagnosing the inter-acting Tsallis Holographic Dark Energy models,” arXivpreprint (2019). arXiv:1905.04299.[88] V. C. Dubey and U. K. Sharma et al. , “Tsallis holo-graphic model of dark energy: Cosmic behaviour,statefinder analysis and ω D − ω (cid:48) D pair in the non-flatuniverse,” (2019). arXiv preprint arXiv:1905.02449.[89] S. Srivastava, V. C. Dubey, U. K. sharma, “Statefinderdiagnosis for Tsallis agegraphic dark energy model with ω D − ω (cid:48) D pair,” Int. J. Mod. Phys. A35