Tsallis holographic dark energy in the brane cosmology
S. Ghaffari, H. Moradpour, J. P. Morais Graça, Valdir B. Bezerra, I. P. Lobo
aa r X i v : . [ phy s i c s . g e n - ph ] O c t Tsallis holographic dark energy in the brane cosmology
S. Ghaffari ∗ , H. Moradpour † , J. P. Morais Gra¸ca ‡ , Valdir B. Bezerra § , I. P. Lobo ¶ Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P. O. Box 55134-441, Maragha, Iran Departamento de F´ısica, Univeidade Federal da Para´ıba,Caixa Postal 5008, CEP 58051-970, Jo˜ao Pessoa, PB, Brazil
We study some cosmological features of Tsallis holographic dark energy (THDE) in Cyclic, DGPand RS II braneworlds. In our setup, a flat FRW universe is considered filled by a pressurelesssource and THDE with the Hubble radius as the IR cutoff, while there is no interaction betweenthem. Our result shows that although suitable behavior can be obtained for the system parameterssuch as the deceleration parameter, the models are not always stable during the cosmic evolutionat the classical level.
I. INTRODUCTION
Due to the weakness of general relativity to describethe current accelerated universe [1, 2], physicists try toeliminate this difficulty by i ) introducing amazing en-ergy sources, called dark energy, ii ) modifying the gen-eral relativity theory or even a combination of these.Braneworld scenario is an interesting approach to mod-ify the Einstein theory, and Dvali-Gabadadze-Porrati(DGP) braneworld, the second model of Randall andSundrum (RS II) and the Cyclic model of Steinhardt andTurok are three pioneering models in this regard [3–5].There is also another Cyclic model motivated by boththe braneworld and loop quantum cosmology scenarios[6–10]. The basic idea behind the braneworld hypothe-sis is that our universe is a brane embedded in a higherdimensional bulk, while only gravity can penetrate thebulk, and as well as the energy-momentum distribution,other forces are limited to the brane [3, 4].In the DGP braneworld model the 4-dimensional FRWuniverse is embedded in a 5D Minkowski bulk. DGPbraneworld has two branches of solutions correspondingto ǫ = +1 and ǫ = −
1. Although, the first case provides aself-accelerating solution for the current universe, it suf-fers from the ghost instability problem [11]. The normalbranch of ǫ = − ∗ sh.ghaff[email protected] † [email protected] ‡ [email protected] § valdir@fisica.ufpb.br ¶ iarley lobo@fisica.ufpb.br a model for dark energy has been proposed called holo-graphic dark energy (HDE) and suffers from the stabilityproblem [14–19]. This idea has been employed in orderto model dark energy by the energy density of quantumfields in vacuum, in the DGP, RSII and cyclic universes[20–25].Since gravity is a long-range interaction, it may sat-isfy the non-extensive probability distributions [26]. Thisview leads to interesting results in gravitational and cos-mological setups [27–37]. Recently, using the Tsallis gen-eralized entropy [26] and holographic hypothesis, a newholographic model for dark energy has been introduced,in which the Hubble radius plays the role of the IR cutoff,as [38] ρ D = BH − δ , (1)where H = ˙ aa is the Hubble parameter. The cosmologicalfeatures of this dark energy model in various cosmologicalsetups can be found in Refs. [39–42].Here, we are interested in studying some cosmologicalconsequences of employing Eq. (1) in the DGP [3], RSII [4] and Cyclic [6, 7] models. Sine the WMAP dataindicates a flat FRW universe, we consider a flat FRWuniverse, in which there is no mutual interaction betweenthe cosmos sectors. In order to achieve this goal, we studysome cosmological features of THDE in Cyclic model inthe next section. Secs. (3) and (4) include its cosmo-logical consequences in the DGP and RS II braneworlds,respectively. The classical stability of the models are alsostudied in the 5th section. The last section is devoted toa summary. II. THDE IN CYCLIC UNIVERSE
Effective field theory of loop quantum cosmology mod-ifies the Friedmann equation as [6] H = ρ m p (1 − ρρ c ) , (2)where ρ is the total energy density of the fluid filling thecosmos, and ρ c denotes the critical density constrainedby quantum gravity and different from the usual criticaldensity ( ρ cr = 3 m p H ). This modified Friedmann equa-tion can also be obtained in the framework of braneworldscenario [7, 10]. In our model, the cosmos includes alsodark matter (DM) and DE, which do not interact mutu-ally, and hence, the total energy-momentum conservationlaw is decomposed as˙ ρ D + 3 H (1 + ω D ) ρ D = 0 , (3)˙ ρ m + 3 Hρ m = 0 → ρ m = ρ (1 + z ) , (4)where ρ is an integral constant, and we used the 1 + z = a relation between the redshift z and the scale factor a while its current time values has been normalized to one. ρ m and ρ D also denote the energy density of DM andDE, respectively, and ω D is the equation of state (EoS)parameter of dark energy. We define the dimensionlessdensity parameters asΩ m = ρ m ρ cr = ρ m m p H Ω D = ρ D ρ cr = ρ D m p H , (5)and insert them in Eq. (2) to obtainΩ D = (1 − Ω m ) + Ω D + Ω mρ c ρ cr − (Ω D + Ω m ) . (6)The use of Eqs. (4) and (6) leads toΩ D ( z → − ≈ Dρ c ρ cr − Ω D . (7)at the z → − D >
1, if ρ c > ρ D (see Ref. [24]for more details). Now, combining Eq. (5) with Eq. (1),we find Ω D = B m p H − δ , (8)for the DE dimensionless density parameter. Now, defin-ing u = Ω m Ω D , using the time derivative of Eq. (2), andcombining the results with Eqs. (5) and (2), we arrive at˙ HH = − u (cid:16) − Ω D (1 + u ) (cid:17) δ − − Ω D (1 + u )) + 2( u + 1) , (9)which can finally be used to writeΩ ′ D = d Ω D d ln a = ˙Ω D H = − u (1 − δ )Ω D (cid:16) − Ω D (1 + u ) (cid:17) ( δ − − Ω D (1 + u )) + u + 1 , (10) where dot denotes derivative with respect to time. Cal-culations of the EoS and parameter of THDE and decel-eration parameter also lead to ω D = − u (2 − δ ) (cid:16) − Ω D (1 + u ) (cid:17) ( δ − − Ω D (1 + u )) + u + 1 . (11)and q = − − ˙ HH = −
1+ 3 u (cid:16) − Ω D (1 + u ) (cid:17) δ − − Ω D (1 + u )) + 2( u + 1) , (12)respectively. In Figs. 1 and 2, the behavior of the dimen- z Ω D -1 0 1 2 300.20.40.60.811.21.4 δ =1.45 δ =1.5 δ =1.55 FIG. 1: Ω D versus z for Ω D = 0 · u = 0 ·
3, and somevalues of δ . sionless density, EoS and deceleration parameters havebeen plotted against redshift z by considering Ω D = 0 · u = 0 · w D ≈ − u ≈ δ = 1 limit [25]. In summary, the phantom line isnot crossed in this model ( w D ≥ − z t from a deceleration phase to an accelerateduniverse lies within the interval 0 · < z t < III. THDE IN DGP BRANEWORLD
For a flat FRW brane embedded in a Minkowski bulk,the Friedmann equation takes the form [43, 44] z ω D δ =1.45 δ =1.5 δ =1.55 z q -1 0 1 2 3-1-0.8-0.6-0.4-0.200.20.40.60.81 δ =1.45 δ =1.5 δ =1.55 FIG. 2: ω D and q versus z for Ω D = 0 ·
73 and some valuesof δ and u = 0 · H = (cid:16)s ρ M + 14 r c + ǫ r c (cid:17) , (13)where ρ includes the energy density of DM, ρ m , and DE, ρ D , on the brane, and r c = M M = G G denotes thecrossover length scale between the small and large dis-tances [43]. It is obvious that this equation is reducedto H = ρ M , (14)for r c ≫
1, nothing but the standard Friedmann equationin flat FRW spacetime. Eq. (13) can also be written as H − ǫr c H = ρ M , (15)which reduces to H = ρ M , (16) for ǫ = − r c ≪ H − [45]. This result clearly provesthat this branch does not give the self-accelerating solu-tion which compels us to consider a DE component onthe brane to describe the current accelerated universe.Using Eq. (5) and Ω r c = H r c , one can rewrite Eq. (15)as Ω m + Ω D + 2 ǫ p Ω r c = 1 . (17)For a THDE (1) with the Hubble radius as IR cut off( L = H − ), by using (5), we obtainΩ D = BH − δ M p . (18)Bearing in mind the time derivative of Eq. (1)˙ ρ D = 2(2 − δ ) ρ D ˙ HH , (19)and combining it with Eq. (18) and its time derivative,one finds Ω ′ D = 2Ω D (1 − δ ) ˙ HH , (20)where prime denotes the derivative respect to x = ln a ,and we used ˙Ω D = H Ω ′ D to write the above relation.Now, combining the time derivative of Eq. (15) withEqs. (4), (5) and (19), we get˙ HH = − − Ω D − ǫ p Ω r c )2( δ − D − ǫ p Ω r c + 2 , (21)which can be inserted into (20), to reach atΩ ′ D = 3Ω D (1 − δ )(Ω D + 2 ǫ p Ω r c − δ − D − ǫ p Ω r c + 1 . (22)In the limiting case r c ≫ r c → D ofTHDE in the Einstein theory [38] is restored, a desiredresult recovering the original HDE (Ω D = const ) for δ =1. The evolution of Ω D as a function of redshift z isplotted in Fig. (3) for different values of the parameter δ , whenever ǫ = 1, Ω D ( z = 0) ≡ Ω D = 0 ·
73 andΩ r c ( z = 0) = 0 ·
002 [46]. Clearly, this figure indicatesthat we have Ω D → D →
1, at the early Universe( z → ∞ ) and the late time ( z → − ω D = − − δ )(1 − Ω D − ǫ p Ω r c )( δ − D − ǫ p Ω r c + 1 , (23)and q = − − Ω D − ǫ p Ω r c )2( δ − D − ǫ p Ω r c + 2 , (24)respectively, which are plotted in Fig. 4. One can eas-ily see that for r c ≫ r c → z Ω D -1 0 1 2 300.10.20.30.40.50.60.70.80.91 δ =2.2 δ =2.4 δ =2.6 FIG. 3: The evolution of Ω D versus z for Ω D = 0 ·
73, Ω r c =0 ·
002 [46] and some values of δ . extra dimension are negligible, the general relativity is re-covered, and hence, Eqs. (23) and (24) decrease to theirrespective relations [38]. It is worth mentioning that, inthe limiting case δ = 1, the relations of Ref. [21], as thedesired result, are obtained. From Fig. 4, it is obviousthat the model can cover the current accelerated uni-verse even in the absence of a interaction between DMand DE. We also see that ω D ( z → − → − z t ) from theacceleration phase to an accelerated phase lies within the0 · < z t < · IV. THDE IN RS II BRANEWORLD
In RS II braneworld scenario, the modified Friedmannequation on the brane is written as H + ka = 8 π M p ρ + 8 π M p ρ Λ (25)where ρ denotes the total energy density of the pressure-less source, ρ m , and DE, ρ Λ , on the brane, and M p = πG is the reduced Planck mass. Following [22], the energydensity of the four dimensional effective DE is given by ρ Λ ≡ ρ Λ4 = M p πM ρ Λ5 + 3 M p π (cid:16) L π − r c (cid:17) , (26)where ρ Λ5 is the 5D bulk holographic dark energy, which-for Tsallis HDE takes the following form ρ Λ5 = 3 c B π M L δ − , (27) z ω D -1 0 1 2 3-1-0.8-0.6-0.4-0.20 δ =2.2 δ =2.4 δ =2.6 z q -1 0 1 2 3-1-0.8-0.6-0.4-0.200.20.4 δ =2.2 δ =2.4 δ =2.6 FIG. 4: The evolution of the EoS parameter ω D and deceletar-ion parameter q versus redshift parameter z for Tsallis HDEin DGP braneworld. We have taken ǫ = 1 and Ω r c = 0 · combined with Eq. (26) to get ρ Λ = 3 Bc M p π L δ − + 3 M p π (cid:16) L π − r c (cid:17) , (28)for the effective 4D THDE density. We can eliminatethe second term in relation (28) for large values of L .Moreover, since ρ Λ ≡ ρ Λ4 [22], we have ρ Λ = 3 Bc M p π H − δ . (29)Using the definition (5), one can write Eq. (25) as follows1 + Ω k = Ω m + 2Ω Λ , (30)where Ω Λ = Bc π H − δ . (31)Since the WMAP data indicates a flat FRW universe [2],we focus on the k = 0 case from now on. In this manner,Eq. (30) indicates that whenever Ω m is negligible, Ω Λ gains its maximum value ( ). Now, it is a matter ofcalculations to show that˙ HH = 3(2Ω Λ − Λ ( δ − , (32)and Ω ′ Λ = Ω Λ (1 − δ ) 3(2Ω Λ − Λ ( δ − . (33)The behavior of Ω D against z is plotted in Fig. 5, wherethe initial condition Ω D ( z = 0) ≡ Ω D = 0 ·
73 has beenconsidered. From this figure we clearly see that at theearly universe ( z → ∞ ) we have Ω D →
0, while at thelate time ( z → − D → · z Ω Λ -1 0 1 2 300.10.20.30.40.5 δ =3.2 δ =3.4 δ =3.6 FIG. 5: Ω D for THDE in RS II braneworld. Here, we havetaken Ω D = 0 ·
73 as the initial condition.
For the EoS and deceleration parameters, one obtains ω Λ = 1 − δ Λ ( δ − , (34)and q = − − ˙ HH = − Λ − Λ ( δ − , (35)respectively. They are also depicted for different valuesof δ in Fig. 6. Our results indicate that the THDE modelwith the Hubble cutoff in the RSII braneworld can modelthe current accelerated universe, and admits the 0 · In this section we would like to study the stability ofmodels against small perturbations by using the squaredof the sound speed ( v s ). In fact, it can be found out byfinding the sign of v s . For v s > v s is given by v s = dpdρ D = ˙ p ˙ ρ D , (36)where P = P D = ω D ρ D , and finally, we get v s = ω D + ˙ ω D ρ D ˙ ρ D . (37) A. THDE in Cyclic univrse By taking the time derivative of Eq. (11) and combin-ing the result with Eqs. (9), (10) and (37), we can finallyobtain the explicit expression of v s for THDE in cyclicuniverse. Since this expression is too long, we do notdemonstrate it here, and we only plot it in Fig. 7 show-ing that, depending on the values of δ , THDE in cycliccosmology can not meet the stability requirement for allvalues of Ω D (or equally z ). Ω D v s2 δ =1.45 δ =1.5 δ =1.55 FIG. 7: v s versus Ω D for THDE in cyclic universe. B. THDE in DGP braneworld In this manner, calculations lead to v s = − − δ )(1 − Ω D − ǫ p Ω r c )( δ − D − ǫ p Ω r c + 1 + (38)Ω D ( δ − (cid:16) ( δ − − ǫ p Ω r c ) − ǫ p Ω r c + 1 (cid:17) (( δ − D − ǫ p Ω r c + 1) , where behavior is shown in Fig 8 against Ω D . Clearly, wesee that v s is ever negative indicating that THDE in DGPbraneworld is always unstable against the perturbationsfor 2 < δ . It is useful to note here that other valuesof δ cannot produce acceptable behavior for the systemparameters including Ω D , q and ω D . C. THDE in RSII braneworld Using Eqs. (37), (32), (33), and the time derivative ofEq. (34), one can find v s = (1 − δ )(1 − D )(1 + 2(2 − δ )Ω D ) , (39) Ω D v s2 δ =2.2 δ =2.4 δ =2.6 FIG. 8: The evolution of v s versus Ω D for THDE model inDGP braneworld, where ǫ = +1 and Ω r c = 0 · where behavior is shown in Fig. 9. We conclude thatTHDE in RSII braneworld is stable for < Ω D < δ > 1. This result is in agreement with Eq. (39),and indeed, this Eq. (39) tells that the model is alsostable for 0 < Ω D < / < δ < Ω D v s2 δ =3.2 δ =3.4 δ =3.6 FIG. 9: v s versus Ω D for THDE in RS II braneworld. VI. CONCLUSION We studied the cosmological consequences of THDEin the Cyclic, DGP and RS II models. In our study, wefocused on a flat FRW brane filled with a pressurelessdark matter and THDE, while there is no mutual inter-action between them. Although all models may describethe current accelerated universe, none of them are alwaysstable against small perturbations during the cosmic evo-lution, at least at the classical level. 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