Interacting entropy-corrected new agegraphic dark energy in non-flat universe
aa r X i v : . [ phy s i c s . g e n - ph ] A ug Interacting entropy-corrected newagegraphic dark energy in non-flat universe
K. Karami , ∗ , A. Sorouri Department of Physics, University of Kurdistan, Pasdaran St., Sanandaj, Iran Research Institute for Astronomy & Astrophysics of Maragha (RIAAM), Maragha, Iran
November 23, 2018
Abstract
Here we consider the entropy-corrected version of the new agegraphic dark energymodel in the non-flat FRW universe. We derive the exact differential equationthat determines the evolution of the entropy-corrected new agegraphic dark energydensity parameter in the presence of interaction with dark matter. We also obtainthe equation of state and deceleration parameters and present a necessary conditionfor the selected model to cross the phantom divide. Moreover, we reconstruct thepotential and the dynamics of the phantom scalar field according to the evolutionarybehavior of the interacting entropy-corrected new agegraphic model.
PACS numbers: 95.36.+x, 04.60.Pp ∗ E-mail: [email protected] Introduction
Type Ia supernovae observational data suggest that the universe is dominated by twodark components: dark matter and dark energy [1]. Dark matter (DM), a matter withoutpressure, is mainly used to explain galactic curves and large-scale structure formation,while dark energy (DE), an exotic energy with negative pressure, is used to explain thepresent cosmic accelerating expansion. However, the nature of DE is still unknown, andpeople have proposed some candidates to describe it (for review see [2, 3] and referencestherein).The holographic DE (HDE) is one of interesting DE candidates which was proposedbased on the holographic principle [4]. According to the holographic principle, the numberof degrees of freedom in a bounded system should be finite and has relations with the areaof its boundary [5]. By applying the holographic principle to cosmology, one can obtainthe upper bound of the entropy contained in the Universe [6]. Following this line, Li [7]argued that for a system with size L and UV cut-off Λ without decaying into a blackhole, it is required that the total energy in a region of size L should not exceed the massof a black hole of the same size, thus L ρ Λ ≤ LM P , where ρ Λ is the quantum zero-pointenergy density caused by UV cut-off Λ and M P is the reduced Planck mass M − P = 8 πG .The largest L allowed is the one saturating this inequality, thus ρ Λ = 3 c M P L − , where c is a numerical constant. The HDE models have been studied widely in the literature[8, 9, 10, 11]. Obviously, in the derivation of HDE, the black hole entropy S BH plays animportant role. As is well known, usually, S BH = A/ (4 G ), where A ∼ L is the area ofhorizon. However, in the literature, this entropy-area relation can be modified to [12] S BH = A G + ˜ α ln A G + ˜ β, (1)where ˜ α and ˜ β are dimensionless constants of order unity. These corrections can appearin the black hole entropy in loop quantum gravity (LQG) [13]. They can also be dueto thermal equilibrium fluctuation, quantum fluctuation, or mass and charge fluctuations(for a good review see [13] and references therein). Using the corrected entropy-arearelation (1), the energy density of the entropy-corrected HDE (ECHDE) can be obtainedas [13] ρ Λ = 3 c M P L − + αL − ln( M P L ) + βL − , (2)where α and β are dimensionless constants of order unity.Recently, the original agegraphic dark energy (OADE) and new agegraphic dark en-ergy (NADE) models were proposed by Cai [14] and Wei & Cai [15], respectively. Thesemodels are based on the uncertainty relation of quantum mechanics as well as the gravita-tional effect in general relativity. Following the line of quantum fluctuations of spacetime,Karolyhazy et al. [16] argued that the distance t in Minkowski spacetime cannot be knownto a better accuracy than δt ∼ t / P t / where t P is the reduced Planck time. Based onKarolyhazy relation, Maziashvili [17] discussed that the energy density of metric fluctua-tions of the Minkowski spacetime is given by ρ Λ ∼ t P t ∼ M P t . (3)2ased on Karolyhazy relation [16] and Maziashvili arguments [17], Cai proposed theOADE model to explain the accelerated expansion of the universe [14]. The ADE modelsassume that the observed DE comes from the spacetime and matter field fluctuations inthe universe [14]. The OADE model had some difficulties. In particular, it cannot justifythe matter-dominated era [14]. This motivated Wei and Cai [15] to propose the NADEmodel, while the time scale is chosen to be the conformal time instead of the age of theuniverse. The evolution behavior of the NADE is very different from that of the OADE.Instead the evolution behavior of the NADE is similar to that of the HDE. But someessential differences exist between them. In particular, the NADE model is free of thedrawback concerning causality problem which exists in the HDE model. The ADE modelshave arisen a lot of enthusiasm recently and have examined and studied in ample detailby [18, 19, 20, 21, 22].Here our aim is to investigate the entropy-corrected version of the interacting NADEmodel in the non-flat universe. This paper is organized as follows. In Section 2, we givea brief review on the NADE and the entropy-corrected NADE (ECNADE) models. InSection 3, we study the ECNADE in a FRW universe with spacial curvature and in thepresence of interaction between DE and DM. In Section 4, we reconstruct the potentialand the dynamics of the phantom scalar field according to the evolutionary behavior ofthe interacting ECNADE model. Section 5 is devoted to conclusions. From Eq. (3), the energy density of the NADE is given by [15] ρ Λ = 3 n M P η , (4)where the numerical factor 3 n is introduced to parameterize some uncertainties, such asthe species of quantum fields in the universe, the effect of curved spacetime (since theenergy density is derived for Minkowski spacetime), and so on. It was found that thecoincidence problem could be solved naturally in the NADE model provided that thesingle model parameter n is of order unity [21]. Also η is conformal time of the FRWuniverse, and given by η = Z d ta = Z d aHa . (5)With the help of quantum corrections to the entropy-area relation (1) in the setup ofLQG, the energy density of the ECNADE is given by [13] ρ Λ = 3 n M P η + αη ln ( M P η ) + βη , (6)which closely mimics to that of ECHDE density (2) and L is replaced with the conformaltime η . Here α and β are dimensionless constants of order unity. In the special case α = β = 0, the above equation yields the well-known NADE density (4).3 Interacting ECNADE and DM in non-flat universe
We consider the Friedmann-Robertson-Walker (FRW) metric for the non-flat universe asd s = − d t + a ( t ) d r − kr + r dΩ ! , (7)where k = 0 , , − k ∼ .
02) [23]. Besides, as usually believed, an early inflation era leads to a flat uni-verse. This is not a necessary consequence if the number of e-foldings is not very large[24]. It is still possible that there is a contribution to the Friedmann equation from thespatial curvature when studying the late universe, though much smaller than other energycomponents according to observations.For the non-flat FRW universe containing the DE and DM, the first Friedmann equa-tion has the following form H + ka = 13 M P ( ρ Λ + ρ m ) , (8)where ρ Λ and ρ m are the energy density of DE and DM, respectively. Let us define thedimensionless energy densities asΩ m = ρ m ρ cr = ρ m M P H , Ω Λ = ρ Λ ρ cr = ρ Λ M P H , Ω k = ka H , (9)then, the first Friedmann equation yieldsΩ m + Ω Λ = 1 + Ω k . (10)We consider a universe containing an interacting ECNADE density ρ Λ and the colddark matter (CDM), with ω m = 0. The energy equations for ECNADE and CDM are˙ ρ Λ + 3 H (1 + ω Λ ) ρ Λ = − Q, (11)˙ ρ m + 3 Hρ m = Q, (12)where following [25], we choose Q = Γ ρ Λ as an interaction term and Γ = 3 b H ( k Ω Λ ) isthe decay rate of the ECNADE component into CDM with a coupling constant b . Thisexpression for the interaction term Q was first introduced in the study of the suitablecoupling between a quintessence scalar field and a pressureless CDM field [26]. Althoughat this point the interaction may look purely phenomenological but different Lagrangianshave been proposed in support of it [27]. Tsujikawa and Sami [27] investigated the cos-mological scaling solutions in a general cosmological background H ∝ ( ρ Λ + ρ m ) ε in-cluding general relativity (GR), Randall-Sundrum (RS) braneworld and Gauss-Bonnet(GB) braneworld. The GR, RS and GB cases correspond to ε = 1, ε = 2 and ε = 2 / p = X /ε g ( Xe ελφ ), where φ is a4calar field with X defined as X = − g µν ∂ µ φ∂ ν φ/ g is an arbitrary function. Also λ ∝ Q , where Q is the coupling term due to the interaction between the scalar filed andthe matter. Tsujikawa and Sami [27] showed that in the absence of the coupling Q be-tween a scalar field and a perfect barotropic fluid, it is not possible to get an accelerationof the universe since the energy density of the field φ decreases in proportional to that ofthe background fluid for scaling solutions. However the presence of the coupling Q allowsto have an accelerated expansion. It should be emphasized that this phenomenologicaldescription has proven viable when contrasted with observations, i.e., SNIa, CMB, largescale structure, H ( z ), and age constraints [28], and recently in galaxy clusters [29]. Thechoice of the interaction between both components was to get a scaling solution to thecoincidence problem such that the universe approaches a stationary stage in which theratio of DE and DM becomes a constant [30]. The dynamics of interacting DE modelswith different Q -classes have been studied in ample detail by [31].From definition ρ Λ = 3 M P H Ω Λ , we getΩ Λ = n H η γ n , (13)where γ n = 1 + 13 n M P η h α ln ( M P η ) + β i . (14)Taking derivative of Eq. (13) with respect to x = ln a , using ˙ η = 1 /a and Ω ′ Λ = ˙Ω Λ /H where prime denotes the derivative with respect to x , one can obtain the equation ofmotion for Ω Λ asΩ ′ Λ = − Λ h ˙ HH + 1 naγ n (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17)i . (15)Taking time derivative of the first Friedmann equation (8) and using Eqs. (6), (9), (10),(12), (13), (14) one can get˙ HH = −
32 (1 − Ω Λ ) + 3 b k ) − Ω k − na (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17) . (16)Substituting this into Eq. (15), one obtainsΩ ′ Λ = Ω Λ h − Ω Λ ) − b (1 + Ω k ) + Ω k + 2 na (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17)(cid:16) Ω Λ − γ n (cid:17)i . (17)Taking time derivative of Eq. (6), using ˙ η = 1 /a and substituting the result in Eq. (11)yields the equation of state (EoS) parameter of the interacting ECNADE as w Λ = − − b (cid:16) k Ω Λ (cid:17) + 23 naγ n (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17) , (18)which shows that the interacting ECNADE can cross the phantom divide, i.e. ω Λ < − nab (1 + Ω k ) > (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17) . (19)5uring the inflation era where H = constant, we have a = e Ht and from Eq. (5) weget η = − /Ha . Since the last two terms in Eq. (6) can be comparable to the firstterm only when η is very small, the corrections make sense only at the late stage of theinflationary expansion of the universe. During the cosmological inflation in the earlyuniverse, ECNADE reduces to the NADE model. The EoS parameter of the interactingECNADE during the inflation era will be w Λ = − − b (cid:16) k Ω Λ (cid:17) + 2 e − Ht nγ n (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH e Ht M P n (cid:17) , (20)with γ n = 1 + H e Ht n M P h α ln (cid:16) M P H e Ht (cid:17) + β i . (21)The deceleration parameter is given by q = − (cid:16) HH (cid:17) . (22)Putting Eq. (16) in the above relation reduces to q = −
32 Ω Λ + 12 (1 − b )(1 + Ω k ) + 1 na (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17) . (23)Note that if we set α = β = 0 then from Eq. (14) γ n = 1. Therefore Eqs. (18) and (23)reduce to ω Λ = − − b (cid:16) k Ω Λ (cid:17) + 2Ω / na , (24) q = −
32 Ω Λ + 12 (1 − b )(1 + Ω k ) + Ω / na , (25)which are the EoS and deceleration parameters of the interacting NADE with CDM inthe non-flat universe [22, 32].Following [14], the OADE density is given by ρ Λ = 3 n M P T , (26)where T is the age of the universe and given by T = Z d t = Z d aHa . (27)Note that appearing the age of the universe T in the energy density of the OADE modelcauses some difficulties. In particular it fails to describe the matter-dominated epochproperly [14]. Similar to the density of ECNADE (6), the density of entropy-correctedOADE (ECOADE) can be written as ρ Λ = 3 n M P T + αT ln ( M P T ) + βT . (28)To obtain the evolution of the density parameter, the EoS and deceleration parametersof the interacting ECOADE in a non-flat FRW universe, one doesn’t need to repeat thecalculations. The only necessary changes are that one must replace η with T in Eq. (14)and put a = 1 in Eqs. (17), (18) and (23). 6 Entropy-corrected new agegraphic phantom model
Here we suggest a correspondence between the interacting ECNADE model with the phan-tom scalar field model. The phantom scalar field model is often regarded as an effectivedescription of an underlying theory of DE [33]. Recent observational data indicates thatthe EoS parameter ω Λ lies in a narrow strip around ω Λ = − − V ( φ ) [33, 34]. Therefore it becomes meaningful to reconstruct V ( φ ) fromsome DE models possessing some significant features of the LQG theory, such as ECHDEand ECNADE models.The energy density and pressure of the phantom scalar field φ are as follows [3] ρ ph = −
12 ˙ φ + V ( φ ) , (29) p ph = −
12 ˙ φ − V ( φ ) . (30)The EoS parameter for the phantom scalar field is given by ω ph = p ph ρ ph = ˙ φ + 2 V ( φ )˙ φ − V ( φ ) . (31)Here like [35], we establish the correspondence between the interacting ECNADE scenarioand the phantom DE model, then equating Eq. (31) with the EoS parameter of interactingECNADE (18), ω ph = ω Λ , and also equating Eq. (29) with (6), ρ ph = ρ Λ , we have˙ φ = − (1 + ω Λ ) ρ Λ , (32) V ( φ ) = 12 (1 − ω Λ ) ρ Λ . (33)Substituting Eqs. (6) and (18) into Eqs. (32) and (33), one can obtain the kinetic energyterm and the phantom potential energy as follows˙ φ = 3 M P H h b (1 + Ω k ) − na (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17)i , (34) V ( φ ) = 32 M P H h Λ + b (1 + Ω k ) − na (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17)i . (35)From Eq. (34) and using ˙ φ = φ ′ H , one can obtain the evolutionary form of the phantomscalar field as φ ( a ) − φ ( a ) = √ M P Z ln a ln a h b (1 + Ω k ) − na (cid:16) Ω Λ γ n (cid:17) / (cid:16) γ n − − αH M P n Ω Λ γ n (cid:17)i / d x, (36)where a is the scale factor at the present time.7 Conclusions
Here we considered the entropy-corrected version of the NADE model which is in in-teraction with CDM in the non-flat FRW universe. However, some experimental datahave implied that our universe is not a perfectly flat universe and that it possesses asmall positive curvature (Ω k ∼ .
02) [23]. Although it is believed that our universe isflat, a contribution to the Friedmann equation from spatial curvature is still possible ifthe number of e-foldings is not very large [24]. The ADE models proposed to explainthe accelerated expansion of the universe, based on the uncertainty relation of quantummechanics as well as the gravitational effect in general relativity [14, 15]. We consideredthe logarithmic correction term to the energy density of NADE model. The addition ofcorrection terms to the energy density of NADE is motivated from the LQG which isone of the promising theories of quantum gravity. Using this modified energy density, wederived the exact differential equation that determines the evolution of the ECNADE den-sity parameter. We also obtained the EoS and deceleration parameters for the interactingECNADE and present a necessary condition for the present model to cross the phantomdivide. Moreover, we established a correspondence between the interacting ECNADEdensity and the phantom scalar field model of DE. We adopted the viewpoint that thescalar field models of DE are effective theories of an underlying theory of DE. Thus, weshould be capable of using the scalar field model to mimic the evolving behavior of theinteracting ECNADE and reconstructing this scalar field model. We reconstructed the po-tential and the dynamics of the phantom scalar field, which describe accelerated expansionof the universe, according to the evolutionary behavior of the interacting ECNADE model.
Acknowledgements
The authors thank the reviewers for very valuable comments. This work has been sup-ported financially by Research Institute for Astronomy & Astrophysics of Maragha (RI-AAM), Maragha, Iran.
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