Interacting entropy-corrected holographic dark energy with apparent horizon as an infrared cutoff
aa r X i v : . [ g r- q c ] F e b Interacting entropy-corrected holographic dark energy withapparent horizon as an infrared cutoff
A. Khodam-Mohammadi ∗ and M. Malekjani † Department of Physics, Faculty of Science,Bu-Ali Sina University, Hamedan 65178, Iran
Abstract
In this work we consider the entropy-corrected version of interacting holographic dark energy(HDE), in the non-flat universe enclosed by apparent horizon. Two corrections of entropy so-calledlogarithmic ’LEC’ and power-law ’PLEC’ in HDE model with apparent horizon as an IR-cutoffare studied. The ratio of dark matter to dark energy densities u , equation of state parameter w D and deceleration parameter q are obtained. We show that the cosmic coincidence is satisfied forboth interacting models. By studying the effect of interaction in EoS parameter, we see that thephantom divide may be crossed and also find that the interacting models can drive an accelerationexpansion at the present and future, while in non-interacting case, this expansion can happenonly at the early time. The graphs of deceleration parameter for interacting models, show thatthe present acceleration expansion is preceded by a sufficiently long period deceleration at past.Moreover, the thermodynamical interpretation of interaction between LECHDE and dark matteris described. We obtain a relation between the interaction term of dark components and thermalfluctuation in a non-flat universe, bounded by the apparent horizon. In limiting case, for ordinaryHDE, the relation of interaction term versus thermal fluctuation is also calculated. ∗ Email:[email protected] † Email:[email protected] . INTRODUCTION The dark energy scenario has attracted a great deal of attention in the last decade. Manycosmological observations reveal that our universe evolves under an acceleration expansion[1]. This expansion may be driven by an unknown energy component with negative pressure,so called, dark energy (DE), which fills ∼
70 percent of energy content of our universe withan effective equation of state (EoS) parameter − . < w eff < − .
72 [2]. Despite ofmany efforts in this subject, the nature of DE is the most mysterious problem in moderncosmology. The first and simplest candidate of dark energy is ΛCDM model, in which w Λ = − w D is not constant, the entropy-corrected dark energy models based onquantum field theory and gravitation have been widely extended by many authors in recentyears [3, 4]. The motivation of these corrections has been based on black hole physics, wheresome gravitational fluctuations and field anomalies can affect the entropy-area law of blackholes. The logarithmic and power-law corrections of entropy are two procedures in dealingwith this fluctuations. First correction has been given by logarithmic fluctuations at thespacetime, in the context of loop quantum gravity (LQG) [5]. The entropy-area relationshipleads to the curvature correction in the Einstein-Hilbert action and vice versa [6, 7]. In thiscase the corrected entropy is given by [8] S BH = A G + ˜ γ ln A G + ˜ β, (1)where ˜ γ and ˜ β are dimensionless constants of order unity. By considering the entropycorrection, the energy density of logarithmic entropy-corrected holographic dark energy(LECHDE) can be given as [9] ρ D = 3 n M p L − + γL − ln( M p L ) + βL − . (2)Three parameters n, β and γ are parameters of model and M P is the reduced Planck mass.The correction terms (two last terms of (2)) are effective only at the early stage of theuniverse and they will be vanished when the universe becomes large, in which ρ ECD → ρ OD ,where ρ O Λ = 3 n M p L − is the dark energy density of ordinary HDE model (more discussion2f HDE model is referred to [10]). In this model, the IR-cutoff ’ L ’ plays an essential role. If L is chosen as particle horizon, the HDE can not make an acceleration expansion [11], while forfuture event horizon, Hubble scale ’ L = H − ’, and apparent horizon (AH) as an IR-cutoff,the HDE can simultaneously drive accelerated expansion and solve the coincidence problem[12–14]. More recently, a model of interacting HDE (i.e. a non gravitational interactionbetween DE and dark matter (DM)) at Ricci scale, in which L = ( ˙ H + 2 H ) − / has beenproposed. The authors performed a detailed discussion on the cosmic coincidence problem,age problem and obtained some observational constraints on their’s model [15].The second class of ECHDE, power-law correction of entropy (PLEC), is appeared indealing with the entanglement of quantum fields in and out of the horizon [16]. In thismodel, the corrected-entropy is given by [3] S = A G [1 − K α A − α/ ] , (3)where α is a dimensionless positive constant and K α = α − α (4 πr c ) α/ − . (4)Here r c is the crossover scale. More detail is referred to [3, 16, 17]. It is worthwhile tomention that in the most acceptable range of 4 > α > A ’s and it falls off rapidly at largevalues of A . Therefore, for large horizon area, the ordinary entropy-area law (first termof ( ?? )) is recovered. However the thermodynamical considerations predict that the case α ≤ S BH ), the Friedmann equation should be modified [3]. In comparison with ordinaryFriedman equation, the energy density of PLECHDE, has been given by [18] ρ D = 3 n M p L − − δM p L − α , (5)where δ and α are the parameters of PLECHDE model. We must mention that the ordinaryHDE is recovered for δ = 0 or α = 2.In historical point of view, laws of black hole thermodynamics have made some relationsbetween thermodynamics and a self gravitating system bounded by a horizon. In this theory,some thermodynamical quantities such as entropy and temperature are purely geometrical3uantities which have been obtained from area and surface gravity of horizon, respectively. Inthe Friedmann-Robertson-Walker (FRW) universe, with horizons, like future event horizonin black hole physics, by studying the thermodynamical quantities and generalized secondlaw (GSL)[19], one can choose the best DE model or the best horizon. For example, ithas been shown that in a non-flat FRW universe, enclosed by apparent horizon, the GSLis governed irrespective of any DE model [14]. The investigation of GSL for LECHDE andPLECHDE models has been performed in [3].Recently, the HDE and agegraphic/new-agegraphic DE models have been extended re-garding the entropy corrections (LECHDE, PLECHDE, PLECNADE) and a thermodynam-ical description of the LECHDE model has been studied [4, 9, 18, 20]. Also at Ref. [14],thermodynamics interpretation of interacting holographic dark energy with AH-IR-cutoff,enclosed by apparent horizon, was studied. These papers give us a strong motivation to studythe LECHDE and PLECHDE models with AH-IR-cutoff in a non-flat universe, enclosed byapparent horizon, which is a generalization of earlier works of Sheykhi et.al. [14, 18]. Itshould be mentioned that, the motivation of a closed universe has been also shown in a suiteCMB experiments [21] and of the cubic correction to the luminosity-distance of supernovameasurements [22].The outline of our paper is as follows: In Sec. II, the interacting LECHDE model withAH-IR-cutoff is studied and the evolution of dark energy, deceleration parameter and EoSparameter are calculated. Also these calculations are performed for PLECHDE model withAH-IR-cutoff in Sec. III. In Sec. IV, the thermodynamical quantities such as entropy andHawking temperature of apparent horizon are obtained only for LECHDE model and thenthe interaction term due to thermal fluctuation is obtained in Sec. V. We finish Our paperwith some concluding remarks. II. INTERACTING “LECHDE” MODEL WITH AH-IR-CUTOFF
The line element of a homogenous and isotropic FRW universe is given by ds = h ab dx a dx b + e r ( dθ + sin θdφ ) , (6)where e r = a ( t ) r, two non-angular metric ( x , x ) = ( t, r ) and two dimensional metric is h ab = diag [ − , a / (1 − Kr )]. Here K = 1 , , − e r A = ( H + K/a ) − / which has been calculatedby the relation h ab ∂ a e r∂ b e r = 0 [23]. This relation implies that the vector ∇ e r is null on theapparent horizon surface. The apparent horizon may be considered as a causal horizon fora dynamical spacetime. Thus one can associate a gravitational entropy and surface gravityto it [24].From Eq. (2), the energy density of LECHDE with apparent horizon, e r A , as an IR-cutoffcan be written as ρ D = 3 n M P e r − A + γ e r − A ln( M P e r A ) + β e r − A . (7)The first Friedmann equation is1 e r A = H + Ka = 13 M P ( ρ m + ρ D ) , (8)where H = ˙ a/a is the Hubble parameter. In a FRW universe, the total energy density ρ = ρ D + ρ m is satisfied in a conservation equation as:˙ ρ + 3 H (1 + w ) ρ = 0 (9)where w = p/ρ is the EoS parameter. Due to non gravitational interaction between darkenergy and pressureless cold dark matter (CDM) with subscript ’ m ’, two energy densities ρ D and ρ m are not conserved separately and the conservation equation can be written as˙ ρ D + 3 H (1 + w D ) ρ D = − Q, (10)˙ ρ m + 3 Hρ m = Q. (11)Here Q is the interaction term which has been usually considered in three forms as [25] Q = Γ ρ D = Hb ρ D Hb ρ m Hb ( ρ m + ρ D ) . (12)In this equation, b is coupling constant. Although a theoretical interpretation of thisinteraction has not been performed yet, as we see from Eqs. (10, 11), the interaction term Q should be as a function of H multiplied to energy density. Therefore in Eq. (12), thesimplest form of Q is considered with a coupling constant b . This term indicates the decayrate of DE to CDM as similar as standard ΛCDM model where vacuum fluctuations can5ecay into matter. In many models the interaction term is necessary in order to solving thecoincidence problem. It has been shown that this interaction can influence the perturbationdynamics, cosmic microwave background (CMB) spectrum and structure formation [26].Differentiating Eq. (7) with respect to cosmic time and using the differentiation of ap-parent horizon with respect to cosmic time, we have˙ − e r A e r − A = H ( ˙ H − Ka ) = 16 M P ( ˙ ρ D + ˙ ρ m ) , (13)where from Eqs. (10, 11) we obtain˙ e r A = H M P e r A ρ D (1 + u + w D ) , (14)˙ ρ D = − Hρ D e r A M P (1 + u + w D )[2 ρ D − γ e r − A − n M P e r − A ] . (15)Here u = ρ m /ρ D is the ratio of energy densities. Also from Eq. (8), we find that 3 M P e r − A =(1 + u ) ρ D where u is governed by u = 3 M P n M P + γ e r − A ln( M P e r A ) + β e r − A − . (16)From Eq. (16), we see that at sufficient large e r A , where ρ D ≈ n M P e r − A , the ratio of energydensities will tend to a constant value u → /n −
1. Also at present time, u varies slowlyup to reach a constant value, u = 1 /n −
1. In Fig. 1, the function u is plotted in versus e r A for fixed γ, n and various β in the Planck mass unit in which M P = 1 / √ πG = 1. Fromthis figure, we conclude that the coincidence problem gets alleviated since for some valuesof model parameters, we get u ∼ O (1) for wide range of e r A (including the present time),and it is growing so that finally reaches to a fixed value of order unity.The deceleration parameter q = − − ˙ H/H may be obtained by using the Friedmannequation and continuity equation as follows [13, 14] q = − (1 + Ω K ) + 32 Ω D (1 + u + w D ) , (17)where Ω K = K/ ( a H ), Ω D = ρ D / (3 M P H ) and Ω m = ρ m / (3 M P H ) are the energy densityparameters. From these dimensionless parameters, the first Friedmann equation can be6 r ~A u β =0.0 β =0.1 β =0.2 β =0.3n=0.8, γ =0.1 FIG. 1: The evolution of u in versus e r A in LECHDE model. The asymptotic value is u = 0 . rewritten as: 1 + Ω K = Ω D + Ω m . Using the third form of interacting term, in whichΓ / H = b (1 + u ) and combining Eq. (15) with (10), the EoS parameter w D is given by w D = − − u (2 ρ D − n M P e r − A − γ e r − A ) − b (1 + u ) ρ D (1 − u ) ρ D − n M P e r − A − γ e r − A . (18)From this equation and Eq. (16), we find e r ′ A = 3 M P e r A h n M P e r A + γ ln( M P e r A )+ β + 3 M P e r A ( b − i / h M P e r A ( n − γ ln( M P e r A ) + 2 β − γ i , (19)where “prime” denotes the differentiation with respect to x = ln a = − ln(1 + z ) in which Hd/dx = d/dt .On the other hand, by using Eqs. (8) and (12), the evolution of dark energy density canbe rewritten as ρ ′ D = − ρ D (cid:2) w D + b (1 + u ) (cid:3) , (20)and then the evolution of Ω D is calculated as:Ω ′ D = − D (cid:20) (1 + w D )(1 − Ω D ) + b (1 + u ) − Ω D u + 23 Ω K (cid:21) . (21)Using Eq. (17), the deceleration parameter is given by q = − (1 + Ω K ) −
32 Ω D (1 + u )[ u − b (1 + u )] ρ D (1 − u ) ρ D − n M P e r − A − γ e r − A . (22)7t is worthwhile to mention that Ω K and Ω D is related byΩ K Ω m = a Ω K Ω m ∴ Ω K = e x Γ(1 − Ω D )1 − e x Γ , (23)where Γ = Ω K / Ω m is a constant value, which from the recent data, is given by Γ ≈ . γ = β = 0, Eqs. (16, 18, 22) reduce to thefollowing simple forms u = 1 /n − , (24) w D = − (1 + 1 u ) Γ3 H , (25) q = − (1 + Ω K ) −
32 Ω D (1 + u )( Γ3 Hu − , (26)which have been also calculated by [14]. In this case, from Eq. (19), the radius of apparenthorizon, e r A , can be obtained as e r A = e r A e M P ( n − b n − ) x = e r A (1 + z ) M P ( n − b − n ) . (27)Here we can choose e r A = 1 at present time: ( x = 0 or vanishing redshift, z = 0). Therefore e r A may be considered as a normalized horizon radius. From Eq. (27), we see that the radiusof apparent horizon is increased by cosmic time provided that | n | > | n | < √ − b .From Eq. (25), we see that, in the absence of interaction, we have w D = 0, but in LECHDEmodel, the EoS parameter may cross the phantom divide ( w D < −
1) even in the absenceof interaction. In Fig. 2, the evolution of the EoS parameter of LECHDE in versus of e r A is studied, both in interacting and non-interacting modes for positive values of β , in thePlanck mass unit. We consider specially the effect of coupling constant on behavior of w D .As it is shown in Fig. 2, by choosing the typical value of parameters of LECHDE model as: γ = 0 . , β = 0 . , n = 0 .
8, two distinct regions of e r A are given as: a : (0 . > e r A > a . Neither of interacting and non-interacting cases can drivean expanding universe ( w D > b : ( e r A > . b . Both of interacting and non-interacting cases may accelerate theexpanding universe and cross the phantom divide. Interacting cases always remain under thequintessence wall, while in non-interacting mode, the EoS parameter grows from phantomregime, w D < −
1, to positive value of EoS parameter, ( w D > − /
3) at small values of8 r A <
1. Therefore the non-interacting case can not drive the late time acceleration in ouruniverse.By solving Eqs. (19, 21, 22, 23) numerically, the behavior of deceleration parameter, q with respect to x = ln ( a ), in LECHDE model, is studied. In Fig. 3 as we can see, the present( x ≈
0) accelerated stage ( q < x <
0, far from x = 0). This is compatible with cosmic structure formation atmatter dominated era and present accelerated expansion.The typical values of γ, β , n are set, so that the function u becomes positive for allstudied regions and gets u ∼ . III. INTERACTING “PLECHDE” MODEL WITH AH-IR-CUTOFF
From Eq. (5), the energy density of PLECHDE with apparent horizon, e r A , as an IR-cutoff, is written as ρ D = 3 n M P e r − A − δM P e r − αA , (28)where using (14, 28), the energy density evolution is given by˙ ρ D = − Hρ D (1 + u + w D ) (cid:20) n − αδ e r − αA (cid:21) . (29)From Eqs. (8) and (28), the ratio of energy densities, u , is given by u = 1 n − δ e r − αA − . (30)Also from Eqs. (28) and (30), as the same as Sec. II, we see that at late time, for α > e r A is large, we have ρ D ≈ n M P e r − A and the ratio of energy densities u , will tendto a constant value u → /n −
1, while this is not valid for α <
2. In Fig. 4, we studythe behavior of u in versus of e r A , for various positive values of δ and fixed value α . Fromthis figure, we see that the function u is descending for δ > u issatisfied for a typical set ( α = 3 , n = 0 . , δ = 0 .
2) at e r A = 1 (present time). In this case u ∼ O (1), only for e r A > .
3. Also the coincidence problem can be solved, since for some9 r ~A w D b =0.0b =0.1b =0.2b =0.3 n=0.8, γ =0.1 a r ~A w D b =0.0b =0.1b =0.2b =0.3n=0.8, γ =0.1 b FIG. 2: The evolution of EoS parameter, w D , versus of e r A in LECHDE model, a : . > e r A > . b : e r A > . values of model parameters, we get u ∼ O (1), at present time, and it finally reaches to afixed value of order unity. 10 x q FIG. 3: The evolution of q in versus x = ln ( a ) in LECHDE model for ( n = 0 . , γ = 0 . , β =0 . , b = . , Γ = 0 .
0 1 2 3 4 5 0.20.40.60.811.21.41.6 r ~A u δ =0.1 δ =0.2 δ =0.3n=0.89, α =3 FIG. 4: The evolution of u in versus of e r A in PLECHDE model. w D , e r ′ A , Ω ′ D and deceleration parameter q are calculated as w D = − − (1 + u )( n − αδ e r − αA − b )1 − ( n − αδ e r − αA ) , (31) e r ′ A = 3 e r A " b − ( n − δ e r − αA )1 − ( n − αδ e r − αA ) , (32)Ω ′ D = − Ω D (cid:20) (1 + u + w D )(3 n − αδ e r − αA − D ) + 2Ω K (cid:21) , (33) q = − (1 + Ω K ) + 3Ω D " u − b (1 + u )1 − ( n − αδ e r − αA ) . (34)The limiting case of Eqs. (30, 31, 34), with δ = 0 or large e r A , has been given by Eqs.(24, 25, 26). Also in this case the eq. (32) reaches to Eq. (27) in the previous section. InPLECHDE model, the EoS parameter may cross the phantom divide ( w D < −
1) even inthe absence of interaction. In Fig. 5, the EoS parameter of PLECHDE is studied both invarious interacting and non-interacting modes. As it is shown in Fig. 5, by choosing thetypical value of parameters of PLECHDE as: ( α = 3 , δ = +0 . , n = 0 . e r A in behavior of EoS parameter as below: a : (0 . > e r A > a . We find: ( w D > b : ( e r A > . b . Both of interacting and non-interacting cases may acceleratethe expansion of the universe and the phantom divide is crossed. Interacting cases alwaysremains under the quintessence regime ( w D < − / w D < −
1, to above the quintessence regime( w D > − /
3) very soon. Therefore, same as previous section, the non-interacting case cannot drive the late time acceleration.Now we want to study the deceleration parameter of PLECHDE model. By solving Eqs.(32, 33, 34, 23), numerically, the behavior of q with respect to x can be studied. In Fig.6, similar to previous case, the present ( x ≈
0) acceleration has been supported by a longperiod deceleration phase at past ( x < α, δ , n are set,so that the function u become positive for all studied regions and gets u ∼ . r ~A w D b =0.0b =0.1b =0.2b =0.3 an=0.89, α =3, δ =0.20 1 2 4 6 8 10−2 −1.5−1 −0.5Q0 r ~A w D b =0.0b =0.1b =0.2b =0.3n=0.89, α =3, δ =0.2b FIG. 5: The evolution of EoS parameter, w D , in versus of e r A in PLECHDE model. a : . > e r A > . δ = 0 . b : e r A > .
09 and δ = 0 .
2. “Q” the Quintessence barrier ( w D = − / IV. THERMODYNAMICS OF NON-INTERACTING LECHDE WITH AH-IR-CUTOFF
In this section we want to associate a thermodynamical description to cosmological hori-zons, similar to black hole physics. In a FRW universe enclosed by an apparent horizon, onecan associate the Hawking temperature to the horizon, which is inversely proportional tosize of the apparent horizon. We know that the FRW universe may consist several cosmicingredients including dark energy, dark matter, radiation and baryonic matter. However13 x q FIG. 6: The evolution of q in versus x = ln ( a ) in PLECHDE model for ( n = 0 . , α = 3 , δ =0 . , b = 0 . , Γ = 0 . many cosmological evident reveal that the dark energy and matter are two dominant com-ponents in our universe. At following, we will consider only LECHDE and CDM componentsin a non-flat FRW universe enclosed by apparent horizon. In a local thermal equilibrium,where there is not any heat flow from the apparent horizon, the temperature of the en-ergy content of the universe ( T ) should be equal to the temperature which is associatedwith apparent horizon ( T h ). In non equilibrium case, the heat will flow outside (inside) theapparent horizon if the temperature of cosmic fluid is hotter (colder) than the apparenthorizon, respectively. The thermal equilibrium state can be accessed at a finite time andtherefore we can consider a unit temperature for whole spacetime (contain DE, CDM andAH). The equilibrium entropy of the LECHDE is connected with its energy and pressure, p D , through the Gibbs law of thermodynamics T dS D = dE D + p D dV, (35)where V = (4 π/ r A is the volume of whole space up to horizon surface and S D is the entropyof DE component. The equilibrium temperature T , can be obtained from the surface gravity( κ H ) of horizon as follows [23] T = | κ H | π = 14 π √− h (cid:12)(cid:12)(cid:12) ∂ a ( √− hh ab ∂ b e r ) (cid:12)(cid:12)(cid:12) . (36)14rom this equation, the temperature of apparent horizon is calculated as T = 12 π e r A − ˙ e r A H e r A ! . (37)Following Cai and Kim [23], the apparent horizon radius e r A should be regarded to have afixed value in thermal equilibrium. It means that ˙ e r A ≈ . Thus the temperature is given by T = 1 / (2 π e r (0) A ) . (38)Now from Eq. (35), we have T dS = ρ D (1 + w D ) dV + V dρ D , (39)and by using Eq. (7), we can obtain dS D d e r A = 83 π ( e r A ) h n M P ( e r A ) − +2 γ ( e r A ) − − ρ D (1 − w D ) i , (40)where superscript (0) denotes that the universe is in a stable thermodynamical equilibriumstate. V. THERMODYNAMICS OF INTERACTING LECHDE WITH AH-IR-CUTOFF
In the presence of interaction, ( Q = 0), the thermal equilibrium is no further maintaindue to thermal fluctuation which has been arose from decaying of dark energy to darkmatter. The conservation equations for ρ m and ρ D , have been given by Eqs. (10, 11).In this case, however the Gibbs law of thermodynamics may hold only approximately fordynamical apparent horizon, the entropy affected under a first order logarithmic correction( S (1) D ) involving temperature T and the heat capacity C , as bellow [27] S (1) D = −
12 ln( CT ) . (41)Hence, the entropy should be modified as: S D = S (0) D + S (1) D . The heat capacity in thermalequilibrium has been defined as: C = T ∂S (0) D /∂T . Using (38), the heat capacity can berewritten as: C = − ( e r A ) ∂S (0) D /∂ e r A . Using Eq. (40) in thermal equilibrium, the correctedterm S (1) D is calculated as S (1) D = −
12 ln (cid:2) ρ D ( e r A ) (1 − w D ) − n M P − γ ( e r A ) − (cid:3) −
12 ln( 23 ) . (42)15imilar to Eq. (40) with interaction, one obtains dS D = 83 π e r A (cid:2) n M P e r − A + 2 γ e r − A − ρ D (1 − w D ) (cid:3) d e r A , (43)where from dS D = dS (0) D + dS (1) D , we can find1 − w D = h n M P e r − A + 2 γ e r − A − π e r A dS (0) D d e r A + dS (1) D d e r A ! i ρ − D . (44)From Eqs. (40, 42), it is obtained dS (0) D d e r A = dS (0) D d e r A d e r A d e r A = 83 π ( e r A ) h n M P ( e r A ) − +2 γ ( e r A ) − − ρ D (1 − w D ) i d e r A d e r A , (45) dS (1) D d e r A = dS (1) D d e r A d e r A d e r A = − n ρ D ( e r A )(1 − w D ) + 4 γ ( e r A ) − +( e r A ) dd e r A [ ρ D (1 − w D )] o / h ρ D ( e r A ) (1 − w D ) − n M P − γ ( e r A ) − i d e r A d e r A , (46)where from (18) and (16), we have1 − w D = (47)4 + 3 u (2 ρ D − n M P e r − A − γ e r − A ) − Γ3 H (1 + u ) ρ D (1 − u ) ρ D − n M P e r − A − γ e r − A , − w D = (48)4 + 3 u ρ D − n M P ( e r A ) − − γ ( e r A ) − (1 − u ) ρ D − n M P ( e r A ) − − γ ( e r A ) − ,du d e r A = − (1 + u ) (cid:20) e r A + dd e r A ln( ρ D ) (cid:21) . (49)Now, we want to find a relation between the interaction term and the thermal fluctuation.For this purpose, by comparing Eqs.(44, 47), the interaction term can be calculated with16espect to thermal fluctuation asΓ3 H = 23(1 + u ) ρ D n (2 ρ D − n M P e r − A − γ e r − A ) (50) (cid:16) (1 + u ρ D − n M P e r − A − γ e r − A (cid:17) + ( d e r A d e r A ) × e r A π e r A [(1 − u ) ρ D − n M P e r − A − γ e r − A ]6 n M P + 2 γ ( e r A ) − − ρ D ( e r A ) (1 − w D ) × h π (cid:0) n M P + 2 γ ( e r A ) − − ρ D ( e r A ) (1 − w D ) (cid:1) +4 γ ( e r A ) − + 2 ρ D (1 − w D ) + e r A dd e r A [ ρ D (1 − w D ) io . In limiting case, for ordinary HDE ( γ = β = 0), where w D = 0 and ρ D = 3 n M P e r − A , fromEqs. (25, 50), we can obtain Γ3 H = 1 − n (cid:20) − e r A dd e r A ln( e r A ) (cid:21) . (51) VI. CONCLUSION
In this paper the logarithmic and power-law entropy-corrected version of interacting HDEwith AH-IR-cutoff in a non-flat universe enclosed by apparent horizon have been studied.In fact we generalized the ordinary HDE model by considering the entropy correction due tofluctuation of spacetime and AH-IR-cutoff. In LECHDE model, corrections are restricted tothe leading order correction which contains the logarithmic of area. In PLECHDE model,the correction is based on the gravitational fluctuations which affect the area law of entropyto a fractional power of area, which is arisen by entanglement of quantum field theory.The ratio of dark matter to dark energy densities u , EoS parameter w D and decelerationparameter q have been obtained. We showed that the cosmic coincidence is satisfied forappropriate model parameters. In dealing with cosmic coincidence problem, we found anappropriate set of values for LECHDE model as: ( γ = 0 . , β = 0 . , n = 0 .
8) and forPLECHDE model as: ( n = 0 . α = 3 , δ = 0 . u ∼ . e r A . Although in the absence of interaction between dark energy and dark matter,these two dark components conserved separately, while by imposing an interaction term,a stable fluctuation around equilibrium is expectable. Therefore, in the interacting case,where the entropy affected under a first order logarithmic correction, we obtained a relationbetween the interaction term and thermal fluctuation in a non-flat universe enclosed by theapparent horizon. Also in limiting case for ordinary HDE, the relation of interaction termversus thermal fluctuation was calculated. Acknowledgments
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