Crossing the phantom divide line in the Holographic dark energy model in a closed universe
aa r X i v : . [ g r- q c ] M a y Crossing the phantom divide line in theHolographic dark energy model in a closed universe
H. Mohseni Sadjadi ∗ Department of Physics, University of Tehran,P. O. B. 14395-547, Tehran 14399-55961, Iran
November 9, 2018
Abstract
Conditions needed to cross the phantom divide line in an interactingholographic dark energy model in closed Friedmann- Robertson-Walkeruniverse are discussed. The probable relationship between this crossingand the coincidence problem is studied.
One of the candidates proposed to explain the present acceleration ofthe universe [1], is the dark energy model which assumes that nearly%70 of the universe is filled of an exotic energy component with nega-tive pressure. Based on observations, the density of (dark) matter anddark energy component must be of the same order today (known ascoincidence problem)[2]. Also, based on recent data, the dark energycomponent seems to have an equation of state parameter w < − w > − E , in a region of size L , must be less than (or equal to)the energy of a black hole of the same size, i.e., E ≤ LM p , where M p is the Planck mass. In terms of energy density, ρ , this inequality canbe rewritten as ρ ≤ M p L − . Based on this result, in [6], an expressionfor a dynamical dark energy (dubbed as the holographic dark energy)was proposed : ρ d = 3 c M p L − , where c is a numerical constant. Dif-ferent choices may be adopted for the infrared cutoff of the universe,e.g., particle horizon, Hubble horizon, future event horizon and so on[7]. In a noninteracting model, if we take the particle horizon as theinfrared cutoff, we are unable to explain the accelerated expansion of ∗ [email protected] he universe [6]. Besides an appropriate equation of state parameterfor dark energy or dark matter cannot be derived if one chooses theHubble horizon as the cutoff [8]. Instead, if we choose the future eventhorizon, although the present accelerated expansion of the universemay be explained [6], but the coincidence problem still unsolved. Thisproblem can be alleviated by considering suitable interaction betweendark matter and holographic dark energy.In this paper we consider a closed Friedmann- Robertson- Walker(FRW) universe (we don’t restrict ourselves to small spatial curvaturelimit) and assume that the universe is composed of two interactingperfect fluids: holographic dark energy and cold (dark) matter. Ageneral (as far as possible) interaction between these components isconsidered. We allow the infrared cutoff to lie between future and par-ticle event horizons. After some general remarks about the propertiesof the model, we discuss the conditions needed to cross the phantomdivide line (transition from quintessence to phantom phase). We showthat this crossing poses some conditions on parameters of the modeland using an example, we show that this can alleviate the coincidenceproblem (at least) at transition epoch.We use units ~ = G = k B = c = 1 throughout the paper. The FRW metric, ds = − dt + a ( t ) (cid:18) dr − r + r ( dθ + sin θdφ ) (cid:19) , (1)describes a homogeneous and isotropic closed space time with scalefactor a ( t ). We assume that this universe is filled with perfect fluidsand its energy momentum tensor is given by T µν = ( P + ρ ) U µ U ν + P g µν , (2)where U µ = (1 , , , ρ and P are energy density and pressureof the total fluid respectively. Using Einstein’s equation, one can obtainFriedmann equations H = 8 π X i ρ i − a ( t )˙ H = − π X i ( P i + ρ i ) + 1 a ( t ) . (3)The subscript i stands for the ith perfect fluid and H = ˙ a ( t ) a ( t ) is theHubble parameter. In this paper the universe is assumed to be com-posed of dark energy component with pressure P d and energy density ρ d , and the cold (dark) matter whose energy density is ρ m . Although hese components, due to their interaction, are not conserved˙ ρ d + 3 H ( ρ d + P d ) = − Q ˙ ρ m + 3 Hρ m = Q, (4)but vanishing of covariant divergence of the energy momentum tensor(2), yields the conservation equation˙ ρ + 3 H ( P + ρ ) = 0 , (5)for the whole system. In this two-component universe the Friedmannequations reduce to H = 8 π ρ m + ρ d ) − a ( t )˙ H = − π ( P d + ρ d + ρ m ) + 1 a ( t ) . (6)The first equation can be rewritten asΩ m + Ω d = 1 + Ω k , (7)where Ω m = ρ m ρ c , Ω d = ρ d ρ c and the geometrical parameter is definedthrough Ω k = a ( t ) H . The critical energy density ρ c is defined by ρ c = H π . Note that for a flat universe, ρ = ρ c and Ω m + Ω d = 1.Different models have been proposed for the dark energy component ofthe universe. Here we adopt holographic dark energy, which in termsof the infrared cutoff of the universe, L , can be expressed as ρ d = 3 c πL . (8)In [9] the infrared cutoff was chosen as L f = a ( t ) sin y f where y f is y f = Z ∞ t dta ( t )= Z r f dr √ − r . (9)In this way L f is the radius of future event horizon measured on thesphere of the horizon [9]. In the presence of bigrip [10] at t = t s , ∞ in(9) must be replaced with t = t s . In the flat case this cutoff reduces to L f = R h = a ( t ) R ∞ t dta ( t ) . A similar choice is to take an infrared cutoffbased on the particle horizon, i. e., L p = a ( t ) sin y p , where y p = Z t dta ( t )= Z r p dr √ − r . (10)As proposed in [11], in general, L can be taken as a combination ofboth L p and L f . In this paper we take the cutoff as: L = αL p + βL f ; 0 ≤ α ≤ , ≤ β ≤ . (11) or α = 0 , β = 1, we get L = L f and L = L p is obtained when β = 0 , α = 1. If we attribute an entropy, S , to the surface A = 4 πL : S = A πL , (12)then the thermodynamics second law implies ˙ L >
0. Using˙ L = HL − β cos y f + α cos y p , (13)we find that the second law of thermodynamics is satisfied when X < , (14)where X = β cos y f − α cos y p c Ω d . (15)Note that y p + y f = Z ∞ dta ( t ) := γ, (16)is a functional of a ( t ) and is time independent: ˙ γ = 0. In the limitΩ k ≪ Ω d , α sin y p + β sin y f = La ( t ) = c ( Ω k Ω d ) (17)implies: y f , y p ≪ k = − H Ω k HH ! , (18)and, the ratio r = Ω m Ω d satisfies˙ r = ( r + 1) (cid:18) Qρ d + 3 Hωr (cid:19) . (19)Note that r determines the ratio of (dark) matter to dark energy den-sity and is of order unity in the present epoch. One can also study thetime evolution of K = Ω k Ω d . Using K = ( Lca ( t ) ) , it is straightforward toshow that in order that ˙ K ≶
0, we must have( α cos γ − β ) cos y f + α sin γ sin y f ≶ . (20)Therefore the behavior of this ratio with respect to the comoving timedepends on the parameters of the model, e.g. for α = 0 , β = 1 (i.e.,when the future event horizon is considered), it is decreasing. In this part we study the ability of our model to describe w = − r (coincidence problem). y using HL = c Ω − d , we arrive at˙ HH + c H L ˙Ω d Ω − d = β cos y f − α cos y p HL − . (21)Hence from w = − − ˙ HH − Ω k k , (22)we find out w = − − X k ) + 13 H (1 + Ω k ) ˙Ω d Ω d . (23)The universe is accelerating for ¨ a >
0, or equivalently when w < − .In this case ˙Ω d < H Ω d X, (24)which by considering the thermodynamics second law results in:˙Ω d < H Ω d , (25)or equivalently, Ω d a ( t ) is a decreasing function of comoving time. w canbe also derived from the equation (19), resulting w = − ˙Ω d H Ω m + ˙Ω k Ω d H Ω m (1 + Ω k ) − Q Hρ d Ω d Ω m . (26)Suppressing ˙Ω d from (23) and (26) leads to w = − Ω d k ) (cid:18) QHρ d + 1 (cid:19) − X Ω d k ) . (27)Note that if w = − d must satisfy the equation (cid:18) X + QHρ d + 1 (cid:19) Ω d = 3 (1 + Ω k ) , (28)at transition time, i.e., this equation must have at least one root. More-over, to cross w = −
1, ˙ w must be negative at w = −
1. To determine˙ w , we note that ˙ X = H (cid:18) X + 12 α (1 + Ω k ) X + Ω k (cid:19) , (29)where α = 1 + 3 w . To derive the above equation we have used1 c Ω d ( − β ˙ y f sin y f + α ˙ y p sin y p ) = H Ω k . (30)Substituting (29) and (30), in time derivative of (27) results in˙ w = − H Ω d k [ X + (cid:18) α k ) + Q Hρ d + 13 (cid:19) X + Ω k α QHρ d ) + 16 H ( QHρ d ˙)] . (31) n studying the divide line crossing, we intend to adopt the validityof thermodynamics second law. Hence it is more convenient to write(31), at transition time, in the form˙ w = − H Ω d k (cid:18) ( X − X −
13 (Ω k − QHρ d − (cid:19) − Ω d k ) (cid:18) QHρ D (cid:19) . . (32)˙ w < X − (cid:18) X − (cid:18) Ω k − QHρ d − (cid:19)(cid:19) > − H (cid:18) QHρ d (cid:19) . . (33)If the universe remains in the phantom phase after the transition,the cosmological evolution may be ended by a big rip singularity [10].But, (28), depending on the interaction Q , and the parameters α and β , may have more than one root for Ω d when w = −
1. This may al-low another transition from phantom to quintessence phase and avoidsbigrip singularity. In this case, in the transition from phantom toquintessence phase, we must have ˙ w > w = − Q . Note that ρ m = rr +1 ρ and ρ d = r +1 ρ . Hence if QHρ d is a function of ρ m and ρ d ,it can be casted into the form q ( ρ, r ), this may prompt us to assume,as a choice, QHρ d =: q ( ρ, r ). X can be expressed in terms r and q as X = 3 r − q . (34)Time evolution of q is given by the equation˙ q = q ,ρ ˙ ρ + q ,r ˙ r = q ,ρ ˙ ρ − Hq ,r ( r + 1)( α + 2 X ) , (35)which at w = −
1, reduces to˙ q = 2 Hq ,r ( r + 1)(1 − X ) . (36)To obtain the above equation we have used ˙ r = − H (1 + r )( α + 2 X ) = − k Ω d ( α + 2 X ) H , and defined q ,x = ∂q∂x . Collecting these results weobtain˙ w = − H Ω d k (cid:18) ( X − X + 13 (1 − Ω k + q − ( r + 1) q ,r )) (cid:19) . (37) X < r < q , and in order to cross the phantom divide line,
X <
13 (Ω k + ( r + 1) q ,r − q −
1) (38)must be hold. As an example consider the interaction term as Q = λ m Hρ m + λ d Hρ d , hence q = λ m r + λ d . Therefore˙ w = − H Ω d k (cid:20) ( X − (cid:18) X + 13 (1 − Ω k + λ d − λ m ) (cid:19)(cid:21) . (39) ere, the validity of thermodynamics second law implies(3 − λ m ) r < λ d , (40)and the condition for crossing w = − − λ m ) r < λ d + 2 λ m + 2Ω k − . (41)So we have to assume(3 − λ m ) r < M in. { λ d , λ d + 2 λ m + 2Ω k − } . (42)For λ m >
3, we have r > M in. { λ d , λ d +2 λ m +2Ω k − } − λ m . (43)If M in. { λ d , λ d +2 λ m +2Ω k − } <
0, the above inequality poses a lowerbound on r at w = − k at transi-tion time. E. g., if we take Ω k = 0 .
02 at transition time, all modelssatisfying 3 < λ m < − M in. { λ d , λ d +2 λ m − . } give rise to r > inaccordance with recent data [2].At the end it is worth to note that in the above example, for X = 1,i.e. when the expansion is adiabatic ˙ S = 0, (29), and (31) imply thatthe higher time derivatives of X and ˙ w are also zero at w = −
1. Inthis situation, w = − t → ∞ . So w = − Qρ d = f ( ρ, r ), then at w = −
1, ( Qρ d H ) . = f ,r ( r + 1)(1 − X ) − Ω k f (see (22)). If X = 1, the sign of f determineswhether w = − Q = λρ m ρ d [12]. Inthis model ( QHρ d ) . = − λρ d [Ω k r + ( X − X = 1,meanwhile w = − λ < Holographic dark energy model in a closed FRW universe (but not nec-essarily with a small spatial curvature), was considered. The infraredcutoff was taken to be lie between particle and future event horizons(see (11)), and dark energy and dark matter were assumed to interactvia a general interaction source (see (4)). The condition of validityof thermodynamics second law for the infrared cutoff was obtained(see (14)). Using equations derived for equation of state parameterof dark energy and its time derivative, condition required for crossingthe phantom divide line was derived (see (33)). By adopting thermo-dynamics second law and restricting the interaction term to specialforms, this condition was reduced to a more compact form (see (38)),revealing the probable relationship between w = − s a consequence of our lack of knowledge about the nature of darkenergy and dark matter, the form of interaction term, Q , is still un-known. Also different choices for the infrared cutoff are used in theliterature. The viability of a model, with a specific Q and a partic-ular infrared cutoff, corresponds to its agreement with astrophysicaldata. Using the general result (33) one can examine whether a pro-posed model (characterized by Q and the cutoff L (defined by (11)) iscompatible with phantom divide line crossing and meanwhile satisfiesthermodynamics second law. However these phenomenological mod-els, can give us some clues to refine our view about the realistic darkenergy model. Acknowledgment
The author would like to thank the University of Tehran for sup-porting him under the grant provided by its Research Council. Thiswork was partially supported by the center of excellence in structureof matter of the Department of Physics.
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