De-Sitter nonlinear sigma model and accelerating universe
aa r X i v : . [ g r- q c ] S e p De Sitter nonlinear sigma model and accelerating universe
Joohan Lee ∗ Department of Physics, University of Seoul, Seoul 130-743 Korea
Tae Hoon Lee † Department of Physics and Institute of Natural Sciences,Soongsil University, Seoul 156-743 Korea
Tae Yoon Moon ‡ and Phillial Oh § Department of Physics and Institute of Basic Science,Sungkyunkwan University, Suwon 440-746 Korea (Dated: October 29, 2018)We consider a cosmology with a noncompact nonlinear sigma model. The target space is of deSitter type and four scalar fields are introduced. The potential is absent but cosmological constantterm Λ is added. One of the scalar fields is time dependent and the remaining three fields have notime dependence but only spatial dependence. We show that a very simple ansatz for the scalarfields results in the accelerating universe with an exponential expansion at late times. It is pointedout that the presence of the energy density and pressure coming from the spatial variation of thethree scalar fields plays an essential role in our analysis which includes Λ = 0 as a special caseand it discriminate from the standard Λ-dominated acceleration. We perform a stability analysisof the solutions and find that some solutions are classically stable and attractor. We also present anonperturbative solution which asymptotically approaches an exponential acceleration and discusspossible cosmological implications in relation to dark energy. It turns out that the equation of stateapproaches asymptotically ω = − ω ∼ − ∓ .
07, which is within the region allowed by the observationaldata. This solution also exhibits a power law expansion at early times, and the energy density ofthe scalar fields mimics that of the stiff matter.
PACS numbers: 11.10.Lm, 95.36.+x, 98.80.-kKeywords: nonlinear sigma model; cosmology; exponential acceleration; dark energy
I. INTRODUCTION
The recent cosmological observations [1] provide many precise data and arouse an explosion of recent interests inthe cosmology. The most recent data and its cosmological interpretation [2] indicate that about 73% of our Universeis made of dark energy, the origin of which is one of the greatest puzzles in the modern cosmology [3].It is highly conceivable that the dark energy is responsible for the late acceleration of the Universe [4] and manycandidates have been proposed. The simplest approach for the accelerating universe is to introduce the cosmologicalconstant [5, 6, 7, 8, 9] for the dark energy. Other approaches [4] include dynamical models of the cosmological constant[10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Among them, most commonly proposed candidates are thequintessence, which is described by a scalar field minimally coupled to Einstein gravity with a potential[12, 15]. Itis shown that the scalar energy density is subdominant in the matter dominated, and then, acceleration takes overat later stage of the cosmological evolution. Later the phantom model with a negative kinetic energy scalar field wasproposed [25] to account for the region where the equation of state is less than ω = −
1, and quintom model where theordinary scalar and the phantom are both introduced [26] to explain the crossing of the ω = − ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]
One of the motivations is that the scalar fields have a geometric origin and the potential term is not necessary. Anothermotivation is to consider the spatial dependence of the scalar fields and examine its consequences. To solve the Einsteinequation, we assume that only one of the scalar fields is time dependent and the remaining three fields have no timedependence but only spatial dependence. We first show that a very simple ansatz can solve the Einstein equationwhich describes the acceleration of the universe with an exponential expansion at late stage. The spatial contributionin combination with the cosmological constant forms an effective cosmological constant and plays important roles inorder to provide the necessary energy density and pressure. We will also show that the acceleration is possible evenwithout the cosmological constant. It seems that this feature of contributing the energy density and pressure comingfrom the spatial variation of the scalar fields was not considered before in relation with dark energy.Then, we perform a stability analysis of the solutions. We find that some of the solutions, depending on the valuesof the parameter given, are classically stable and attractor solutions. We will consider two cases where the targetspace has signature (+ , − , − , − ) or ( − , + , + , +). Especially, in the (+ , − , − , − ) case, the linear stability analysis fails,but we are able to find out that there exists a nonperturbative solution which asymptotically approaches the de Sitteracceleration, but at early times it is a power law expansion. In this case, the cosmological constant term is uniquelyfixed in terms of the other parameter. These features differentiate the present analysis from the standard Λ-dominatedlate-time exponential acceleration.One might think that adding a cosmological constant term with the scalar fields could be ad hoc , but it seemsthat at present, the dynamical models of the dark energy is not completely successful in solving the cosmologicalconstant problem and many of them require some kind of fine tuning anyhow. Nevertheless, the aim of this paper isnot to explain the smallness of the cosmological constant (the fine tuning problem in our approach is mentioned in theConclusion and Discussion section), but to focus on the late-time exponential acceleration of the universe and stabilityof its behavior. It turns out that the accelerating universe requires some bound on the original cosmological constantterm. In ( − , + , + , +) case, it require that the original cosmological constant term must be negative for stability, stillthe acceleration is possible and can be led by the scalar fields.The paper is organized as follows. In Sec. II, we present noncompact nonlinear sigma model coupled with Einsteingravity with a cosmological constant term and discuss the ansatz which solves the equations in some generality. InSec. III, we describe the exponential accelerating solution with de Sitter target space. In Sec. IV, the stabilityanalysis is performed and allowed range of the cosmological constant is classified. In Sec. V, a nonperturbativesolution is obtained and possible cosmological implications in relation to dark energy is given. Section VI includesthe conclusion and discussion. II. THE ACTION AND COSMOLOGICAL CONSTANT
We consider an action in which the Einstein gravity is coupled to a nonlinear sigma model with a cosmologicalconstant term (in units of M p = 1): S = Z d x √− g [ 12 R − g µν λ G αβ (Φ) ∂ µ Φ α ∂ ν Φ β − Λ + L matter ] (1)where Φ α = ( φ, σ i ) ( i = 1 , , G αβ is the metric of the noncompact target space, λ is the self-coupling constant ofthe nonlinear sigma model and it is assumed to be positive. Λ is the cosmological constant. The equations of motionare given by R µν = 2 λ G αβ ∂ µ Φ α ∂ ν Φ β + Λ g µν + ˆ T µν (2)1 √− g ∂ µ [ √− gg µν G αβ ∂ ν Φ β ] = 12 ∂G βγ ∂ Φ α g µν ∂ µ Φ β ∂ ν Φ γ , (3)where ˆ T µν (= T µν − g µν T /
2) is assumed to take the perfect fluid form; T µν = ( − ρ m , p m , p m , p m ) . (4)The matter sector satisfies the continuity equation; ∇ µ T µν = 0 . If we ignore the matter part, we can solve the Eqs. (2) and (3) with the following ansatz φ = t, σ i = x i . (5)To check whether Eq. (2) can be solved (without ˆ T µν ) with this ansatz, first note that if Λ = 0, g µν ( t, x i ) = + G µν ( φ, σ i )satisfies the equation as long as the scalar curvature of the space-time metric g µν and that of the target space G αβ are constants. Then, we can add an cosmological constant Λ which has the same sign with these curvatures and theequation can still be satisfied [29, 30]. This has the effect of scaling the space-time metric via g µν → (1 + 4Λ /R ) g µν ,where R is the scalar curvature constant. The cosmological constant could even have some value of the opposite signwith these curvatures as long as the absolute value of the cosmological constant is smaller than that of the targetspace, i.e., | Λ | < | R | /
4. This point can be extended further. Suppose g µν ( t, x i ) = − G µν ( φ, σ i ). Then, the scalarcurvatures of the space-time and target space have opposite signature and the equation cannot be satisfied with Λ = 0.But if we add Λ such that the sign is the same as the space-time scalar curvature constant and the absolute valueis greater than the scalar curvature of the target space, the equation can be satisfied. Also, one can check that themetric ansatz g µν ( x ) = ± G µν ( φ, σ i ) satisfies the Eq. (3). In summary, we find that the metric ansatz and (5) solve(2) and (3), and we have R µν = ( ± λ + Λ) g µν , (6)as long as the constant curvature condition is satisfied and without ˆ T µν .The σ i = x i ansatz first appeared in higher dimensional gravity theory in association with spontaneous compactifi-cation of the extra dimensions [29, 30]. It does not break the isotropy and homogeneity of the universe as long as wedo not introduce the potential for the σ fields. Also the φ = t has been exploited to unify early-time and late-timeuniverse based on phantom cosmology [31]. Note that the quantity Λ eff ≡ ± /λ + Λ plays the role of the effectivecosmological constant and there is curvature constant restriction on the value of Λ; ± /λ + Λ must have the samesignature as that of the space-time scalar curvature constant. The above aspect of the ansatz (5), (6) is quite a generalfeature of the nonlinear sigma model coupled to gravity. In this paper, we will consider the de sitter target space with G ( ǫ ) αβ = ǫ ( 1 , − e ξφ , − e ξφ , − e ξφ ) (7)with ǫ = ∓ ξ being an arbitrary positive constant. It turns out that the spatial ansatz (5) provides contributionof the energy density and pressure such as to reveal diverse aspects of the late time exponential acceleration, notpresent in the standard cosmological constant dominated acceleration. We will also find that the allowed value of thecosmological constant divides further if required the stability. III. DE SITTER SOLUTION
To discuss the cosmological implication of the solution (5), (6) with the de Sitter target space metric (7), weintroduce the standard space-time metric via ds = − dt + a ( t ) dx i dx i . (8)With H = ˙ a/a , the equation of motion (2) becomes H = 23 λ [ ǫ ( 12 ˙ φ − e ξφ ˙ σ i + 12 a ( ∂ i φ ) − a e ξφ ( ∂ i σ j ) ) + λ ρ m (9)˙ H = − λ ǫ [ ˙ φ − e ξφ ˙ σ i − a ( ∂ i φ ) − a e ξφ ( ∂ i σ j ) ] −
12 (1 + ω m ) ρ m , (10)where ω m = p m /ρ m . The continuity equation implies ρ m ∝ a − ω m ) . Plugging the ansatz σ i = x i and φ ≡ φ ( t ) intothe above equations, Eqs. (3), (9) and (10) become0 = ¨ φ + 3 H ˙ φ − ξ e ξφ a , (11) H = 23 λ [ ǫ ( 12 ˙ φ − a e ξφ ) + λ ρ m (12)˙ H = − λ ǫ [ ˙ φ − a e ξφ ] −
12 (1 + ω m ) ρ m . (13)The second terms in both (12) and (13) are the contributions coming from the spatial variations of σ i which is essentialfor the subsequent analysis.The scalar dominance requires a check of whether the matter contribution term can be ignored at late times. Oursolution corresponds to a linearly increasing scalar field with positive ξ such that the kinetic energy terms and thesecond terms in both (12) and (13) are constant. Therefore, the contribution of the matter density which decreasesas a − ω m ) becomes negligible at late times and we ignore the matter part here after. Now, substitution of φ = t leads to a ( t ) = e ξt , ξ = r − ǫ λ + Λ3 (Λ > ǫλ ) . (14)In the above equation, we fixed the initial values by a (0) = 1 and φ (0) = 0. Later, we will relax these initial conditionsand accommodate more general conditions. This describes a de Sitter expansion of the universe. From here on, wewill always assume σ i = x i and study the time-dependent behavior of the Eqs. (11), (12), (13). Note that for the ǫ = +1 case, the cosmological constant term has to be bigger than some positive value. In contrast, for the ǫ = − eff = 3 ξ in Eq. (14). IV. STABILITY
To check the stability of the above solution, we first consider the following quantities,2 ξφ − N = X, N = ln ( a ) (15)Plugging (15) into (11) ∼ (13), we obtain 3 H + ˙ H = − ǫλ e X + Λ (16)¨ X + 3 H ˙ X − (6 ξ + 4 ǫλ ) e X + 2Λ = 0 (17)The solution (14) corresponds to X = 0 with H = ξ, ˙ φ = 1. In order to accommodate more initial conditions, weconsider the solution X = X (0) ≡ ln f . Then, the effective cosmological constant becomes Λ eff ( f ) = − ǫf /λ + Λ,and the solution (14) is replaced by φ = p f t + φ (0) , a ( t ) = a (0) e √ fξt , ξ = s − ǫ λ + Λ3 f , (18)with 2 ξφ (0) − a (0) = ln f, Λ > ǫfλ . Note that the exponent ξ behaves under the change of the initial conditionswhen Λ = 0 as follows; f → g, e √ fξt → e √ gξt .The linear perturbation of Eq. (17) leads to δ ¨ X + 3 p f ξδ ˙ X − (6 ξ + 4 ǫλ ) f δX = 0 (19)Introducing δX ∼ e γt , Eq. (19) yields γ + 3 p f ξγ − (6 ξ + 4 ǫλ ) f = 0 (20)The solutions for the Eq. (20) are γ + / p f = − ξ + q ξ + ǫλ γ − / p f = − ξ − q ξ + ǫλ Name ǫ ξ
Stability Λ(A) +1 ξ > γ + > , γ − <
0, Unstable, saddle point Λ > f/λ (B) − λ ≤ ξ < / λ γ ± < − f/ λ ≤ Λ < − ξ > / λ γ + > , γ − < , Unstable, saddle point Λ > − ξ < / λ γ ± , Imaginary, stable, attractor − f/λ < Λ < − f/ λ (E) − ξ = 2 / λ γ + = 0 , γ − < ξ = 1 / λ Nonperturbative Λ = 3 f/λ φ = √ f t + φ + A ln (1 + Ce − √ fγt )TABLE I: Various accelerating solutions and their stability From these equations, we have the following cases:(A). For the ǫ = +1 case, γ + > γ − < ξ . Hence, the solution is unstable. It corresponds to a saddlepoint. The cosmological constant has to be positive.For the ǫ = − λ ≤ ξ < λ , γ ± are both negative and the solution is stable and an attractor. For ξ = λ ,the root is degenerate with γ + / √ f = γ − / √ f = − ξ/ ξ > λ , γ + > γ − <
0, so the solution is unstable and corresponds to a saddle point.(D). In the case ξ < λ , γ ± becomes imaginary and the perturbation is oscillatory and it is an attractor.The linear perturbation can be integrated explicitly. For (A), (B) and (C), Eq. (19) yields δX = Ae γ + t + Be γ − t (23) δH = − ǫfλ ( Ae γ + t γ + + 6 √ f ξ + Be γ − t γ − + 6 √ f ξ − C e − √ fξt ) , (24)where C = A/ ( γ + + 6 √ f ξ ) + B/ ( γ − + 6 √ f ξ ). Note that when ǫ = +1, the second term in Eq. (24) diverges for thevalue ξ = 1 / λ . This might imply that linear perturbation fails in this case and in fact, there exist a nonperturbativesolution as will be discussed in the next section. For case (D), we have δX = D cos( ωt + θ ) e − √ fξt (25)where ω = f ( λ − ξ ) and δH = − Ee − √ fξt + F ( t ) e − √ fξt (26)where E = 36 Df p f ξ cos θ /λ (81 f ξ + 4 ω ) + 8 Df ω sin θ /λ (81 f ξ + 4 ω ) ,F ( t ) = 36 Df p f ξ cos ( ωt + θ ) /λ (81 f ξ + 4 ω ) + 8 Df ω sin ( ωt + θ ) /λ (81 f ξ + 4 ω ) . (27)We comment on the case ξ = 2 / λ separately. It corresponds to when the cosmological constant Λ is zero. Inthis case, one of the roots γ + of Eq. (21) becomes zero, and the other root γ − is negative. Its stability is indecisiveat this level. These solutions, their stability and contents of the cosmological constant are summarized in Table I. Itis interesting to note that in the ǫ = − V. NONPERTURBATIVE SOLUTION
It turns out that in the ǫ = +1 case, an explicit nonperturbative solution can be found. To see that, let us firstassume e ξφ = √ f a. Then, Eq. (12) suggests that for nontrivial solution, we must have H = 13 λ ˙ φ , λ Λ = 3 f (28)The first of the above equation yields ξ = 1 / λ . This value of ξ was the one where linear perturbation failed inthe previous section. Substituting the ansatz a = √ f e ξφ into (11), we obtain¨ φ + 3 ξ ˙ φ − f ξ = 0 (29)Note that this equation describes particle motion where constant external force and velocity square dependent frictionalforce are acting. When ˙ φ (0) < f , the constant force term dictates the particle motion at early times and it acceleratesuntil the velocity reaches the terminal velocity ˙ φ ( ∞ ) = √ f . When ˙ φ (0) > f , the friction term dominates at earlytimes and it decelerates until the velocity reaches the terminal velocity ˙ φ ( ∞ ) = √ f . We can find the solution for the above Eq. (29) as follows φ ( t ) = p f t + φ (0) − ξ ln(1 + C ) + 13 ξ ln(1 + Ce − √ fξt ) , (30)with ξ = q λ and a ( t ) = a (0) e √ fξt ( 1 + Ce − √ fξt C ) . (31)The constant C remains arbitrary as long as the validity of the solution is confined within the region 1 + Ce − √ fγt > e ξφ = √ f a with φ (0) and ˙ φ (0) = √ f (1 − C ) / (1 + C ) . It indicates that starting with an arbitrary value of C, thesolution converges rapidly to φ ( t ) = √ f t and a ( t ) = e √ fξt . There is a wide range of initial conditions in which thesolution rapidly converges to de Sitter acceleration.To see the solution more closely, let us divide the case with C >
C < . First, note that the condition1 + Ce − √ fγt > | C | , i.e., | C | < C < . When
C >
0, it could be arbitrary.We introduce a time scale defined by | C | = e − √ fξt ∗ . Then, Eq. (30) can be written as φ ( t ) = ( ξ ln(sinh(3 √ f ξ ( t + t ∗ ))) + ˜ φ (0) , ( C < , < t ∗ < ∞ ) ξ ln(cosh(3 √ f ξ ( t + t ∗ ))) + ˜ φ (0) , ( C > , − ∞ < t ∗ < ∞ ) (32)where ˜ φ (0) = φ (0) − ln(sinh(3 √ f ξt ∗ )) / ξ for C < φ (0) = φ (0) − ln(cosh(3 √ f ξt ∗ )) / ξ for C > a ( t ) = ( ˜ a (0)(sinh(3 √ f ξ ( t + t ∗ ))) , ( C < , < t ∗ < ∞ )˜ a (0)(cosh(3 √ f ξ ( t + t ∗ ))) , ( C > , − ∞ < t ∗ < ∞ ) (33)where ˜ a (0) = a (0) / (sinh(3 √ f ξt ∗ )) for C < a (0) = a (0) / (cosh(3 √ f ξt ∗ )) for C > φ | t = − t ∗ becomes singular in (32), but our initial time is chosen to be 0, and it is outside the range of dynamics. Had wechosen our initial time to be t i , the singularity would occur at t = t i − t ∗ . We will assume that | C | ∼ t ∗ isnearly the initial time.Let us assume that the initial time t = 0 is chosen when the universe is still at the matter-dominated epoch andexamine the early-time behavior of the solutions (32) and (33). Then, for C < φ ( t ) ∼ ξ ln( t + t ∗ ) + ˜ φ , a ( t ) ∼ ( t + t ∗ ) . (34)The logarithmic time-dependent φ field [11, 12, 16, 17] also appears in the quintessence with exponential potential.The energy density ρ φ = λ ˙ φ ∼ ( t + t ∗ ) − ∼ a − scales the same as the stiff matter density and is known as a scalingsolution. In our case, the scaling behavior holds only at early times, and as time goes by, the full solution (32) willtake over. For C > φ field and the scale factor a ( t ) remains constant upto first order in time, and as time goes by, both quantities begins to grow. There is no scaling behavior in this case.Note that for both cases, Ω φ = ρ φ ρ c = 1 due to the first condition of Eq. (28). This is because in the above solution,matter contribution was neglected. It would be interesting to check whether the dominance of the energy density ofscalar fields emerges from the matter-dominated epoch when the matter contributions are included.Let us discuss some issues of dark energy with this solution. First, we have the acceleration given by¨ aa = f ξ ± e − √ fξ ( t + t ∗ ) + e − √ fξ ( t + t ∗ ) (1 ± e − √ fξ ( t + t ∗ ) ) , (35)where + in the numerator is for C > − is for C <
0. We see that for
C >
0, it is always accelerating. For
C <
0, there is a transition time where the scale factor changes from deceleration to acceleration. It is given with t ∗ ∼ t tr = 16 √ f ξ ln(5 + 2 √ ∼ . t . (36)where t is the current age of the universe [32]. This fix √ f ξ ∼ . /t . Next, equation of state with this value of √ f ξ is given by ω = − ∓ e − √ fξ ( t + t ∗ ) (1 ∓ e − √ fξ ( t + t ∗ ) ) ∼ − ∓ . . (37)For C > ω approaches − ω is singular at t = − t ∗ and again it is outsidethe range of dynamics. For C <
0, it approaches − ω = −
1, but asymptotically approaches the ω = − VI. CONCLUSION AND DISCUSSION
We presented a de Sitter nonlinear sigma model coupled to Einstein gravity in order to describe the currentacceleration of the Universe. It has some characteristic features as follows. Out of the four scalar fields, only one ofthem is time-dependent and the remaining three fields have only spatial dependence. If the time dependent scalar fieldis phantom, then the remaining fields are ordinary scalar fields, or vice versa. The formal case could also be thoughtof as the dilatonic phantom coupled with triplet of scalar fields, whereas the latter case as the dilaton coupled withtriplet of phantom scalar fields. Since kinetic energy of both positive and negative sign exists, it could be thoughtof as a quintom model [26] with dilaton interaction between the two sectors. But the quintom model only considerstime-dependent fields. A specific form of the potential is not needed to achieve the late-time exponential acceleration,but introduction of a potential could produce subdominant behavior of the scalar fields as in the quintessence. It issuspected that the potential does not modify the late-time exponential behavior, because it does not change the Eq.(13).We find that a simple ansatz provides the constant energy density and results in an accelerating universe withan exponential expansion. The balance between the pressures coming from the time-dependent field and spatial-dependent fields makes it possible to achieve the exponential acceleration. It is pointed out that the target space ofEuclidean de Sitter space with signature (+ , + , + , +) cannot produce such balance and exponential acceleration of theuniverse. The model has essentially two parameters, ξ and the cosmological constant term Λ. √ f ξM p plays the role ofthe Hubble constant and is a function of the strength of the self-coupling constant and the cosmological constant termΛ. Consider, for example, the nonperturbative case with ξ = 1 / λ and Λ = 3 f /λ . Recall f = e ξφ (0) /M p /a (0)and let us assume a (0) ∼
1. We mention a couple of cases where √ f ξM p ∼ − M p and Λ ∼ − M p can berealized. In the first case with φ (0) ∼ − M p and λ ∼ / ξφ (0) ∼ − M p . If the scale when the nonlinearsigma model sets in is of the order of Gev with φ (0) ∼ − GeV , this requires extremely weak coupling constant with λ ∼ − . In this case, we have ξφ (0) ∼ − M p . These are fine tunings which can yield the small Hubble constantand the cosmological constant.Stability analysis shows some of the solutions, depending on the values of the parameter ξ , are classically stable andattractor solutions. They require that the original cosmological constant term must be negative, still the accelerationis possible led by the scalar fields. In one case, where the cosmological constant term is uniquely fixed, there isa nonperturbative solution which asymptotically approaches the de Sitter phase of acceleration. This solution alsoexhibits a power law expansion at early times, and the energy density of the scalar fields mimics the matter energydensity. It remains to be seen whether the stability survives when the analysis is extended to spatial variations.The present analysis indicates that the acceleration phase can be dominated by the nonlinear sigma model. Weonly focused on the late time behavior except the nonperturbative case. To show whether this behavior of scalardominance can emerge from matter-dominated epoch, the analysis has to be extended including the contributionof matter density at early times which was neglected. Finally, whether the de Sitter nonlinear sigma model couldcome from particle physics as an effective low energy field theory remains to be seen. These aspects needs furtherinvestigation. Note added : After the completion of this work, we became aware of Ref. [33] where the ansatz (5) each multipliedby some constant factors to have Minkowski background also appeared in the cosmological context of the Lorentzviolating massive graviton models [34]. These models deal with flat background metric. However, in our de Sitterbackground solution, a linear perturbation of the metric in the action (1) does not result in any massive gravitonmode even though the ansatz (5) spontaneously break the diffeomorphism invariance. This can be readily seen bychecking that the mass term which is of the second order in the perturbations cancels out in the second term of theaction in (1). Moreover, the de Sitter background solution allows the modification where each of the ansatz in (5) canbe multiplied by a same constant factor only which still does not yield Lorentz violating mass term. Perhaps, it couldbe possible to generate a mass term by a suitable deformation of the target space metric in the action (1), which isbeyond the scope of this work, but nevertheless whose implications would be worthwhile to be explored in detail.
VII. ACKNOWLEDGEMENT
We thank the anonymous referee for pointing out Ref. [33] to us. We also thank S. T. Hong for early participationand Y.-Y. Keum for useful information on dark energy. The work of THL was supported by the Korea ResearchFoundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-313-C00165). The work of PO is supported by the Science Research Center Program of the Korea Science andEngineering Foundation through the Center for Quantum Spacetime(CQUeST) of Sogang University with Grant No.R11-2005-021.
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