Dynamics of phantom model with O(N) symmetry in loop quantum cosmology
aa r X i v : . [ g r- q c ] J un Dynamics of phantom model with O(N) symmetry inloop quantum cosmology
Zu-Yao Sun a ,Chun-Xiao Yue b , You-Gen Shen c , Chang-Bo Sun da College of Arts and Sciences, Shanghai Maritime University, Shanghai 200135, China. b Shanghai Jianqiao College, Shanghai 201319, China. c Key Laboratory for Research in Galaxies and Cosmology, Shanghai AstronomicalObservatory, Chinese Academy of Sciences, 80 Nandan RD, Shanghai 200030, China d College of science, Shaanxi university of science and tecnology, Xi’an, Shanxi 710021,China
Abstract
Many astrophysical data show that the expansion of our universe is accelerating.In this paper, we study the model of phantom with O(N) symmetry in backgroundof loop quantum cosmology(LQC). We investigate the phase-space stability of thecorresponding autonomous system and find no stable node but only 2 saddle pointsin the field of real numbers. The dynamics is similar to the single-field phantommodel in LQC[1]. The effect of O( N ) symmetry just influence the detail of theuniverse’s evolution. This is a sharp contrast with the result in general relativity, inwhich the dynamics of scalar fields models with O( N ) symmetry are quite differentfrom the single-field models[2],[3],[4]. e-mail: [email protected] e-mail: [email protected] Introduction
Recently, many astrophysical data, such as WMAP[5], type Ia supernovae[6] and largescale galaxy surveys[7], show that our universe is almost flat, and currently undergoinga period of accelerating expansion which is driven by a yet unknown dark energy. About70% of the energy density in our universe is dark energy nowadays. There are manycandidates of dark energy such as the cosmological constant[8], quintessence[9, 10, 11,12, 13, 2], phantom[14, 15, 16, 3], quintom[17, 4] etc. The equation-of-state parameter w = p/ρ plays an important role in these models. The parameter of quintessence andphantom lies in the range − / > w > − w < − w could cross -1.Li et al. generalized the quintessence and phantom models to fields possess O( N )symmetry[2, 3]. Setare et al. extended the generalization to the quintom model[4]. Inthe above papers they showed that the behaviors of the dynamic system with O( N )symmetry are quite different from the single-field models.Many works on dark energy have been done in the framework of Einstein’s classicalgeneral relativity(GR). However, most physicists believe that the gravity should be quan-tized in a ultimate theory. Thus, we should consider the quantum effects in the evolutionof our universe. Loop quantum gravity(LQG) is a outstanding candidate of quantumgravity theory. LQG is a background independent and non-perturbative theory. Atthe quantum level, the classical spacetime continuum is replaced by a discrete quantumgeometry and the operators corresponding to geometrical quantities have discrete eigen-values. Many topics on cosmology are discussed within the framework of LQG in variousliterature in which it is known as Loop Quantum Cosmology (LQC)[1, 18, 19, 20, 21]. Re-cently, Samart et al. [1] have studied the phantom dynamics in LQC. The work of quintomand hessence model in loop quantum cosmology have been done by Wei et al. [19]. Theyfound the behaviors is different from the ones in standard FRW, such as the avoidanceof future singularities.In this paper, we investigate the evolution of universe dominated by phantom withO( N ) symmetry. The phase-space stability of the corresponding autonomous system hasbeen discussed. We find there’s no stable attractor but only 2 saddle points, this resultis similar to the single-field phantom in LQC. In order to research the evolution of ourmodel, we numerically study the system in exponential potential, and draw 2 graph todisplay the evolution of Hubble parameter H and energy density ρ to cosmic time t .The figures reveal that in LQC even for a universe dominated by phantom, the Hubbleparameter H may turn negative and oscillate. Thus, the universe will enter a oscillatoryregime. The relation between ρ and H is also shown in a graph.1 Loop quantum cosmology
Loop quantum cosmology(for review, see Ref.[22, 23]) can describe the homogeneousand isotropic spacetimes by effective modified Friedmann equation for flat universe. Thequantum effects could be reflected by adding a correction term ρ/ρ c into the eqution.In Einstein’s general relativity, the standard Friedmann equation which describes a flatuniverse is: H = κ ρ (2.1)Instead of Eq. (2.1), the modified Friedmann equation for a flat universe in LQC is givenby [24, 18] H = κ ρ (1 − ρρ c ) (2.2)As in GR, H = ˙ a/a is Hubble parameter, ρ is the total energy density, and a dot denotesthe derivative with respect to cosmic time t. ρ c is critical loop quantum density: ρ c = √ π γ G ~ , (2.3)Where γ is the dimensionless Barbero-Immirzi parameter and is suggested that γ ≃ . ρ + 3 H ( ρ + p ) = 0 (2.4)We obtain the effective modified Raychaudhuri equation˙ H = − κ ρ + p ) (1 − ρρ c ) (2.5)where p is the total pressure. In LQC the accelerating condition is¨ aa = ˙ H + H > O(N) Phantom
We consider the flat Robertson-Walker metric ds = dt − a ( t ) (cid:0) dx + dy + dz (cid:1) (3.7)The Lagrangian density for the Phantom with O( N ) symmetry is given by[3] L Φ = − g µν ( ∂ µ Φ a )( ∂ ν Φ a ) − V ( | Φ a | ) (3.8)where Φ a is the component of the scalar field, a = 1 , , · · · , N . In order to impose theO( N ) symmetries, following[3], we write it in the formΦ = R ( t ) cos ϕ ( t )Φ = R ( t ) sin ϕ ( t ) cos ϕ ( t )Φ = R ( t ) sin ϕ ( t ) sin ϕ ( t ) cos ϕ ( t ) (3.9) · · · · · · Φ N − = R ( t ) sin ϕ ( t ) · · · sin ϕ N − ( t ) cos ϕ N − ( t )Φ N = R ( t ) sin ϕ ( t ) · · · sin ϕ N − ( t ) sin ϕ N − ( t )Thus, we have | Φ a | = R and assume that the potential V( | Φ a | ) depends only on R .The radial equation of scalar fields is¨ R + 3 H ˙ R − Ω a R − ∂V ( R ) ∂R = 0 (3.10)The “angular components” contribute a effective term Ω a R to the system’s dynamics.The energy density ρ and the pressure p of the O( N ) phantom are given as ρ = −
12 ( ˙ R + Ω a R ) + V ( R ) (3.11)and p = −
12 ( ˙ R + Ω a R ) − V ( R ) (3.12)Hence, the equation-of-state for the O( N ) phantom is w = − ( ˙ R + Ω a R ) − V ( R ) − ( ˙ R + Ω a R ) + V ( R ) (3.13)Different from GR, the accelerating condition for universe in LQC is not w < − / EVOLUTION AND STABILITY OF THE MODEL
In this section we discuss the dynamics of universe dominated by phantom with O( N )symmetry in LQC. In order to get a possible evolution of the model, we choose exponentialpotential V = V e − λκR , (4.14)The evolution of universe is governed by Eq.(2.2),(2.5),(3.10). To investigate the stabilityof the model, we introduce the following dimensionless variables x = κ ˙ R √ H , y = κ p V ( R ) √ H , z = κ √ H Ω a R , (4.15) m = ρρ c , ξ = 1 κR , N = ln a (4.16)Using these variables, Eq.(2.2),(2.5),(3.10) can be written as the following autonomoussystem: dxdN = − x + √ ξz − r λy − (3 x + 3 xz )(1 − m ) dydN = − r xyλ − (3 x y + 3 yz )(1 − m ) dzdN = − z − √ xzξ − (3 x z + 3 z )(1 − m ) (4.17) dmdN = − m (1 + − x − y − z − x + y − z ) dξdN = −√ xξ In the following, it is straightforward to analyze the critical points as well as theirstability. We can get critical points by setting the right hand of the equation(4.17) tozero. As presented in table 1, in the field of real numbers there are only 2 real criticalpoints: A and B . We expand the variables around the critical point A and B in the form x = x c + δx, y = y c + δy, z = z c + δz, m = m c + δm, ξ = ξ c + δξ , where δx, δy, δz, δm, δξ are perturbations of the variables near the critical points and we consider them forming acolumn vector denoted as U. Substituting the above expression into (4.17) and neglectinghigher order terms in the perturbations, we can obtain the equations for the perturbationsup to first order as: U ′ = M · U , (4.18)where the prime denotes differentiation with respect to N. M is a 5 × x c y c z c m c ξ c A − λ √ − √ λ √ − λ √ √ λ √ w AccelerationA − λ , ( − − λ ) , ( − − λ ) , λ , 0 saddle point − − λ YesB − λ , ( − − λ ) , ( − − λ ) , λ , 0 saddle point − − λ YesTable 2: The eigenvalues of the matrix M and the properties of critical points in theautonomous system (4.17).( x c , y c , z c , m c , ξ c ) is given by M = ∂x ′ ∂x ∂x ′ ∂y ∂x ′ ∂z ∂x ′ ∂m ∂x ′ ∂ξ∂y ′ ∂x ∂y ′ ∂y ∂y ′ ∂z ∂y ′ ∂m ∂y ′ ∂ξ∂z ′ ∂x ∂z ′ ∂y ∂z ′ ∂z ∂z ′ ∂m ∂z ′ ∂ξ∂m ′ ∂x ∂m ′ ∂y ∂m ′ ∂z ∂m ′ ∂m ∂m ′ ∂ξ∂ξ ′ ∂x ∂ξ ′ ∂y ∂ξ ′ ∂z ∂ξ ′ ∂m ∂ξ ′ ∂ξ ( x = x c ,y = y c ,z = z c ,m = m c ,ξ = ξ c ) . (4.19)The eigenvalue of (4.19) determine the type and stability of the critical points. Wepresent them in table 2. As shown, both critical A and B are saddle points, it followsimmediately that the phase trajectory is very sensitive to initial conditions given to thesystem. This result is similar to the single-field phantom in LQC[1]. It is interesting tocontrast the influence of O( N ) symmetry in GR and LQC. Several works[2][3][4] showsthat the dynamics of quintessence, phantom and quintom with O( N ) symmetry in GRis quite different from the single-field model.In the following, we study the possible evolution of our model numerically. In orderto do this, we set the value of some parameters: λ = − .
1, Ω = 1, κ = √ π , ρ c =1 . V = 1 .
2, and draw 3 graphs: Fig.1, Fig.2, Fig.3. Fig.1 shows the evolution ofHubble parameter H with respect to cosmic time t . From Fig.1, we found H may takemaximum value while ρ = ρ c /
2, and then decline even to negative interval. After H takingminimum value, it would bounce and then oscillate. Thus, the accelerating expansion ofour universe would end in finite time and undergoes contraction from halting( H = 0).Then the universe enters an oscillating stage. Fig.2 shows the evolution of cosmic total5nergy density ρ , it oscillate while t goes by. In fact, from Fig.3 we can get some insightinto the evolution of the model. Fig.3 reveals the relation between H and ρ . As time goesby, the trajectory makes a closed region, and the oscillation of H and ρ are restricted ina certain interval. Figure 1: The evolution of H with respect to t for λ = − .
1, Ω = 1, κ = √ π , ρ c = 1 . V = 1 . Ρ Figure 2: The evolution of cosmic total energy density ρ with respect to t for λ = − . κ = √ π , ρ c = 1 . V = 1 . In this paper, we investigate the dynamics of phantom model with O( N ) symmetryin background of loop quantum cosmology. We discuss its stability by analyzing theautonomous system Eq.(4.17). It shows that there is no attractor in the system, but6 .2 0.4 0.6 0.8 1 1.2 1.4 Ρ -1.5-1-0.50.511.5H Figure 3: The relation between H and ρ while t goes from 0 to 14. Set values are λ = − .
1, Ω = 1, κ = √ π , ρ c = 1 . V = 1 . N ) symmetry just influence the detail of the universe’s evolution. It is a sharpcontrast with the result in GR, in which the dynamics of scalar fields models with O( N )symmetry are quite different from the single-field models. Finally, to show some insightinto our model, we draw 3 graphs by Numerical analysis. The possible oscillation of H and ρ are revealed in Fig.1. and Fig.2. Fig.3 shows that as time goes by, H and ρ arerestricted in a certain interval, and the trajectory makes a closed region. Acknowledgements
The work has been supported by the National Natural Science Foundation of China(Grant No. 10973014) and special fund for academic development of young teachers inshanghai colleges and universities. (Grant No. shs-07025.).