On the Dynamics of Bianchi IX cosmological models
aa r X i v : . [ g r- q c ] J a n On the Dynamics of Bianchi IX cosmological models
Hossein Farajollahi ∗ and Arvin Ravanpak † Department of Physics, University of Guilan, Rasht, Iran (Dated: October 26, 2018)
Abstract
A cosmological description of the universe is proposed in the context of Hamiltonian formulationof a Bianchi IX cosmology minimally coupled to a massless scalar field. The classical and quantumresults are studied with special attention to the case of closed Friedmann-Robertson-Walker model.
PACS numbers: 04.20.-q; 04.20.Cv; 04.60.Ds; 98.80.QcKeywords: General Relativity; Quantum Cosmology; Time; Bianchi IX; FRW. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Any theoretical scheme of gravity must address a variety of conceptual issues includingthe problem of time and identification of dynamical observables [1], [2], [3], [4], [5], [6], [7],[8], [9], [10]. Studying cosmological models instead of general relativity helps us to overcomethe problems related to the infinite number of degrees of freedom in the theory and pay moreattention to the issues arising from the time reparametrization invariance of the theory; suchas the identification of a dynamical time and also construction of observables for the theory[11], [12], [13], [14]. In particular, Bianchi IX cosmological model is an interesting candidateto test the possible solutions of the above problems [15], [16], [17].The problem of time is the most known difficulty of the Wheeler-DeWitt (WDW) quan-tum geometrodynamics which is a theoretical basis for modern quantum cosmology [18],[19]. Time in quantum mechanics and general relativity are drastically different from eachother; in quantum mechanics time is a global and absolute parameter, in general relativitya local and dynamical variable [20]. As a result, the wave function in quantum formulationis time independent, i.e., the universe has a static picture.A common candidate for a dynamical time in classical and quantum model is a masslessscalar field coupled to gravity [15], [14], [21]. In this work we apply this to Bianchi IXcosmological model and investigate its classical and quantum implementation. Due to thepresence of the minimally coupled scalar field term in the formulation, we show that thedynamics of the metric functions can be obtained using a time variable, as a function of thescalar field.The article is organized as follows: In section two the Hamiltonian formulation of BianchiIX model coupled to a massless scalar field is studied, with particular attention to the closedFRW model. In section three the canonical quantization of the model is presented and thewave function of the universe is discussed. Again special attention is given to the quantumFRW cosmological solutions with closed curvature. Section four presents conclusion drawnfrom this work. 2 . THE CLASSICAL MODEL
The characteristic feature of the Bianchi IX model is the existence of the simply transitiveisometry group. The infinitesimal generators of this group are the three linearly independentspacelike killing vectors, ξ i , which obey[ ξ i , ξ j ] = C ijk ξ k , (1)where i , j and k runs from 1 to 3. The generators of these transformations are ξ = x ∂ − x ∂ ,ξ = x ∂ − x ∂ , (2) ξ = x ∂ − x ∂ . One may write the metric of Bianchi IX in terms of forms , ds = N dt − h ij σ i σ j , (3)where N is the lapse function, h ij is the 3-metric and the σ i are the left invariant of SU (2): σ = cosψdθ + sinψsinθdφ,σ = − sinψdθ + cosψsinθdφ, (4) σ = dψ + cosθdφ, with 0 ≤ ψ ≤ π , 0 ≤ θ ≤ π and 0 ≤ φ ≤ π .Without lose of generality, we can assume that the metric h ij is diagonal. Using theMisner variables to parameterize the metric h ij = e α ( e β ) ij , (5)where α and β are functions of time only. The matrix β ij = diag [ β + + √ β − , β + − √ β − , − β + ] , (6)with the property T rβ = 0, ensures that the 3-volume of the hypersurface depends only onthe conformal factor α . The variables β + and β − describe the anisotropy of the spacetime,3nd, in particular, if they equal to zero, the model reduces to the ordinary closed FRWmodel.The Hilbert action for the model, including the massless scalar field minimally coupledto gravity, is given by S H = Z √− g ( R − g µν φ ,µ φ ,ν ) d x, (7)where µ and ν runs from 1 to 4 and √− g = N √ h . If one assume that the scalar fieldis spatially homogeneous, the Lagrangian for the model in terms of the Misner variablesbecomes L = − e α N ( ˙ α − ˙ β − ˙ β − ) + N e α V ( β + , β − ) − e α N ˙ φ , (8)where V ( β + , β − ) = e − β + − e − β + cosh(2 √ β − ) + 2 e β + (cosh(4 √ β − ) − . (9)For the Hamiltonian formulation of the theory, let’s determine the conjugate momenta tothe dynamical variables α , β + , β − , φ , and N : P α = ∂L∂ ˙ α = − e α ˙ αN , (10) P + = ∂L∂ ˙ β + = 12 e α ˙ β + N , (11) P − = ∂L∂ ˙ β − = 12 e α ˙ β − N , (12) P φ = ∂L∂ ˙ φ = − e α ˙ φN , (13) P N = ∂L∂ ˙ N = 0 . (14)Obviously, the variable N is not a canonical variable as its canonical conjugate momenta iszero. The canonical Hamiltonian can then be written as H c = e − α N
24 ( − P α + P + P − ) − N e α V ( β + , β − )2 − N e − α P φ , (15)with the total Hamiltonian to be H t = H c + λP N . (16)Since P N ≈ H = e − α
24 [ P α − P − P − + 12 e α V ( β + , β − )] + e − α P φ ≈ . (17)4y choosing N = N e − α and introducing a new variable a = e α , ( a is scale factor) we canredefine the secondary constraint as H = 124 [ a P a − P − P − + 12 a V ( β + , β − )] + P φ ≈ , (18)where P a = ∂L∂ ˙ a . Thus H t = λP N + N H , (19)in which according to Dirac procedure, both of these constraints are first class.Now, we define a new time variable and its conjugate momenta as [11]: T = Z Σ t φP φ √ hd x, (20)Π T = P φ , (21)where the integration is over the spatial hypersurface Σ t . The Hamiltonian constraint interms of the new variables becomes H = H ∗ + Π T ≈ , (22)where H ∗ is the Hamiltonian constraint without scalar field.The equation of motion for T , dTdt = Z Σ t N √ hd x, (23)shows that the time variable equals the four-volume enclosed between the initial and finalhypersurfaces, which is necessarily positive and monotonically increasing and can play therole of a cosmological time. Even though, the time variable is not a Dirac observable, it canbe used to play the role of a cosmological time. Besides, the conjugate momentum to thisvariable is a Dirac observable.Classically, We can analyze in which sense the dynamical time, T , can be regarded asan appropriate time variable for the theory. Near the cosmological singularity ( a →
0) thepotential term V ( β + , β − ) can be neglected and so we have H = 124 [ a P a − P − P − ] + Π T ≈ . (24)5n such approximation we obtain the new equations of motion a ′ = 112 a P a , (25) P a ′ = − aP a , (26) P + ′ = P −′ = 0 , (27) β + ′ = − P + , (28) β −′ = − P − , (29)which , ( ... ) ′ , denotes derivative of ( ... ), with respect to T . The solutions to the above systemfor the scale factor a and its conjugate momenta are a = a e CT , (30) P a = P e − CT , (31)where a = a | T =0 , P = P | T =0 and C is a constant. A positive C with positive a provides anaccelerating expanded universe, with a positive constant Hubble, H = C . We have recovereda monotonic dependence of our new variable T with respect to the isotropic variable of theUniverse a and therefore T shows to be a relational time variable for the gravitationaldynamics. In this case, the universe expands exponentially according to the cosmologicaltime variable, and naturally accelerating without a beginning singularity.In the presence of the potential term, i.e., far from the singularity, the equations of motionare a ′ = 112 a P a , (32) P a ′ = − aP a − a [ e − β + − e − β + cosh(2 √ β − )+ 2 e β + (cosh(4 √ β − ) − , (33) β + ′ = − P + , (34) β −′ = − P − , (35) P + ′ = − a [ − e − β + + e − β + cosh(2 √ β − )+ e β + (cosh(4 √ β − ) − , (36) P −′ = − √ a [ − e − β + sinh(2 √ β − ) + e β + sinh(4 √ β − )] . (37)Although, it is hard to solve these set of coupled nonlinear differential equations analytically,a numerical solution of the equations shows, in particular, the dynamic of the scale factor in6 T a FIG. 1: dynamics of scale factor with respect to T terms of the time variable T (Fig.1). Again, We have recovered a monotonic dependence ofour new variable T with respect to the isotropic variable of the Universe a and the universeexpansion begins with no singularity. Isotropic Bianchi type IX :In particular, we are more interested to find the behavior of the isotropic variables a and p a when β + = 0 and β − = 0. We find the equations of motions as, a ′ = a P a , (38)and P ′ a = − aP a − a . (39)The nontrivial solution of the above coupled non linear differential equations (38) and(39), for the scale factor gives a ( T ) = 2 √ c e √ c ( T + c ) (4 c + e √ c ( T + c ) ) , (40)where c and c are positive constants of integrations. This solution shows that the universehas no singularity at all. Besides, the universe shrinks to a big crunch as T goes to infinitywhile it reaches a maximum size during its history. Note that, in the isotropic limit, theBianchi IX model contains the closed FRW model and the behavior of the scale factor a isas expected for this model. 7 . THE QUANTUM MODEL In quantization of the theory, we obtain the Wheeler-DeWitt equation b H ψ = 0 , (41)or c H ∗ ψ = − Π T ψ. (42)So, we get a Schr¨odinger like equation c H ∗ ψ = i ~ ∂∂T ψ, (43)where the new variable plays the role of a dynamical time for the theory. Explicitly, theSchr¨odinger equation can be written as:124 [ a ∂ ∂a − ∂ ∂β − ∂ ∂β − ] ψ = i ~ ∂∂T ψ, (44)where we have ignored the potential term in this regime.In fact, in quantum cosmology, the Universe is described by a single wave function ψ providing puzzling interpretations when analyzing the differences between ordinary quantummechanics and quantum cosmology. If we consider the universe wave function as: ψ ( a, β + , β − ; T ) = A ( a ) B + ( β + ) B − ( β − ) T ( T ) , (45)then, by using separation of variables method we obtain i ~ T d T dT = E, (46)1 B + d B + dβ + m = 0 , (47)1 B − d B − dβ − + n = 0 , (48) a A d A da + l = 0 , (49)where l = m + n + k , k = πE ~ and E , k , m , n and l are arbitrary constants. Thesolutions to these equations are: T ( T ) = exp( − iET ~ ) , (50) B + ( β + ) = c cos( mβ + ) + c sin( mβ + ) , (51) B − ( β − ) = c ′ cos( nβ − ) + c ′ sin( nβ − ) , (52) A ( a ) = √ a ( c ′′ cos( p (1 − l ) ln a c ′′ sin( p (1 − l ) ln a , (53)8here c , c , c ′ , c ′ , c ′′ and c ′′ are normalisation coefficients.For the Hamiltonian operator to be self adjoint the wave function must satisfy one of thefollowing boundary conditions [22], [23] ψ ( a, β + , β − , T ) | a =0 = 0 , (54) ∂ψ ( a, β + , β − , T ) ∂a | a =0 = 0 . (55)The first boundary condition is satisfied while the second one leads to infinity. Obviously,the wave function is not square integrable and in order to obtain a possible physical solutionwe may construct wave packets as [23] ψ ( a, β + , β − ; T ) = Z A ( E ) ψ E ( a, β + , β − ; T ) dE. (56)However, the wave packets are also problematic and a satisfactory solution to the WDWequation is not easy to obtain. Isotropic Bianchi type IX :In the case of isotropic universe, the associated Wheeler-DeWitt equation, i.e., theSchr¨odinger-like equation is − ∂ a Ψ − a Ψ − i a ∂ T Ψ = 0 . (57)The solution is Ψ E ( a, T ) = e − iET √ a (cid:2) c J + ν (3 a ) + c J − ν (3 a ) (cid:3) , (58)where c and c are constants of integrations, J + ν and J − ν are Bessel functions and ν = √ − E/
4. Since J − ν grows exponentially to infinity as a goes to zero, one must set c = 0,and consequently the first boundary condition, (54), is satisfied. In this isotropic case thewave function is still not square integrable and even using wave packets does not give us asatisfactory wave function for the universe.
4. CONCLUSION
In this work, we present the Hamiltonian formulation of the Bianchi type IX cosmologicalmodel minimally coupled to a scalar field. We show that the dynamics of the metric functionscan be obtained using a time variable, T ( t ), as a function of the scalar field.9e then apply the results for the classical and quantum models. We find that the classicalmodel has solutions which avoid the usual initial cosmological singularity. Particularly, in apositive curvature spacetime, the solution also shows a big crunch in future. In the quantumdescription of the model, however, a satisfactory wave function is difficult to be physicallyinterpreted. Even in the case of isotropic Bianchi type IX model, it is not easy to find anormalisable solution to the WDW equation. [1] T. P. Shestakova and C. Simeone, Grav. Cosmol. 10, 161-176 (2004).[2] F. H. Gaioli and E. T. Garcia Alvarez, Gen. Rel. Grav. 26, 1267-1275 (1994).[3] A. Carlini and J. Greensite, Phys. Rev. D52, 936-960 (1995).[4] R. M. Wald, Phys. Rev. D48, 2377-2381 (1993).[5] S. A. Gogilidze, A. M. Khvedelidze, V. V. Papoyan, Yu. G. Palii and V. N. Pervushin, Grav.Cosmol. 3, 17-23 (1997).[6] M. Henneaux and C. Teitelboim, Phys. Lett. B 222, 195 (1989).[7] W. Unruh and R. Wald, Phys. Rev. D 40, 2598 (1989).[8] K. V. Kuchaˇr, Phys. Rev. D 43, 3332 (1991).[9] J. D. Brown and K. V. Kuchaˇr, Phys. Rev. D51, 5600-5629 (1995).[10] H. Farajollahi, Gen. Rel. Grav. 37, 383-390 (2005).[11] H. Farajollahi, Int. J. Theor. Phys. 47, 1479-1489 (2008).[12] C. Rovelli, Phys. Rev. D65, 044017 (2002).[13] A. Ashtekar, R. Tate and C. Uggla, Int. J. Mod. Phys. D2, 15-50 (1993).[14] A. M. Khvedelidze and Yu. G. Palii, Class. Quant. Grav. 18, 1767-1786 (2001).[15] G. Montani, M. V. Battisti, R. Benini and G. Imponente, Int. J. Mod. Phys. A23, 2353-2503(2008).[16] V. A. Savchenko, T. P. Shestakova and G. M. Vereshkov, Int. J. Mod. Phys. A15, 3207-3220(2000).[17] D. Marolf, Class. Quant. Grav. 12, 1441-1454 (1995).[18] A. Kheyfets, W. A. Miller and R. Vaulin, Class. Quant. Grav. 23, 4333-4352 (2006).[19] J. Feinberg and Y. Peleg, Phys. Rev. D52, 1988-2000 (1995).[20] R. M. Wald, Class. Quant. Grav. 16 A177-A190 (1999).
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