Classical and Quantum Cosmology of an Accelerating Model Universe with Compactification of Extra Dimensions
aa r X i v : . [ g r- q c ] F e b Classical and Quantum Cosmology of anAccelerating Model Universe withCompactification of Extra Dimensions
F. Darabi ∗ Department of Physics, Azarbaijan University of Tarbiat Moallem, 53714-161, Tabriz, Iran .
Abstract
We study a (4 + D )-dimensional Kaluza-Klein cosmology with a Robertson-Walkertype metric having two scale factors a and R , corresponding to D -dimensional internalspace and 4-dimensional universe, respectively. By introducing an exotic matter in theform of perfect fluid with an special equation of state, as the space-time part of the higherdimensional energy-momentum tensor, a four dimensional effective decaying cosmologicalterm appears as λ ∼ R − m with 0 ≤ m ≤
2, playing the role of an evolving dark energyin the universe. By taking m = 2, which has some interesting implications in reconcilingobservations with inflationary models and is consistent with quantum tunneling, theresulting Einstein’s field equations yield the exponential solutions for the scale factors a and R . These exponential behaviors may account for the dynamical compactification ofextra dimensions and the accelerating expansion of the 4-dimensional universe in termsof Hubble parameter, H . The acceleration of the universe may be explained by thenegative pressure of the exotic matter. It is shown that the rate of compactificationof higher dimensions as well as expansion of 4-dimensional universe depends on thedimension, D . We then obtain the corresponding Wheeler-DeWitt equation and find thegeneral exact solutions in D -dimensions. A good correspondence between the solutionsof classical Einstein’s equations and the solutions of quantum Wheeler-DeWitt equationin any dimension, D , is obtained based on Hartle’s point of view concerning the classicallimits of quantum cosmology. ∗ e-mail: [email protected] Introduction
Cosmological models with a cosmological term Λ are currently serious candidates to describethe dynamics of our four dimensional universe. The history of cosmological term dates backto Einstein, and its original role was to allow static homogeneous solutions to Einstein’s equa-tions in the presence of matter which turned out to be unnecessary when the expansion of theuniverse was discovered . However, particle physicists then realized that the non-vanishingcosmological constant can be interpreted as a measure of the energy density of the vacuumwhich turned out to be the sum of a number of apparently disjoint contributions of quantumfields. In fact, a dynamical characteristic for the vacuum energy density (cosmological term)was attributed by quantum field theorists since the developments in particle physics and in-flationary scenarios. According to modern quantum field theory, the structure of a vacuumturns out to be interrelated with some spontaneous symmetry-breaking effects through thecondensation of quantum (scalar) fields. This phenomenon gives rise to a non-vanishing vac-uum energy density of the form < T µν > = − < ρ > g µν . Therefore, the observed (or effective)cosmological term receives an extra contribution from < T µν > as follows:Λ = λ + 8 πG < ρ >, where λ is the bare cosmological constant and G is the gravitational constant. From quantumfield theory we may expect < ρ > ≈ M P l ≈ × GeV ( M P l is the Planck mass), or anotherenergy scale related to some spontaneous symmetry breaking effect such as M SUSY or M W eak .Therefore, the bare cosmological constant receives potential contributions from these massscales resulting in a large effective cosmological term. However, the experimental upper boundon the present value of the cosmological term, Λ, provided by measurements of the Hubbleconstant, H , reads numerically as | Λ | πG ≤ − g/cm ≈ − GeV , which is too far from the expectation of quantum field theory . The question of why theobserved vacuum energy is so small in comparison to the scales of particle physics is known asthe cosmological constant problem. It is generally thought to be easier to imagine an unknownmechanism which would set Λ exactly to zero than one which would suppress it by just theright amount to yield an observationally tiny cosmological constant. If Λ is a dynamicalvariable (or vacuum parameter), then it is natural to suppose that in an expanding universethe cosmological term relaxes to the present tiny value by some relaxation mechanism whichmay be provided by a time-varying vacuum with a rolling scalar field [2].There are still other possibilities to be advocated. In recent years, several attempts in thesedirections have been done, in the context of quantum cosmology [3]. One plausible explanationfor a tiny cosmological term is to suppose that Λ is dynamically evolving and not constant, i.e.,Λ ∝ R − m , where R is the scale factor of the universe and m is a parameter. So, as the universeexpands from its small size in the early universe, the initially large effective cosmological termevolves and reduces to its present small value[4].The study of Λ-decaying cosmological models has recently been the subject of particularinterest both from classical and quantum aspects. The Λ decaying models may serve as poten-2ial candidates to solve this problem by decaying the large value of the cosmological constantΛ to its present observed value.Also, there are strong (astronomical) observational motivations for considering cosmologicalmodels in which Λ is dynamically decreasing as Λ ∝ R − m . Some models assume a priori afixed value for the parameter m . The case m = 2, corresponding to the cosmic string matter,has mostly been taken based on dimensional considerations by some authors [5]. The case m ≈ m = 3 corresponding to the ordinarymatter [7]. There are also some other models in which the value of m is not fixed a priori and the numerical bounds on the value of m is estimated by observational data or obtainedby calculation of the quantum tunnelling rate [8]. Other aspects of Λ-decaying models havealso been discussed with no specific numerical bounds on m [9]. It is clear that the functionaldependence Λ ∝ R − m is phenomenological and does not result from the first principles ofparticle physics. However, for some domain for example, 0 ≤ m <
3, the decaying lawΛ ∝ R − m deserves further investigation. One important reason is that the age of the universe,in these models, is always larger than the age obtained in the standard Einstein-de Sittercosmology, or the one we get in an open universe. Therefore, if we are interested in solvingthe age problem, the decaying Λ term appears to be a good candidate. In fact, according tothe ansatz Λ ∝ R − m , one may suppose the natural value < ρ > ≈ M P l to be the value of Λ atthe Planck time when R was of the order of the Planck length. Theoretically this ansatz doesnot directly solve the cosmological constant problem, but it relates this problem to the ageproblem of why our universe is so old and have a radius R much larger than the Planck length.In other words, this ansatz reduces two above problems to one problem of “Why our universecould have escaped the death at the Planck time”, which seems to be the most natural fateof a baby-universe in quantum cosmology? One may assume that the value of Λ in the earlyuniverse might have been much bigger than its present value and large enough to drive somesymmetry breakings which might have occurred in the early universe.On the other hand, the idea that our 4-dimensional universe might have emerged froma higher dimensional space-time is now receiving much attention [10] where the compactifi-cation of higher dimensions plays a key role. However, the question of how and why thiscompactification occurs remains as an open problem. From string theory we know that thecompactification may take place provided that the higher dimensional manifold admits specialproperties, namely if the geometry of the manifold allows, for example, the existence of suitableKilling vectors. However, it is difficult to understand why such manifolds are preferred andwhether other possible mechanisms for compactification do exist. In cosmology, on the otherhand, different kinds of compactifications could be considered. For example, in an approach,called dynamical compactification , the extra dimensions evolve in time towards very smallsizes and the extra-dimensional universe reduces to an effective four-dimensional one. Thistype of compactification was considered in the context of Modern Kaluza-Klein theories [18].It is then a natural question that how an effective four dimensional universe evolve in timeand whether the resulting cosmology is similar to the standard Friedmann-Robertson-Walkerfour dimensional universe without extra dimensions.Meanwhile, the recent distance measurements of type Ia supernova suggest strongly anaccelerating universe [11]. This accelerating expansion is generally believed to be driven by an3nergy source called dark energy which provides negative pressure, such as a positive cosmo-logical constant [12], or a slowly evolving real scalar field called quintessence [13]. Moreover,the basic conclusion from all previous observations that ∼
70 percent of the energy density ofthe universe is in a dark energy sector, has been confirmed after the recent WMAP [14].To model a universe based on these considerations one may start from a fundamental theoryincluding both gravity and standard model of particle physics. In this regards, it is interestingto begin with ten or eleven-dimensional space-time of superstring/M-theory, in which case oneneeds a compactification of ten or eleven-dimensional supergravity theory where an effective4-dimensional cosmology undergoes acceleration. However, it has been known for some timethat it is difficult to derive such a cosmology and has been considered that there is a no-gotheorem that excludes such a possibility, if one takes the internal space to be time-independentand compact without boundary [15]. However, it has recently been shown that one may avoidthis no-go theorem by giving up the condition of time-independence of the internal space; anda solution of the vacuum Einstein equations with compact hyperbolic internal space has beenproposed based on this model [16]. Similar accelerating cosmologies can also be obtained forSM2 and SD2 branes , not only for hyperbolic but also for flat internal space [17].On the other hand, from cosmological point of view, it is not so difficult to find cosmolog-ical models in which the 4-dimensional universe undergoes an accelerating expansion and theinternal space contracts with time, exhibiting the dynamical compactification [18], [19], [20].In [20], for instance, it is shown that using a more general metric, as compared to Ref.[18],and introducing matter without specifying its nature, the size of compact space evolves as aninverse power of the radius of the universe. The Friedmann-Robertson-Walker equations ofthe standard four-dimensional cosmology is obtained using an effective pressure expressed interms of the components of the higher dimensional energy-momentum tensor, and the negativevalue of this pressure may explain the acceleration of our present universe.To the author’s knowledge the question of Λ-decaying cosmological model has not receivedmuch attention in higher dimensional Kaluza-Klein cosmologies. Moreover, the exotic matterhas not been considered as an alternative candidate to produce the acceleration of the universe.The purpose of the present chapter is to study a (4 + D )-dimensional Kaluza-Klein cosmology,with an extended Robertson-Walker type metric, in this context [1]. As we are concernedwith cosmological solutions, which are intrinsically time dependent, we may suppose that theinternal space is also time dependent. It is shown that by taking this higher dimensionalmetric and introducing a 4-dimensional exotic matter, a decaying cosmological term Λ ∼ R − m with 0 ≤ m ≤ m = 2 the resultingfield equations yield the exponential solutions for the scale factors of the four-dimensionaluniverse and the internal space. These solutions may account for the accelerating universeand dynamical compactification of extra dimensions, driven by the negative pressure of theexotic matter . It should be noted, however, that the solutions in principle describe typicalinflation rather than the recently observed acceleration of the universe which is known to takeplace in an ordinary matter dominated universe. Nevertheless, regarding the fact that about70 percent of the total energy density of the universe is of dark energy type with negative A similar work [21] has already been done in which the same extended FRW metric was chosen witha radiation fluid occupying all the extended space-time. They found an inflation for 3-dimensions and acontraction for the D remaining spatial dimensions. , we introduce the classical cosmology modelby taking a higher dimensional Robertson-Walker type metric and a higher dimensional matterwhose non-zero part is a four-dimensional exotic matter. In section , we obtain the Einsteinequations for the two scale factors. In section , we solve the Einstein equations and obtainthe solutions. In section , we study the corresponding quantum cosmology and derive theWheeler-DeWitt equation. In section , the exact solutions of the Wheeler-DeWitt equationis obtained. Finally, in section , we show a good correspondence between the classical andquantum cosmology. The chapter is ended with concluding remarks. To begin with, we study the metric considered in [22] in which the space-time is assumed to beof Robertson-Walker type having a (3+1)-dimensional space-time part and an internal spacewith dimension D . We adopt a real chart { t, r i , ρ a } with t , r i , and ρ a denoting the time, spacecoordinates and internal space dimensions, respectively. We, therefore, take ds = − N ( t ) dt + R ( t ) dr i dr i (1 + kr ) + a ( t ) dρ a dρ a (1 + k ′ ρ ) , (1)where N ( t ) is the lapse function, R ( t ) and a ( t ) are the scale factor of the universe and the radiusof internal space, respectively; r ≡ r i r i ( i = 1 , , , ρ ≡ ρ a ρ a ( a = 1 , ...D ), and k, k ′ = 0 , ± D -dimensional space. Weassume the internal space to be flat with compact topology S D , which means k ′ = 0. Thisassumption is motivated by the possibility of the compact spaces to be flat or hyperbolic in“ accelerating cosmologies from compactification ” scenarios, as discussed in Introduction.The form of energy-momentum tensor is dictated by Einstein’s equations and by the sym-metries of the metric (1). Therefore, we may assume T AB = ( − ρ, p, p, p, p D , p D , ..., p D ) , (2)where A and B run over both the space-time coordinates and the internal space dimensions.Now, we examine the case for which the pressure along all the extra dimensions vanishes,namely p D = 0. In so doing, we are motivated by the brane world scenarios where the matteris to be confined to the 4-dimensional universe, so that all components of T AB is set to zero butthe space-time components [23] and it means no matter escapes through the extra dimensions. There is a little difference between this metric and that of [22], in that here the lapse function is generallyconsidered as N ( t ) instead of taking N = 1.
5e assume the energy-momentum tensor T µν of space-time to be an exotic χ fluid with theequation of state p χ = ( m − ρ χ , (3)where p χ and ρ χ are the pressure and density of the fluid, respectively and the parameter m isrestricted to the range 0 ≤ m ≤ ≤ m ≤ .Using standard techniques we obtain the scalar curvature corresponding to the metric (1) R = − RaN ¨ R + 6 Ra ˙ N ˙ R − R ¨ aN + 2 R ˙ N ˙ a − aN k + aN k r − aN ˙ R − R ˙ R ˙ aNR N a , and then substitute it into the dimensionally extended Einstein-Hilbert action ( without higherdimensional cosmological term) plus a matter term indicating the above mentioned exotic fluid.This leads to the effective Lagrangian L = 12 N Ra D ˙ R + D ( D − N R a D − ˙ a + D N R a D − ˙ R ˙ a − kN Ra D + 16 N ρ χ R a D , (4)where a dot represents differentiation with respect to t . We now take a closed ( k = 1) uni-verse. Although the flat universe ( k = 0) is almost favored by observations, we will show anequivalence between ( k = 1) and ( k = 0) universes. One may obtain the continuity equationby using the contracted Bianchi identity in (4+ D ) dimensions, namely ∇ M G MN = ∇ M T MN = 0 , together with the assumption that the matter is confined to (3+1)-dimensional space-time as T ab = T µa = 0 , which gives rise to ∇ µ T µν = 0 , or ˙ ρ χ R + 3( p χ + ρ χ ) ˙ R = 0 . (5)It is easily shown that substituting the equation of state (3) into the continuity equation (5)leads to the following behavior of the energy density in a closed ( k = 1) Friedmann-Robertson-Walker universe [24] ρ χ ( R ) = ρ χ ( R ) (cid:18) R R (cid:19) m, (6)where R is the value of the scale factor at an arbitrary reference time t . Given Einstein equations, this condition on the energy-momentum tensor implies a condition on Riccitensor as R ≥ We take the Planck units, G = c = ¯ h = 1 ≡ ρ χ ( R ) , (7)which leads to L = 12 N Ra D ˙ R + D ( D − N R a D − ˙ a + D N R a D − ˙ R ˙ a − N Ra D + 16 N Λ R a D , (8)where the cosmological term is now decaying with the scale factor R asΛ( R ) = Λ( R ) (cid:18) R R (cid:19) m. (9)Note that Λ is now playing the role of an evolving dark energy [26] in 4-dimensions, becausewe did not consider explicitly a (4 + D ) dimensional cosmological term in the action, and Λappears merely due to the specific choice of the equation of state (3) for the exotic matter.The decaying Λ term may also explain the smallness of the present value of the cosmologicalconstant since as the universe evolves from its small to large sizes the large initial value of Λdecays to small values. This phenomenon may somehow alleviate the cosmological constantproblem.Of particular interest, to us, among the different values of m is m = 2 which has some inter-esting implications in reconciling observations with inflationary models [27], and is consistentwith quantum tunnelling [8]. We take m = 2 and set the initial values of R and Λ( R ) asΛ( R ) R = 3 , Λ( R ) = 3 R , (10)leading to a positive cosmological term which, according to (7), guarantees the weak energycondition ρ χ > N ( t ), in principle, is also an arbitrary function of time due to the factthat Einstein’s general relativity is a reparametrization invariant theory. We, therefore, takethe gauge N ( t ) = R ( t ) a D ( t ) . (11)Now, the Lagrangian becomes L = 12 ˙ R R + D ( D − a a + D R ˙ aRa , (12)where Eq.(10) has been used. It is seen that the parameters k and Λ are effectively removedfrom the Lagrangian and this implies that although k and Λ are not zero in this model thecorresponding 4-dimensional universe is equivalent to a flat universe with a zero cosmological7erm. In other words, we do not distinguish between our familiar 4-dimensional universe, whichseems to be flat and without any exotic fluid, and a closed universe filled with an exotic fluid.We now define the new variables X = log R , Y = log a. (13)The lagrangian (12) is written as L = 12 ˙ X + D ( D − Y + D X ˙ Y . (14)The equations of motion are obtained ¨ X + D Y = 0 , (15)¨ X + D −
13 ¨ Y = 0 . (16)Combining the equations (15) and (16) we obtain¨ X = 0 , (17)¨ Y = 0 . (18) The solutions for X and Y in Eqs. (17) and (18) are obtained X = A t + γ, (19) Y = B t + δ, (20)and the solutions for R ( t ) and a ( t ) are then as follows R ( t ) = Ae αt , (21) a ( t ) = Be βt , (22)where the constants “ A , B , γ and δ ” or “ A , B , α and β ” should be obtained, in principle,in terms of the initial conditions. It is a reasonable assumption that the size of all spatialdimensions be the same at t = 0. Moreover, it may be assumed that this size would be thePlanck size l p in accordance with quantum cosmological considerations. Therefore, we take R (0) = a (0) = l P so that A = B = l p , and R ( t ) = l p e αt , (23) a ( t ) = l p e βt . (24)8t is important to note that the constants α, β are not independent, and a relation may beobtained between them. This is done by imposing the zero energy condition H = 0 which isthe well-known result in cosmology due to the existence of arbitrary laps function N ( t ) in thetheory. The Hamiltonian constraint is obtained through the Legender transformation of theLagrangian (14) H = 12 ˙ X + D ( D − Y + D X ˙ Y = 0 , (25)which is written in terms of α and β as H = 12 α + D ( D − β + D αβ = 0 . (26)This constraint is satisfied only for α ≤ , β ≥ α ≥ , β ≤ D = 1, the case α = 0 or β = 0 gives rise to time independent scale factors, namely R = a = l P , which is not physically viable since we know, at least based on observations, thescale factor of the universe is time dependent. We, therefore, choose α > , β < D = 1, we find ( β = arbitrary α = 0 or α = − β. (27)The former is not physically viable, since it predicts no time evolution for the universe. Thelatter, however, may predict exponential expansion for R ( t ), and exponential contraction for a ( t ), both with the same exponent α > D >
1, we find α ± = Dβ − ± s −
23 (1 − D ) , (28)which gives two positive values for α indicating two possible expanding universes provided β < α ± ,for a given negative value of β , become larger for higher dimensions. Therefore, the universeexpands more rapidly in both possibilities. On the contrary, for a given positive value of α ,indicating an expanding universe, the parameter β may take two negative values β ± = 2 αD − ± s −
23 (1 − D ) − , (29)indicating two ways of compactification. Moreover, they become smaller for higher dimensions,exhibiting lower rates of compactification.To find the constants α, β we first obtain the Hubble parameter for R ( t ) H = ˙ RR = α, (30)9y which the constant α is fixed. The observed positive value of H will then justify ourprevious assumption, α >
0. We may, therefore, write the solutions (23) and (24) in terms ofthe Hubble parameter H as R ( t ) = l p e Ht , (31) a ( t ) = l p e − Ht , (32)for D = 1, and R ( t ) = l p e Ht , (33) a ( t ) ± = l p e HtD h − ± √ − (1 − D ) i − , (34)and R ± ( t ) = l p e Dβt h − ± √ − (1 − D ) i , (35) a ( t ) = l p e βt . (36)for D >
H >
0, it is seen that the solution corresponding to D = 1 may predict anaccelerating (de Sitter) universe and a contracting internal space with exactly the same rates.For D >
1, in Eqs.(33) and (34), for a given
H > R ( t ) the exponent in a ( t ) takes two negative values and becomes smaller for higher dimensions. This means thatwhile the 4-dimensional (de Sitter) universe is expanding by the rate H , the higher dimensionsmay be compactified in two possible ways with different rates of compactification as a functionof dimension, D . In Eqs.(35) and (36), on the other hand, for a given β < R ( t ) takes two positive values which become larger for higher dimensions. This also meansthat while the extra dimensions contract by the rate β , the universe may be expanded in twopossible ways with different expansion rates as a function of D .It is easy to show that the Lagrangian (14) ( or the equations of motion ) is invariant underthe simultaneous transformation R → R − , a → a − , (37)which is consistent with the time reversal t → − t . Therefore, four different phases of “ expansion-contraction ” for R ( t ) and a ( t ) are distinguished, Eqs.(33) - (36). One may prefer the “ expand-ing R ( t ) - contracting a ( t )” phase to “ expanding a ( t ) - contracting R ( t )” one, considering thepresent status of the 4D universe . For the special case D = 3, both the Lagrangian (14) and the Hamiltonian constraint (25) are invariantunder the transformation a → R , R → a. Therefore, we have a dynamical symmetry between R and a , namely a ↔ R. In this case there is no real line of demarcation between a and R to single out one of them as the real scalefactor of the universe. This is because the internal space is flat k ′ = 0 and according to (12) one may assumethe 4D universe with k, Λ = 0 to be equivalent to the one in which k = Λ = 0. Therefore, both have the sametopology S . q for the scale factor R is obtained q = − ¨ RR ˙ R = − . (38)Observational evidences not only do not rule out the negative deceleration parameter but alsoputs the limits on the present value of q as − ≤ q < k = 1) universe the cosmological term Λ decaysexponentially with time t as Λ( t ) = 3 l − p e − Ht , (39)whereas in the contraction phase ( t → − t ) it grows exponentially to large values so thatat t = 0 it becomes extremely large, of the order of M p . This huge value of Λ may beextinguished rapidly by assuming a sufficiently large Hubble parameter H , consistent with thepresent observations, to alleviate the cosmological constant problem. An appropriate quantum mechanical description of the universe is likely to be afforded byquantum cosmology which was introduced and developed by DeWitt [28]. In quantum cos-mology the universe, as a whole, is treated quantum mechanically and is described by a singlewave function, Ψ( h ij , φ ), defined on a manifold ( superapace ) of all possible three geometries andall matter field configurations. The wave function Ψ( h ij , φ ) has no explicit time dependencedue to the fact that there is no a real time parameter external to the universe. Therefore,there is no Schr¨ o dinger wave equation but the operator version of the Hamiltonian constraintof the Dirac canonical quantization procedure [29], namely vanishing of the variation of theEinstein-Hilbert action S with respect to the arbitrary lapse function NH = δSδN = 0 , which is written ˆ H Ψ( h ij , φ ) = 0 . This equation is known as the Wheeler-DeWitt (WDW) equation. The goal of quantumcosmology by solving the WDW equation is to understand the origin and evolution of theuniverse, quantum mechanically. As a differential equation, the WDW equation has an infinitenumber of solutions. To get a unique viable solution , we should also respect the question ofboundary condition in quantum cosmology which is of prime importance in obtaining therelevant solutions for the WDW equation.In principle, it is very difficult to solve the WDW equation in the superspace due to thelarge number of degrees of freedom. In practice, one has to freeze out of all but a finitenumber of degrees of freedom of the gravitational and matter fields. This procedure is knownas quantization in minisuperspace , and will be used in the following discussion.11he minisuperspace in our model is two-dimensional with gravitational variables X and Y .To obtain the Wheeler-DeWitt equation, in this minisuperspace, we start with the Lagrangian(14). The conjugate momenta corresponding to X and Y are obtained P X = ∂L∂ ˙ X = ˙ X + D Y , (40) P Y = ∂L∂ ˙ Y = D X + D ( D − Y , (41)from which we obtain ˙ X = 6 D + 2 (cid:20) P X (cid:18) − D (cid:19) + P Y (cid:21) , (42)˙ Y = 6 D ( D − " P Y − D ) D + 2 − P X D (1 − D ) D + 2 . (43)Substituting Eqs.(42), (43) into the Hamiltonian constraint (25), we obtain H = (1 − D ) P X − D P Y + 6 P X P Y = 0 . (44)Now, we may use the following quantum mechanical replacements P X → − i ∂∂X , P Y → − i ∂∂Y , by which the Wheeler-DeWitt equation is obtained " ( D − ∂ ∂X + 6 D ∂ ∂Y − ∂∂X ∂∂Y Ψ( X, Y ) = 0 , (45)where Ψ( X, Y ) is the wave function of the universe in the (
X, Y ) mini-superspace.We introduce the following change of variables x = X (1 − DD + 3 ) + DD + 3 Y , y = X − YD + 3 , (46)by which the Wheeler-DeWitt equation takes a simple form ( − ∂ ∂x + D + 2 D ∂ ∂y ) Ψ( x, y ) = 0 . (47)Now, we can separate the variables as Ψ( x, y ) = φ ( x ) ψ ( y ) to obtain the following equations ∂ φ ( x ) ∂x = γ φ ( x ) , (48) ∂ ψ ( y ) ∂y = γDD + 2 ψ ( y ) , (49)where we assume γ >
0. 12
Solutions of Wheeler-DeWitt equation
The solutions of Eqs.(48), (49) in terms of x, y are as follows φ ( x ) = e ± √ γ x , (50) ψ ( y ) = e ± q γDD +2 y , (51)leading to the four possible solutions for Ψ( x, y ) asΨ ± D ( x, y ) = A ± e ± √ γ x ± q γDD +2 y , (52)Ψ ± D ( x, y ) = B ± e ± √ γ x ∓ q γDD +2 y , (53)or alternative solutions in terms of X, Y asΨ ± D ( x, y ) = A ± e ± √ γ ( X + DYD +3 ) ± q γDD +2 ( X − YD +3 ) , (54)Ψ ± D ( x, y ) = B ± e ± √ γ ( X + DYD +3 ) ∓ q γDD +2 ( X − YD +3 ) , (55)where A ± , B ± are the normalization constants. We may also write down the solutions in termsof R and a Ψ ± D ( R, a ) = A ± R ± D +3 (cid:16) √ γ + q γDD +2 (cid:17) a ± D +3 (cid:16) √ γ D − q γDD +2 (cid:17) , (56)Ψ ± D ( R, a ) = B ± R ± D +3 (cid:16) √ γ − q γDD +2 (cid:17) a ± D +3 (cid:16) √ γ D + q γDD +2 (cid:17) . (57)It is now important to impose the good boundary conditions on the above solutions to singleout the physical ones. In so doing, we may impose the following conditionΨ D ( R → ∞ , a → ∞ ) = 0 , (58)which requires the wave function of the universe to be normalizable. This means that ourminisuperspace model has no classical solutions that expand simultaneously to infinite valuesof a and R , as Eqs.(31)-(36) show. Then, one may take the following solutionsΨ ± D ( R, a ) = C ± R − D +3 (cid:16) √ γ ± q γDD +2 (cid:17) a − D +3 (cid:16) √ γ D ∓ q γDD +2 (cid:17) , (59)where C ± are the normalization constants and the exponents of R and a are negative for anyvalue of D .One may obtain the solutions (59) in ( X, Y ) mini-superspace asΨ ± D ( x, y ) = C ± e − √ γ ( X + DYD +3 ) ∓ q γDD +2 ( X − YD +3 ) . (60) For D = 3, there is a exchange symmetry Ψ( R, a ) ↔ Ψ( a, R ) under the exchange a ↔ R . For D = 1, the exponent of “ a ” corresponding to Ψ + becomes zero so that Ψ + depends only on R withthe condition Ψ + ( R → ∞ ) → Correspondence between classical and quantum cos-mology
One of the most interesting topics in the context of quantum cosmology is the mechanismsthrough which the classical cosmology may emerge from quantum theory. When does aWheeler-DeWitt wave function predict a classical space-time? Quantum cosmology is thequantum mechanics of an isolated system (universe). It is not possible to use the Copenhageninterpretation, which needs the existence of an external observer, since here the observer ispart of the system. Indeed, any attempt in constructing a viable quantum gravity requiresunderstanding the connections between classical and quantum physics. Much work has beendone in this direction over the past decade. Actually, there is some tendency towards usingsemiclassical approximations in dividing the behaviour of the wave function into two types,oscillatory or exponential which are supposed to correspond to classically allowed or forbiddenregions. Hartle [30] has put forward a simple rule for applying quantum mechanics to a singlesystem (universe):
If the wave function is sufficiently peaked about some region in the config-uration space we predict to observe a correlation between the observables which characterizethis region . Halliwell [31] has shown that the oscillatory semiclassical WKB wave function ispeaked about a region of the minisuperspace in which the correlation between the coordinateand momentum holds good and stresses that both correlation and decoherence are necessarybefore one can say a system is classical. Using Wigner functions, Habib and Laflamme [32] havestudied the mutual compatibility of these requirements and shown that some form of coarsegraining is necessary for classical prediction from WKB wave functions. Alternatively, Gaus-sian or coherent states with sharply peaked wave functions are often used to obtain classicallimits by constructing wave packets.In the investigation of classical limits, we first take D = 1 and look for a correspondencebetween classical and quantum solutions. Using Eqs.(31) and (32) in the Planck units, thecorresponding classical locus in ( R, a ) configuration space, is Ra = 1 , (61)whereas in ( X, Y ) coordinates we have X + Y = 0 . (62)We now consider the wave functions (60) in ( X, Y ) mini-superspace for D = 1Ψ +1 ( X, Y ) = C + e − √ γ X , (63)Ψ − ( X, Y ) = C − e − √ γ X + Y . (64)The above wave functions, in their present form, are not square integrable as is required forthe wave functions to predict the classical limit. However, one may take the absolute value ofthe exponents to make the wave functions square integrableΨ +1 ( X, Y ) = C + e −| √ γ X | , (65)14 − ( X, Y ) = C − e −| √ γ X + Y | . (66)We next consider the general case D >
1. Eliminating the parameter t in Eqs. (33) and (34)the classical loci in terms of R, a are obtained a ± = R D h − ± √ − (1 − D ) i − . (67)The corresponding forms of these loci in terms of X, Y are Y + = 2 D X − s −
23 (1 − D ) − , (68) Y − = 2 D X − − s −
23 (1 − D ) − . (69)The wave functions (60) also are not square integrable, so we may replace the exponents bytheir absolute values Ψ ± D ( x, y ) = C ± e − (cid:12)(cid:12)(cid:12) √ γ ( X + DYD +3 ) ∓ q γDD +2 ( X − YD +3 ) (cid:12)(cid:12)(cid:12) , (70)to make them square integrable. Now, following Hartle’s point of view, we try to make corre-spondence between the classical loci and the wave functions.Figures 1 - 6 show respectively the 2D plots of the typical wave functions Ψ +1 - Ψ +6 in termsof ( X, Y ) for γ = 10 − ; Figures 23 - 28 show the corresponding 3D plots, respectively. On theother hand, Figures 12 - 17 show the classical loci corresponding to D = 1 −
6, respectively.It is seen that the 2D and 3D plots of the wave functions Ψ +1 - Ψ +6 are exactly peaked on theclassical loci.In the same way, Figures 7 - 11 show respectively the 2D plots of the wave functions Ψ − -Ψ − . Figures 29 - 33 show the corresponding 3D plots, respectively. Figures 18 - 22 show theclassical loci for D = 2 −
6, respectively. Again, an exact correspondence is seen between the2D and 3D plots of the wave functions Ψ − - Ψ − and the classical loci. This procedure willapply for all D . 15 oncluding remarks First, we have studied a (4+ D )-dimensional classical Kaluza-Klein cosmology with a Robertson-Walker type metric having two scale factors, R for the universe and a for the higher dimensionalspace. By introducing a typical exotic matter with the equation of state p χ = ( m − ρ χ in4-dimensions, a decaying cosmological term is obtained effectively as λ ∼ R − m . By taking m = 2, the corresponding Einstein field equations are obtained and we find exponential solu-tions for R and a in terms of the Hubble parameter H . These exponential solutions indicatethe accelerating expansion of the universe and dynamical compactification of extra dimen-sions, respectively. It turns out that the rate of compactification of extra dimensions as wellas expansion of the universe depends on the number of extra dimensions, D . The more extradimensions, the less rate of compactification and the more rate of acceleration. It is worthnoting that the model is free of initial singularity problem because both R and a are non-zeroat t = 0, resulting in a finite Ricci scalar.Although the model describes in principle a closed universe with non-vanishing cosmologicalconstant, it is equivalent to a flat universe with zero cosmological constant. Therefore, onemay assume that we are really living in a closed universe with Λ = 0 , but it effectively appearsas a flat universe with Λ = 0. Note that we have not considered ordinary matter sources inthe model except an exotic matter source which is to be considered as a source of dark energy.Therefore, it seems the solutions to describe typical inflation rather than the recently observedacceleration of the universe which is known to take place in an ordinary matter dominateduniverse. However, if the large percent of the matter sources in the universe would be of darkenergy type (as the present observations strongly recommend), then one may keep the resultshere even in the presence of other matter source, keeping in mind that the relevant contributionto the total matter source of the universe is the dark energy.A question may arise on the fact that no physics is supposed to exist below the plancklength whereas for the contracting solution, the scale factor a ( t ) goes to zero starting from l p .However, it is not a major problem because we have not considered elements of quantum gravitytheory in this model and merely studied a model based on general relativity which is supposedto be valid in any scale without limitation. The scale l p , in this paper, is not introduced withina quantum gravity model (action); it just appears as a typical initial condition, in the middle ofa classical model, based on the quantum cosmological consideration. One may choose anotherscale based on some other physical considerations.We have also studied the corresponding quantum cosmology, through the Wheeler-DeWittequation, and obtained the exact solutions. Based on Hartle’point of view on the correspon-dence between the classical and quantum solutions, we have shown by 2D and 3D plots ofthe wave functions a good correspondence between the classical and quantum cosmologicalsolutions for any D , provided that the wave functions vanish for the infinite scale factors.There is no such a correspondence if another boundary condition, other than stated, is taken.Therefore, this correspondence guaranties that the chosen boundary condition is a good one.16 eferences [1] This work is the extended version of the published paper, Class.Quant.Grav. 20, 3385(2003) [2]
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J. J. Halliwell, Phys. Rev. D [32] S. Habib, R. Laflamme, Phys. Rev. D igure captions FIG. 1. 2D plot of Ψ +1 in terms of ( X, Y ) for γ = 10 − FIG. 2. 2D plot of Ψ +2 in terms of ( X, Y ) for γ = 10 − FIG. 3. 2D plot of Ψ +3 in terms of ( X, Y ) for γ = 10 − FIG. 4. 2D plot of Ψ +4 in terms of ( X, Y ) for γ = 10 − FIG. 5. 2D plot of Ψ +5 in terms of ( X, Y ) for γ = 10 − FIG. 6. 2D plot of Ψ +6 in terms of ( X, Y ) for γ = 10 − FIG. 7. 2D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 8. 2D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 9. 2D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 10. 2D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 11. 2D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 12. Classical locus X + Y = 0 for D = 1FIG. 13. Classical locus Y − = D X [ − − q − (1 − D )] − for D = 2FIG. 14. Classical locus Y − = D X [ − − q − (1 − D )] − for D = 3FIG. 15. Classical locus Y − = D X [ − − q − (1 − D )] − for D = 4FIG. 16. Classical locus Y − = D X [ − − q − (1 − D )] − for D = 5FIG. 17. Classical locus Y − = D X [ − − q − (1 − D )] − for D = 6FIG. 18. Classical locus Y + = D X [ − q − (1 − D )] − for D = 2FIG. 19. Classical locus Y + = D X [ − q − (1 − D )] − for D = 3FIG. 20. Classical locus Y + = D X [ − q − (1 − D )] − for D = 4FIG. 21. Classical locus Y + = D X [ − q − (1 − D )] − for D = 5FIG. 22. Classical locus Y + = D X [ − q − (1 − D )] − for D = 6FIG. 23. 3D plot of Ψ +1 in terms of ( X, Y ) for γ = 10 − FIG. 24. 3D plot of Ψ +2 in terms of ( X, Y ) for γ = 10 − FIG. 25. 3D plot of Ψ +3 in terms of ( X, Y ) for γ = 10 − FIG. 26. 3D plot of Ψ +4 in terms of ( X, Y ) for γ = 10 − FIG. 27. 3D plot of Ψ +5 in terms of ( X, Y ) for γ = 10 − FIG. 28. 3D plot of Ψ +6 in terms of ( X, Y ) for γ = 10 − FIG. 29. 3D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 30. 3D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 31. 3D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 32. 3D plot of Ψ − in terms of ( X, Y ) for γ = 10 − FIG. 33. 3D plot of Ψ − in terms of ( X, Y ) for γ = 10 −20