WKB-type Approximation to Noncommutative Quantum Cosmology E. Mena, O. Obregon, and M. Sabido ∗ † ‡ Instituto de F´ısica de la Universidad de Guanajuato, A.P. E-143, C.P. 37150, León, Guanajuato, México (Dated: February 7, 2008) In this work, we develop and apply the WKB approximation to several examples of noncommutative quantum cosmology, obtaining the time evolution of the noncommutative universe, this is done starting from a noncommutative quantum formulation of cosmology where the noncommutativity is introduced by a deformation on the minisuperspace variables. This procedure gives a straightforward algorithm to incorporate noncommutativity to cosmology and inflation. PACSnumbers: 02.40.Gh,04.60.Kz,98.80.Qc 7 0 I. INTRODUCTION classical cosmological scenario [10, 11]. In [12] the au- 0 thors avoid the difficulties of analyzing noncommuta- 2 tivecosmologicalmodels,thatwouldarisewhenworking There has been a lot of interest in the old idea of non- n with a noncommutative theory of gravity [5]. Their pro- commutative space-time [1], and an immense amount of a posal introduces the effects of noncommutativity at the J workhasbeendoneonthesubject. Thisrenewedinterest 6 is a consequence of the developments in M-Theory and quantum level, namely quantum cosmology, by deform- ing the minisuperspace through a Moyal deformation of 1 StringTheory,whereinthedescriptionofthelowenergy the Wheeler-DeWittequation. Itis thenpossibletopro- excitations of open strings, in the presence of a Neveu- 1 Schwarz (NS) constant background B field, a noncom- ceedasin noncommutativequantummechanics[13]. On v − the other hand, noncommutativity among the fields is mutative effective low energy gauge theory action [2, 3] 7 a consequence of the usual noncommutative space-time appears in a natural way. Along the lines of noncommu- 9 [3, 4] also for gravitational fields [5]. So the mentioned tative gauge theory, noncommutative theories of gravity 0 proposal is an effective noncommutativity in quantum 1 have been constructed [5]. All of these formulations of cosmology. 0 gravity on noncommutative space-time are highly non- 7 linear and a direct calculation of cosmological models is TheaimofthispaperistoapplyaWKBtypemethod 0 incredibly difficult. One may naively expect noncommu- to noncommutative quantum cosmology, and find the / c tative effects to be present at the Planck scale so it is noncommutative classical solutions, avoiding in this way q almostimpossibleto detectthem, butdue tothe UV/IR the difficult task to solve these cosmological models in - mixing [6, 7], the effects of noncommutative might be the complicated framework of noncommutative gravity r g important at the cosmologicalscale. [5]. We know how to introduce noncommutativity at a v: A simplifiedapproachto study the veryearlyuniverse quantum level, by taking into account the changes that i isquantumcosmology(QC),whichmeansthatthe grav- the Moyal product of functions induces on the quantum X itational and matter variables have been reduced to a equation (i.e. Schröedinger equation), and from there r finite number of degrees of freedom (these models were calculatethe effects ofnoncommutativityatthe classical a extensivelystudiedby meansofHamiltonianmethods in level. Thisalsohastheadvantagethatforsomenoncom- the 1970’s, for reviews see [8, 9]); for homogenous cos- mutaivemodelsforwhichthequantumsolutionscannot mological models the metric depends only on time, this be found, the corresponding noncommutative classical permits to integrate the space dependence and obtain solutionsariseveryeasilyfromthisformulation. Thepro- a model with a finite dimensional configuration space, cedureispresentedthroughaseriesofexamples: firstthe minisuperspace, whosevariablesarethe 3-metriccompo- Kanstowski-Sachs cosmological (KS) model is presented nents. Unfortunately this construction is plagued with in detail, and the formalism developed for this model, is severaldrawbacks,there is no easy way to study the dy- thenappliedtotheFriedmann-Robertson-Walker(FRW) namical evolution of the system; also the wave functions universecoupledtoascalarfieldφandcosmologicalcon- usuallycannotbenormalized. Onewaytoextractuseful stant Λ, the noncommutative quantum and classical so- dynamical information is through a WKB type method. lutionsfordifferentcasesarepresented. Themethodhas the advantage that broadens the noncommutative cos- In the last few years there have been severalattempts mological models that can be solved. This is of particu- to study the possible effects of noncommutativity in the lar interest in conection with inflation; in [11] the effects of noncommutativity during inflation are explored, but noncommutativityisonlyincorporatedtothescalarfield neglectingthe gravitationalsector,using the methodde- ∗Electronicaddress: emena@fisica.ugto.mx †Electronicaddress: octavio@fisica.ugto.mx velopedinthispaper,noncommutativitycanbeincorpo- ‡Electronicaddress: msabido@fisica.ugto.mx ratedonbothsectors. Finallytheprocedureisappliedto 2 a toy model of string effective quantum cosmology [14]. the WKB approximation is reached in the limit This work is organized as follows. In section 2 we review several quantum cosmological models via the ∂2S1(β) ∂S1(β) 2 ∂2S2(Ω) ∂S2(Ω) 2 << , << , Wheeler-DeWitt equation (WDW) and find the corre- ∂β2 ∂β ∂Ω2 ∂Ω (cid:12) (cid:12) (cid:18) (cid:19) (cid:12) (cid:12) (cid:18) (cid:19) spondingwavefunctionformostofthesemodels,thenwe (cid:12) (cid:12) (cid:12) (cid:12) (6) (cid:12) (cid:12) (cid:12) (cid:12) obtaintheclassicalsolutionsusingaWKBtypeapproxi- a(cid:12)nd gives(cid:12)the Einstein-Hamilt(cid:12)on-Jacobi(cid:12)(EHJ) equation mation. Insection3werepeatthesametreatmenttothe noncommutative counterpartsofthe examples presented ∂S2(Ω) 2+ ∂S1(β) 2 48e 2√3Ω =0, (7) − in section 2, this is achieved through a noncommutative − ∂Ω ∂β − (cid:18) (cid:19) (cid:18) (cid:19) deformationoftheminisuperspacevariables. Finally,the solvingEq.(7)onegetsthefunctionsS ,S ,andcalculate last section is devoted to discussion and outlook. 1 2 the temporal evolution. First we fix the value of N(t)= 24e √3β 2√3Ω, by using (2) and the definition for the − − II. QUANTUM COSMOLOGY AND THE WKB momenta Πβ = dSd1β(β) and ΠΩ = dSd2Ω(Ω) we obtain the APPROXIMATION classical solutions 1 48 OurgoalistopresentaWKBtypemethodfornoncom- Ω(t) = ln cosh2 2√3P (t t ) , mutativequantumcosmology. Westartbyreviewingthe 2√3 "Pβ20 (cid:16) β0 − 0 (cid:17)# quantumcosmologicalmodelsinwhichweareinterested, β(t) = β +2P (t t ), (8) and find the classical evolution through a WKB type 0 β0 − 0 approximation. The models presented are: Kantowski- where β and P are the initial conditions. These solu- 0 β0 Sachscosmology,FRWcosmologywithcosmologicalcon- tions are the same that we get by solving the field equa- stant coupled to a scalar field, and a cosmologicalmodel tions of general relativity. in the framework of string theory. B. FRW cosmology with scalar field and Λ A. Kantowski-Sachs Cosmology The next set of examples correspond to homogeneous The first example we are interested is the Kantowski- and isotropic universes,the so called FRW universe cou- Sachs universe, this is one of the simplest anisotropic pled to a scalar field and cosmological constant. The cosmological models. The Kantowski-Sachs line element FRW metric is given by: is [12] dr2 ds2 = N2dt2+e2α(t) +r2(dϑ2+sin2ϑdϕ2) , − 1 kr2 ds2 =−N2dt2+e2√3βdr2+e−2√3(β+Ω) dϑ2+sin2ϑdϕ2 , (cid:20) − ((cid:21)9) (1) (cid:0) (cid:1) where a(t) = eα(t) is the scale factor, N(t) is the lapse fromthegeneralrelativityLagrangianwefindthecanon- function, and k is the curvature constant that takes the ical momenta, values 0,+1, 1, which correspond to a flat, closed and − open universes, respectively. The Lagrangian we are to 12 12 ΠΩ = e−√3β−2√3ΩΩ˙, Πβ = e−√3β−2√3Ωβ˙, (2) workoniscomposedbythegravitysectorandthematter −N N sector,whichfortheFRWuniverseendowedwithascalar using canonical quantization, and a particular factor or- field and cosmologicalconstant Λ is dering, we get the WDW equation, through the usual identifications ΠΩ =−i∂∂Ω and Πβ =−i∂∂β we get tot = g+ φ =e3α 6α˙2 1φ˙2 N 2Λ+6ke−2α , L L L " N − 2 N − # ∂2 ∂2 (cid:0) ((cid:1)10) 48e 2√3Ω ψ(Ω,β)=0. (3) ∂Ω2 − ∂β2 − − the corresponding canonical momenta are (cid:20) (cid:21) The solution to this equation is given by Π = ∂L =12e3αα˙ , Π = ∂L e3α φ˙ . (11) α ∂α˙ N φ ∂φ˙ − N ψ =e iν√3βK 4e √3Ω , (4) ± iν − ProceedingasbeforetheWDWequationisobtainedfrom (cid:16) (cid:17) the classical Hamiltonian. By the variation of (10) with whereν is the separationconstantandK arethe mod- iv respect to N, ∂ /∂N =0, implies the well-knownresult ified Bessel functions. L =0. We now proceedto apply the WKB type method. For H this we propose the wave function e−3αN 1 ∂2 + 1 ∂2 +e6α 2Λ+6ke−2α Ψ(α,φ)=0. −24∂α2 2∂φ2 (cid:20) (cid:21) Ψ(β,Ω) ei(S1(β)+S2(Ω)), (5) (cid:0) (cid:1) (12) ≈ 3 Now that we have the complete framework and the procedure, these classical solutions can be derived by corresponding WDW equation, we can proceed to study solving Einstein’s field equations. We can expect that different cases. this approximationincludes all the gravitationaldegrees of freedom of the particular cosmological model under Intable1 wecanseethe differentcasesthatwesolved study. This almost trivial observation is central to the [18], all of them are calculated by using the WKB type ideas we are presenting in the next section. case Quantum Solution Classical Solution k=0, Λ6=0 ψ =e±iν√23φKiν 4 Λ3e3α φ(t)=φ0−Pφ0t, and J for Λ(cid:16)

WKB-type Approximation to Noncommutative Quantum Cosmology

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