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, Le´on, Guanajuato, M´exico (Dated: February 7, 2008) In this work, we develop and apply the WKB approximation to several examples of noncommu- tative quantum cosmology, obtaining the time evolution of the noncommutative universe, this is done starting from a noncommutative quantum formulation of cosmology where the noncommu- tativity 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. Schr¨oedinger 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) <q0 (cid:17) α(t)= 1ln Pφ20 + 1ln sech √3P (t t ) , ν 6 4Λ 3 2 φ0 − 0 k6=0, Λ=0 ψ(1) =e±i√ν3φKiν 6e2α for k=1, (cid:18)φ(t)(cid:19)=φ0−P(cid:16)φ0(t−h t0), i(cid:17) ψ(2) =e±i√ν3φJν 6(cid:0)e2α (cid:1)for k=−1 α(t)= 41ln P12φ2k0 + 21ln sech √13Pφ0(t−t0) k=0, Λ=0 Unk(cid:0)nown(cid:1) φ(t)=φ P (t (cid:20)t ),(cid:21) (cid:16) dα(ht) = 1i(cid:17)(t t ), 6 6 0− φ0 − 0 √Pφ0−2e6α(2Λ+6ke−2α) √12 − 0 R Table1: ClassicalandquantumsolutionsfortheFRWuniversecoupledtoascalarfieldφ. ForthecaseΛ=0k=0, 6 6 the classical solution for the scale factor is given in an implicit expression. We have fixed the lapse function to N(t)=e3α. C. Stringy quantum cosmology Theclassicalsolutionsforthescalefactorandthedilaton are Ourfinalexampleisrelatedtothegracefulexitofpre- 1 P2 P bigbangcosmology[14],thismodelisbasedonthegravi- φ¯(τ) = ln β0 sech2 β0(m 2)(τ τ ) , dilaton effective action in 1+3 dimensions m−2 "V0λ2s (cid:18)2λs − − 0 (cid:19)# P β β(τ) = β + (τ τ ), (17) S = λs d4x√ ge φ(R+∂ φ∂νφ+V), (13) 0 λs − 0 − µ − 2 − Z form=0andm=4,thesolutionshavebeenobtainedin [14],andareusedinconnectionto the gracefulexitfrom in this expression λ is the fundamental string length, s pre-big bang cosmology in quantum string cosmology. φ is the dilaton field with V the possible dilaton poten- tial. Working with an isotropic background, and setting a(t)=eβ(t)/√3, after integrating by parts, we get III. NONCOMMUTATIVE QUANTUM COSMOLOGY AND THE WKB TYPE S = λs dτ φ¯2 β2+Ve 2φ¯ , (14) APPROXIMATION − 2 ′ − ′ − Z (cid:16) (cid:17) Inthissectionweconstructnoncommutativequantum we have used the time parametrization [19] dt = e φ¯dτ, cosmologyfortheexamplespresentedintheprevioussec- − thegaugeg =1,anddefinedφ¯=φ ln d3x √3β. tionandcalculatetheclassicalevolutionviaaWKBtype 00 − λ3s − approximation. To get the classical cosmological solu- From this action we calculate the canoRni(cid:16)cal m(cid:17)omenta, tions would be a very difficult task in any model of non- Πβ =λsβ′ and Πφ¯ =−λsφ¯′. From the classicalhamilto- commutative gravity [5], as a consequence of the highly nian we find the WDW equation nonlinear character of the field equations. We will fol- low the original proposal of noncommutative quantum ∂∂φ¯22 − ∂∂β22 +λ2sV(φ¯,β)e−2φ¯ Ψ(φ¯,β)=0, (15) ctoosgmeotltohgeydthesaitrewdacsladsesvicealolpseodluitnio[1n2s].. TThheisfiwrsitllnaollnocwomus- (cid:20) (cid:21) mutative example that we present is the noncommuta- tive KS followed by the noncommutative FRW universe inparticularforapotentialofthe formV(φ¯)= V emφ¯, − 0 coupled to a scalar field, and finally stringy noncommu- the quantum solution is tative quantum cosmology. First we present in quite a generalform, the constructionofnoncommutative quan- Ψ(φ¯,β)=e±−im2−2νβKiν 2mλs√V20e(m2−2)φ¯ . (16) ttuhme cclaosssmicoalloegvyoaluntdiotnh.e WKB type method to calculate (cid:20) − (cid:21) 4 Let us start with a generic form for the commutative assuming that we can write Ψ(Ω,β) = e√3νβX(Ω) the WDW equation, this is defined in the minisuperspace equation for X(Ω) is variables x,y. As mentioned in [12] a noncommutative deformation of the minisuperspace variables is assumed d2 +48e 3iνθe 2√3Ω+3ν2 X(Ω)=0, (26) − − [x,y]=iθ, (18) −dΩ2 (cid:20) (cid:21) this noncommutativity [20] can be formulated in terms then the solution of the NCWDW equation is of noncommutative minisuperspace functions with the Moyal product of functions f(x,y)⋆g(x,y)=f(x,y)eiθ2“←∂−x−∂→y−←∂−y−∂→x”g(x,y). (19) Ψ(Ω,β)=e±i√3νβKiν 4e−√3Ω±23νθ . (27) (cid:16) (cid:17) Then the noncommutative WDW equation can be writ- Usually the next step is to construct a “Gaussian” wave ten as packet and do the physics with the new wave function. Π2+Π2 V(x,y) ⋆Ψ(x,y)=0. (20) This is not necessary for our purposes, as we are inter- − x y− ested on the classical solutions, by applying the WKB Weknowfr(cid:0)omnoncommutative(cid:1)quantummechanics[13], type method outlined in the previous section. Using that the symplectic structure is modified changing the equations (5), and (6) we find the solutions for S (β) commutator algebra. It is possible to return to the orig- 1 and S (Ω) which have the form inal commutative variables and usual commutation rela- 2 tions if we introduce the following change of variables S (β)=P β, (28) θ θ 1 β0 x x+ Π and y y Π . (21) 1 → 2 y → − 2 x S2(Ω)=−√3 Pβ20 −48e−√3θPβ0e−2√3Ω The efects of the Moyalstar product are reflected in the q WDW equation, only through the potential +Pβ0arctanh Pβ20 −48e−√3θPβ0e−2√3Ω , V(x,y)⋆Ψ(x,y)=V(x+ θΠy,y θΠx), (22) √3 q Pβ0  2 − 2   takingthisintoaccountandusingtheusualsubstitutions thenthedeformationofthemomentaprovideuswiththe Π = i∂ we arrive to qµ − qµ noncommutative classical solutions ∂2 ∂2 θ ∂ θ ∂ V x i ,y+i Ψ(x,y)=0, ∂x2 − ∂y2 − − 2∂y 2∂x 1 48 θ th(cid:20)is is the noncomm(cid:18)utative WDW equat(cid:19)io(cid:21)n (NCWD(W23)) Ω(t) = 2√3ln"Pβ20 cosh22√3Pβ0(t−t0)#− 2Pβ0, and it’s solutions give the quantum description of the β(t) = β0+2Pβ0(t−t0) noncommutative Universe. We can use the NCWDW to θ P tanh2 2√3P (t t ) , (29) find the temporal evolution of our noncommutative cos- − 2 β0 β0 − 0 mology by a WKB type procedure. For this we propose h i that the noncommutative wave function has the form thesesolutionshavealreadybeenobtainedin[16]. Inthat ΨNC(β,Ω) ei(SNC1(β)+SNC2(Ω)),which in the limit paper the authors deform the symplectic structure at a ≈ ∂2S (β) ∂S (β) 2 classical level changing the Poisson brackets. NC1 NC1 << , ∂β2 ∂β (cid:12) (cid:12) (cid:18) (cid:19) (cid:12)(cid:12)∂2SNC2(Ω)(cid:12)(cid:12) ∂SNC2(Ω) 2 (cid:12) (cid:12) << , (24) B. Noncommutative FRW cosmology with scalar ∂Ω2 ∂Ω (cid:12) (cid:12) (cid:18) (cid:19) field and Λ (cid:12) (cid:12) yielding the(cid:12)noncommut(cid:12)ative Einstein-Hamilton-Jacobi (cid:12) (cid:12) equation(NCEHJ),thatgivesthesolutionstoS and NC1 We can use the NCWKB type method to FRW uni- S . After the identification Π = ∂(SNC1) and NC2 xNC ∂x verse coupled to a scalar field. Proceeding as before the ΠyNC = ∂(S∂NyC2) together with the definitions of the corresponding NCWDW equation is canonicalmomentaandEq.(21)we canfindthe time de- pendent solutions for x and y. 1 ∂2 1 ∂2 In the rest of this section we will apply this ideas to + +e6(α−iθ2∂∂φ) 2Λ+6ke−2(α−iθ2∂∂φ) Ψ=0. −24∂α2 2∂φ2 the examples that have already been presented. (cid:20) (cid:16) (3(cid:17)0)(cid:21) A. Noncommutative Kantowski-Sachs Cosmology From the NCWDW equation, we use the method devel- oped in the previous sections and calculate the classical Using the method outlined in the preceding para- evolutionbyappliyingtheNCWKBtypemethod. These graphs,appliedtoEq.(3) wefindthe NCWDW equation results are presented in the next table ∂2 ∂2 48e−2√3(Ω−iθ2∂∂β) Ψ(Ω,β)=0, (25) ∂Ω2 − ∂β2 − (cid:20) (cid:21) 5 case NC Quantum Solution NC Classical Solution k=0, Λ6=0 ψ =e±iν√23φKiν 4 Λ3e3(α−23νθ) φ(t)=φ0−Pφ0t−√3θPφ0tanh √23Pφ0(t−t0) , and J fhorqΛ<0 i α(t)= θP + 1ln Pφ20 + 1ln se(cid:16)ch √3P (t t(cid:17)) , ν 2 φ0 6 4Λ 3 2 φ0 − 0 k6=0, Λ=0 ψ(1) =e±i√ν3φKiν 6e2(α−θ2ν) for k=1 φ(t)=φ0−Pφ0(t(cid:18)−t0)(cid:19)−√3θP(cid:16)φ0tanhh P√φ30(t−t0) i,(cid:17) ψ(2) =e±iν/√3φJν h6e2(α−θ2ν) i, for k=−1 α(t)= θ2Pφ0 + 41ln P12φ2k0 + 21ln sech √(cid:16)13Pφ0(t−t0)(cid:17) k=0, Λ=0 Unhknown i φ(t)=φ P ((cid:20)t t(cid:21))+6θ (cid:16)e6α hΛ+2e 2α dt,i(cid:17) 6 6 0− φ0 − 0 − dα(t) = 1 (t t ), R rPφ0−2e6α+3θPφ0“2Λ+6ke−2α−RθPφ0”(cid:0) √12 −(cid:1)0 Table 2: Classical and quantum solutions for noncommutative FRW universe coupled to a scalar field. For these models noncommutativity is introduced in the gravitationaland matter sectors. As in the commutative scenario, for Λ=0andk=0thenoncommutativeclassicalsolutionisgiveninanimplicitform,andthereisnotaclosedanalytical qu6antum solu6 tions. As in the commutative case we have fixed the value of the lapse function N(t)=e3α. C. Stringy noncommutative quantum cosmology IV. CONCLUSIONS AND OUTLOOK As in the previous examples we introduce the non- In this paper we have presented the NCWKB type commutative relation [φ¯,β] = iθ, and from the classical method for noncommutative quantum cosmology and hamiltonian we find the NCWDW equation with this procedure, found the noncommutative classi- cal solutions for several noncommutative quantum cos- mological models. ∂∂φ¯22 − ∂∂β22 −λ2sV(φ¯,β)e(m−2)(φ¯−iθ2∂∂β) Ψ(φ¯,β)=0. Noncommutativity is a proposal that originally (cid:20) (cid:21) emergedatthe quantumlevel,moreoverspace-timenon- (31) commutativity has as a consequence that the fields do The noncommutative wave function is not commute [3, 4, 5], by this reason we incorporate noncommutativity in the minisuperspace variables in a Ψ(φ¯,β)=e±−im2−2νβKiν 2mλs√V20e(m−2)(φ¯∓m4−2θν) , smimecihlaarnimcsa.nBneyrmaesaintsisofctohnesiWdeKreBd ainppsrtoaxnidmaardtioqnuaonnttuhme (cid:20) − ((cid:21)32) correspondingNCWDW equation,one gets the noncom- using the NCWKB type method the classical solutions mutative generalized Einstein-Hamilton-Jacobi equation for the noncommutative stringy cosmology are (NCEHJ), from which the classicalevolutionof the non- commutative model is obtained. The examples we stud- ied were the Kantowski-Sachs cosmological model, the 1 P2 P FRWuniversewithcosmologicalconstantandcoupledto φ¯(τ) = ln β0 sech2 β0(m 2)(τ τ ) m−2 "V0λ2s (cid:18)2λs − − 0 (cid:19)# aInstchaelacrofimemldu,taantdiveasscterninagrioq,utahnetuclmascsiocsamlsoololugticioanlsmfooudnedl. θ from the WKB-type method are solutions to the corre- P , − 2 β0 sponding Einsteins field equations. Due to the complex- Pβ ity of the noncommutative theories of gravity [5], classi- β(τ) = β + (τ τ ) 0 0 λ − cal solutions to the noncommutative field equations are s P P almost impossible to find, but in the approach of non- + θ β0tanh β0(m 2)(τ τ0) , (33) commutative quantum cosmology and by means of the 2 2λ − − (cid:20) s (cid:21) WKB-type procedure, they can be easily constructed. Also the quantum evolution of the system is not needed the classicalevolutionforstring cosmologycanbe calcu- tofindtheclassicalbehavior,fromtable2wecanseethat lated for m = 0 and m = 4. An interesting issue con- for the case Λ = 0 and k = 0 the wave function can not 6 6 cernstheBfieldthatisturnedoffinthestringcosmology be analitacally calculated, but still the noncommutative model[14]anddoesnotcontributetotheeffectiveaction. effects can be incorporated and the classical evolution In open string theory, however noncommutativity arises is found implicitly. This procedure gives a straightfor- precisely in the low energy limit of string theory in the ward algorithm to incorporate noncommutative effects presence ofa constantB field. The θ parameterwe have to cosmological models. In this approach the effects of introduced in the minisuperspace could then be under- noncommutativity are encoded in the potential through stoodasakindofB-fieldrelatedwiththeNeveu-Schwarz the Moyal product of functions Eq. (22). We only need B-field. the NCWDW equation and the approximations (6), to 6 get the NCEHJ and from it, the noncommutative classi- Acknowledgments cal behavior can easily be constructed. As already men- tioned,in[11]theeffectsofnoncommutativitywerestud- iedinconnectionwithinflation,butthenoncommutative deformation was only done in the matter sector neglect- ingthegravitysector. Theproceduredevelopedherehas We will like to thank M. P. Ryan for enlightening dis- the advantagethatwecanimplementnoncommutativity cussions on quantum cosmology and G. Garc´ıa for com- inbothsectorsinastraightforwardwayandfindtheclas- ments about the manuscript. This work was partially sicalsolutions(i.e. inflationarymodels). These ideasare supported by CONACYT grants 47641 and 51306, and being explored and will be reported elsewhere. PROMEPgrantsUGTO-CA-3andPROMEP-PTC-085. [1] H.Snyder,Phys.Rev. 71, 38 (1947). [9] M.MacCallum,in: GeneralRelativity: AnEinsteinCen- [2] A. Connes, M. R. Douglas, and A. Schwarz, JHEP tenarySurvey,editedbyS.HawkingandW.Israel(Cam- 9802:003 (1998). bridge University Press, Cambridge, England, 1979). [3] N.Seiberg and E. Witten, JHEP 9909:032 (1999). [10] J. M. Romero and J. A. Santiago, Mod. Phys. Lett. A [4] B. Jurco, L. Moller, S. Schraml, P. Schuppand J. Wess, 20 (2005) 78. Eur. Phys.J. C 21, 383 (2001),hep-th/0104153. [11] R.BrandenbergerandP.M.Ho,Phys.Rev.D66(2002) [5] H. Garc´ıa-Compe´an, O. Obreg´on, C. Ram´ırez and 023517; Q. G. Huang and M. Li, Nucl. Phys. B 713 M. Sabido, Phys. Rev. D 68 (2003) 044015; H. Garc´ıa- (2005) 219; Q. G. Huang and M. Li, JHEP 0306 (2003) Compe´an,O.Obreg´on,C.Ram´ırezandM.Sabido,Phys. 014; H. s. Kim, G. S. Lee, H. W. Lee and Y. S. Myung, Rev. D 68 (2003) 045010; M. Ban˜ados, O. Chandia, N. Phys. Rev. D 70 (2004) 043521; H. s. Kim, G. S. Lee Grandi, F.A. Schaposnik and G.A. Silva, Phys. Rev. D and Y. S. Myung, Mod. Phys. Lett. A 20 (2005) 271; 64(2001)084012;H.NishinoandS.Rajpoot,Phys.Lett. D. J. Liu and X. Z. Li, Phys.Rev. D 70 (2004) 123504. B 532 (2002) 334; V.P. Nair, “Gravitational Fields on a [12] H.Garc´ıa-Compe´an, O.Obreg´onand C.Ram´ırez, Phys. NoncommutativeSpace”,hep-th/0112114;S.Cacciatori, Rev. Lett.88, 161301 (2002). D.Klemm,L.MartucciandD.Zanon,Phys.Lett.B536 [13] J. Gamboa, M. Loewe and J. C. Rojas., Phys. Rev. D (2002)101;S.Cacciatori,A.H.Chamseddine,D.Klemm, 64,067901;M. Chaichian, M. M. Sheikh-Jabbari, and A. L. Martucci, W.A. Sabra and D. Zanon, “Noncommuta- Tureanu., Phys. Rev. Lett. 86, 2716. tive Gravity in Two Dimensions”, hep-th/0203038; Y. [14] M. Gasperini, J. Maharana and G. Veneziano, Nucl. Abe and V.P. Nair,“Noncommutative Gravity: Fuzzy Phys. B 472 (1996) 349. SphereandOthers”,hep-th/0212270;M.A.Cardellaand [15] D. V. Ahluwalia, Phys. Lett. A 275, 31 (2000). Daniela Zanon, “Noncommutative Deformation of Four- [16] G. D. Barbosa and N. Pinto-Neto, Phys. Rev. D 70 dimensional Einstein Gravity”, hep-th/0212071; A.H. (2004) 103512. Chamseddine, “Invariant Actions for Noncommutative [17] L. O. Pimentel and C. Mora, Gen. Rel. Grav. 37, 817 Gravity”, hep-th/0202137; J.W. Moffat, Phys. Lett. B (2005). 491(2000)345;Phys.Lett.B493(2000)142;V.O.Riv- [18] The case k 6=0, Λ6=0 does not have a closed analytical elles, Phys. Lett. B 558 (2003) 19; E. Harikumar and solution to theWDW equation. V.O. Rivelles, arXiv:hep-th/0607115. [19] The prime denotes differentiation respect to τ. [6] S. Minwalla, M. Van Raamsdonk and N. Seiberg, JHEP [20] Thiscommutationrelationimpliesanduncertaintyprin- 0002, 020 (2000). ciplegivingandabsoluteminimaldistanceinminisuper- [7] M.R.DouglasandN.A.Nekrasov,Rev.Mod.Phys.73, space, in [15] the same conclusion arises by a different 977 (2001). approach. [8] M.P. Ryan, in: Hamiltonian Cosmology (Springer, Berlin, 1972).