Horizon Problem Remediation via Deformed Phase Space
aa r X i v : . [ g r- q c ] J un Horizon Problem Remediation via Deformed PhaseSpace
S.M. M. Rasouli ∗ , Mehrdad Farhoudi † and Nima Khosravi ‡ Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, IranApril 26, 2011
Abstract
We investigate the effects of a special kind of dynamical deformation between the momentaof the scalar field of the Brans–Dicke theory and the scale factor of the FRW metric. Thisspecial choice of deformation includes linearly a deformation parameter. We trace the deformationfootprints in the cosmological equations of motion when the BD coupling parameter goes to infinity.One class of the solutions gives a constant scale factor in the late time that confirms the previousresult obtained via another approach in the literature. This effect can be interpreted as a quantumgravity footprint in the coarse grained explanation. The another class of the solutions removes thebig bang singularity, and the accelerating expansion region has an infinite temporal range whichovercomes the horizon problem. After this epoch, there is a graceful exiting by which the universeenters in the radiation dominated era.PACS number: 04 . .F y ; 04 . .Kd ; 02 . .Gh ; 98 . .Qc Keywords: Deformed Phase Space; Brans–Dicke Theory; Noncommutative Phase Space; Quantum Cosmology.
General relativity ( GR ) and quantum theory, as generally believed, are two prominent paradigmsto illustrate the nature. The idea of GR was inspired by the general covariance, the principle ofequivalence and also the Mach ideas. However after the formulation of GR established by Einstein,its full satisfaction with the Mach principle has been a matter of debate. That is, although thematter content of the universe affects the geometry, but there are still vacuum solutions for GR,contrary to the strong version of Mach idea that states if there is no matter then, there will be nogeometry. This proposition has made further considerations for alternative gravitational theoriesto be more Machian. One approach to this purpose has been stated by the scalar–tensor theorieswhich among them, the Brans–Dicke ( BD ) theory is the simplest one [1]. In this scope, a scalarfield plays the role of Newtonian gravitational constant and makes the theory to be more Machian. Also in the BD theory, there exists an adjustable dimensionless parameter which, in principle, canbe fixed by observations. On the other hand, with suitable boundary conditions, GR is deduciblefrom the BD theory when making this parameter goes to infinity limit (though not always, see, e.g.,Refs. [3, 4, 5, 6]). In this view, the BD theory presents a modified version of GR (at least for tracelessenergy–momentum tensor [6]). Hence in this sense, there may also exist some other features in the BD ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] However, the other motivation for development of this variant of GR was the Dirac large number hypothesis [2]. ω goes to infinity is a more realistic one.The quantum theory is mainly established to justify the behavior of very small scales. Thus, itshould be considered for small scale behaviors of GR (which is an excellent theory for large scalestructures) as well. For example, in the standard cosmology, the universe has been commenced bya big bang in the very early universe. It is generally believed that, in the big bang, the scale ofthe universe is almost zero, thus it is predictable that the quantum behaviors to be significant inthis regime. Indeed, there have been considerable attempts to combine quantum theory and GRin order to achieve a quantum gravity theory [7, 8, 9, 10]. One of these approaches is deformationof the phase space structure [11] that introduces, if not all but at least, a part of the quantumeffects. This mechanism has been employed in the context of cosmology [12, 13], in affecting thesmall and even large scale behaviors, by removing the singularity and by the coarse grained effects,respectively. In this approach, usually a (length) parameter, which can be interpreted as the Planck(length) constant, presents the quantum regime. However, it is crucial to recover the standard resultsby taking the appropriate limits of this parameter. The another significant aspect of this kind ofmechanism is its correspondence to the other methods of quantization, which is not only interestingbut also indispensable, see, e.g., Refs. [14, 15].As the scalar–tensor gravitational theories involve more degrees of freedom, they can give morenumber of solutions than GR [16]. Hence, we purpose to investigate deformation of the phase spacestructure in this context by studying the flat FRW metric and employing noncommutativity betweenthe momenta of the BD scalar field and the scale factor. Then, we obtain the BD dynamical equations,however for simplicity and to be able to compare the outcomes with the corresponding results in GR,we solve them when the BD coupling parameter goes to infinity. Though, it is important to notethat such a procedure does not make a precise transition to GR when the trace of energy–momentumvanishes [6], but we should also emphasize that if one just starts the procedure by the GR formalism,one cannot achieve such a wide classes of solutions, as expected.In the next section, we briefly describe the BD theory in the context of Hamiltonian formalismwhich is essential for introducing deformation (noncommutativity) in the phase space. In Section 3,we derive the BD equations of motion in the presence of a special kind of deformation, then deducetheir cosmological implications when the BD coupling parameter goes to infinity. Then, we discussthe solutions for different signs of the deformation parameter and the integration constants of thesolutions while highlighting their effects. Finally, we will end up the work by conclusions in the lastsection, while two appendixes have also been furnished. In this section, we review the BD theory in the Hamiltonian formalism. However, in order to studythe cosmological behavior, we consider the spatially flat FRW metric as the background geometry,namely ds = − N ( t ) dt + a ( t ) (cid:16) dx + dy + dz (cid:17) , (1)where N ( t ) is a lapse function and a ( t ) is the scale factor.The BD Lagrangian density in the Jordan frame [1, 17, 18] in vacuum is given by L = √− g (cid:18) φR − ωφ g µν φ ,µ φ ,ν (cid:19) , (2)where the Greek indices run from zero to three. Also, the φ is the BD scalar field, R is the Ricciscalar and ω is the BD coupling constant that is supposed to be bigger than − / L = − N − a ˙ a φ − N − a ˙ a ˙ φ + ωN − a φ − ˙ φ , (3)where the dot represents derivative with respect to the time and a total time derivative term hasbeen neglected. Thus, the corresponding Hamiltonian is H = Nχ (cid:18) − ω a − φ − P a + 12 a − φP φ − a − P a P φ (cid:19) , (4)where χ ≡ ω + 3. As the momentum conjugate to N ( t ) vanishes, one has to add it as a constraintto the above Hamiltonian. Therefore, the Dirac Hamiltonian becomes H = H + λP N , (5)where λ is a Lagrange multiplier. Here, we consider the ordinary phase space structure described bythe usual ordinary Poisson brackets { a, P a } = { φ, P φ } = { N, P N } = 1 , (6)where the other brackets vanish. Therefore, the equations of motion with respect to the Hamiltonian(5) are ˙ a = { a, H} = − Nχ (cid:18) ω a − φ − P a + 12 a − P φ (cid:19) , (7)˙ P a = { P a , H} = − Nχ (cid:18) ω a − φ − P a − a − φP φ + a − P a P φ (cid:19) , (8)˙ φ = { φ, H} = Nχ (cid:18) a − φP φ − a − P a (cid:19) , (9)˙ P φ = { P φ , H} = − Nχ (cid:18) ω a − φ − P a + 12 a − P φ (cid:19) , (10)˙ N = { N, H} = λ, (11)˙ P N = { P N , H} = 1 χ (cid:18) ω a − φ − P a − a − φP φ + 12 a − P a P φ (cid:19) . (12)Let us work in the comoving gauge, that is we fix the gauge by N = 1. Also, to satisfy theconstraint P N = 0 at all times, the secondary constraint ˙ P N = 0 should also be satisfied. Hence, byEq. (12), one obtains P φ = 12 (cid:18) ± r χ (cid:19) aφ − P a . (13)Now, differentiating Eq. (7) with respect to the time, while using Eqs. (8), (9) and (13), leads to¨ a = − (cid:18) χχ ± √ χ (cid:19) a − ˙ a . (14)In addition to the trivial static solution, one can get solutions as a ( t ) = C ( t − t ini . ) q ± where C andthe initial time t ini . are integration constants, and q ± is q ± = 23 χ − (cid:20) χ − ± r χ (cid:21) , (15)when χ = 1 / ω = − / t ini . = 0, and the constant C canbe fixed by the scale of the universe at an appropriate definite time. Also, one can easily obtain˙ φφ = ∓ √ χχ ± √ χ ! ˙ aa , (16)3ith solutions φ ( t ) = C t s ± , where C is an integration constant and s ± is given by s ± = 2(1 ∓ √ χ )3 χ − , (17)when again χ = 1 /
3. Through the Hamiltonian approach, we have actually rederived the O’Hanlonand Tupper solution [23, 24], as expected. This solution has a big bang singularity when t tends tozero. Also, we should emphasize that the behavior of the scale factor and the scalar field depends onthe BD coupling parameter. In particular, when ω goes to infinity, the non–trivial solution is notthe same as the corresponding result of GR (i.e. the Minkowski space–time), for in this case one gets a ( t ) = C t / and φ = constant , (18)where the scale factor has a decelerated expanding behavior.In the case χ = 1 /
3, the solutions for upper sign are a = C ( t − t ′ ) / and φ = C ( t − t ′ ) − ,and for lower sign are a = a exp( t/t ′′ ) and φ = φ exp( − t/t ′′ ), where C , C , a , φ , t ′ and t ′′ areconstants.In the next section, we investigate a modified version of the above formalism by introducing anoncommutative model. As mentioned in the introduction, deformation of the phase space can present a sort of tracingquantum footprints in a given model [11]. Of course, a general deformation makes equations verydifficult and even unsolvable. Hence, it is customary to pick a proper choice which not only makescalculations be possible, but also gives non–trivial results. Indeed, by appealing to the simplicityprinciple (or the Occam’s razor), simplifications are usually performed in most toy models, and evenin real ones, before a consistent and complete theory is deduced. Thus in this work, we just considera dynamical deformation between the conjugate momentum sector as { P ′ a , P ′ φ } = lφ ′ ( t ) (19)and leave the other Poisson brackets among the primed parameters (corresponding to those appearedin relation (6)) unchanged. Hence, the Jacobi identity is still satisfied. In general, noncommutativitybetween the momenta is a kind of generalization of the usual noncommutativity between the spatialcoordinates [25]. In addition, noncommutativity between the momenta, in effective, has similaritywith the behavior of a charged particle in the presence of a magnetic field. In the scope of gravitationaltheories, this kind of noncommutativity can be interpreted as a gravitomagnetic field [26]. Also, onemay find a clue in the string theory, especially in the flux compactification, as mentioned in Ref. [27].The dynamical behavior of deformation as time dependence has also been employed in the literature,e.g. the κ –Minkowskian spacetime [28, 29] and the generalized uncertainty principle [30, 31], thatare considered also in the cosmological phase space [32, 33]. In addition, the main reason for thepeculiarity of the chosen term in the right hand side of (19) is the dimensionality analysis that isdescribed in more details in the Appendix A.The minimally deformed (noncommutative) version of Hamiltonian (4) is achieved by replacingthe unprimed variables with the primed ones, namely H ′ = N ′ χ (cid:18) − ω a ′− φ ′− P ′ a + 12 a ′− φ ′ P ′ φ − a ′− P ′ a P ′ φ (cid:19) . (20)However, it is more convenient to re–introduce the new variables by applying the standard transfor-mation [25, 34] P ′ φ = P φ − laφ, (21) One could easily obtain the solution from Eq. (14) for this limit, however for the sake of completeness, we havefirstly derived explicit solutions for any ω too. P ′ φ from (21) into Hamiltonian (20) gives H nc0 = H + N lχ (cid:18) l a − φ − a − φ P φ + 12 a − φP a (cid:19) , (22)where we have also substituted H ′ , as a function of the primed variables, with H nc0 , as a functionof the unprimed ones via the employed transformation. Once again, the noncommutative DiracHamiltonian is H nc = H nc0 + λP N . (23)Therefore, the equations of motion become˙ a = { a, H nc } = Nχ (cid:18) − ω a − φ − P a − a − P φ + l a − φ (cid:19) , (24)˙ P a = { P a , H nc } = − Nχ ω a − φ − P a − a − φP φ + a − P a P φ − l a − φ + 2 la − φ P φ − l a − φP a ! , (25)˙ φ = { φ, H nc } = Nχ (cid:18) a − φP φ − a − P a − la − φ (cid:19) , (26)˙ P φ = { P φ , H nc } = − Nχ (cid:18) ω a − φ − P a + 12 a − P φ + 32 l a − φ − la − φP φ + l a − P a (cid:19) , (27)˙ N = { N, H nc } = λ , (28)˙ P N = { P N , H nc } =+ 1 χ ω a − φ − P a − a − φP φ + 12 a − P a P φ − l a − φ + la − φ P φ − l a − φP a ! . (29)In the comoving gauge, i.e. N = 1, the secondary constraint ˙ P N = 0 gives P φ = a (cid:20) lφ + (cid:18) ± r χ (cid:19) φ − P a (cid:21) . (30)Employing Eqs. (24)–(27) and (30), and performing a little manipulation lead to¨ a = − (cid:18) χχ ± √ χ (cid:19) a − ˙ a ± √ χ la − φ ˙ a (31)and again ˙ φφ = ∓ √ χχ ± √ χ ! ˙ aa . (32)Solution of Eq. (32) can be in the form φ = φ a ξ , (33)where φ is a constant and ξ is ξ ≡ ∓ √ χχ ± √ χ . (34)Substituting (33) into (31) yields¨ a = − (cid:18) χχ ± √ χ (cid:19) a − ˙ a ± φ √ χ la ξ − ˙ a. (35)Note that, when the deformation parameter l tends to zero, all noncommutative equations reduce totheir corresponding ones in the previous section. 5ow, as proposed, we are interested to investigate effects when ω goes to infinity, though again itdoes not mean that it makes transition to the standard GR, as has been shown for the commutativecase in the previous section. However, as it is obvious from Eq. (35), in order to obtain such effects,it crucially depends on how the constant φ is, or can be, related to the ω . Actually for this purpose,the value of constant φ (which represents different initial conditions) proportional to √ χ can be areasonable one. A particular motivation for it, however, is the new term in Eq. (35) (in comparisonto Eq. (14)) which should not vanish in the limit χ → ∞ on the one hand. Namely, if it wouldvanish, one would not be able to see any new effects in comparison to the original undeformed theoryin the limit ω → ∞ . On the other hand, the new term also should not become infinite. However, aconsequence of this choice is that when ω tends to infinity, then φ goes to infinity as well. Though,this brings a technical problem, i.e. it makes some ambiguities in the behavior of φ –field in solution(33) when φ −→ ∞ and ξ −→
0. In this limit, φ is a time independent (constant) variable which isinfinity. To study the inconvenience caused by this divergence of φ , the renormalization argumentmay assist in the following manner.It is well–known that the procedure of renormalization occurs in the quantum field theoreticallevel. However in our toy model, this may indicate itself naively only in the deformation parameteras the only presenter of quantum regime in this work. Hence, as the first option, the deformationparameter l ≡ l bare can be re–defined in an appropriate way that makes the transition from Eq. (35)to Eq. (36) being possible. That is, it can be re–defined as ℓ renormalized ≡ φ l bare / √ χ (see belowEq. (36)) with a finite φ . Then, when χ −→ ∞ , the l bare deformation parameter goes to infinity suchthat the ℓ renormalized deformation parameter becomes a finite constant. Also, there is an alternativeapproach which is considered in the Appendix B.Therefore, by taking ω goes to infinity and choosing the minus sign in Eq. (35), one gets¨ a = − a − ˙ a − ℓa − ˙ a, (36)where φ l = √ χℓ , which fixes the new parameter ℓ with dimensionality L − . Substituting a ¨ a =( a ˙ a ˙) − a ˙ a into Eq. (36) gives ( a ˙ a ˙) = − ℓ ˙ a , (37)that yields a ˙ a = − ℓa + C , where C is an integration constant with dimensionality [ C ] = L − . Then,one easily obtains 12 a + Cℓ a + C ℓ ln (cid:12)(cid:12)(cid:12) a − Cℓ (cid:12)(cid:12)(cid:12) = ℓ ( − t + t ) , (38)where t is an integration constant too. Obviously, the above equation is invariant under the transfor-mation ( ℓ, C, t ) → ( − ℓ, − C, − t ). This symmetry makes a counterpart relevant between the solutionsand, consequently reduces the number of investigations for different cases by half. Thus, in the follow-ing categorization, we consider the two probable options (the Case I and Case II) of the logarithmicterm in Eq. (38) only for interesting cases of different signs of the ℓ and C , without probing thecounterpart solutions. Also, for the sake of completeness, we explicitly investigate the solutions when ℓ tends to zero in Case III. a − Cℓ > As mentioned, we investigate this case for different signs of the ℓ and C . ℓ & C For convenience, assume ˜ ℓ ≡ − ℓ > b ≡ − C >
0, thus Eq. (38) reads12 a + ϑa + ϑ ln ( a − ϑ ) = ˜ ℓ ( t − t ) , (39) This choice is not restrictive, for in the following we will consider different signs for the ℓ . For example, the case ℓ <
C < ℓ >
C > t → − t . ϑ ≡ C/ℓ = b/ ˜ ℓ > a > ϑ . This means that the initial value of thescale factor cannot be zero and indeed, the universe has been started with a non–vanishing size.Therefore, one may interpret that the existence of a deformation parameter, as an indicator whichusually presents the quantum corrections to models, removes the big bang singularity. Actually, thisresult is a common expectation in the quantum cosmological models.Then, by differentiating (39), for a > ϑ , we obviously get˙ a = ˜ ℓa ( a − ϑ ) > a = − ˜ ℓ a ( a − ϑ )( a − ϑ ) . (41)Thus, the sign of ¨ a depends on two different regions, ϑ < a < ϑ and a > ϑ , which we investigate inthe following. Region ϑ < a < ϑ In this region, the expansion is an accelerated one in similar to the inflationary phase. Though,it is not exactly as the standard inflationary phase, but it can solve the horizon problem as will bediscussed in the following. Usually, a successful candidate for the standard inflation should satisfy twoessential properties among the other ones, namely the 60 e–fold duration and a graceful exit from thisepoch. Our model naturally satisfies the latter requirement, for the scale factor transits to the nextregion, i.e. a > ϑ , where it decelerates. However, at first glance, it looks that it does not satisfy theformer condition as it has much less than 60 e–fold duration. Indeed, the number of e–fold definition,i.e. N = a final /a initial , for our model is N = 2 ϑ/ϑ = 2. Nevertheless, its result is comparable to thestandard inflation one by presenting a solution for the horizon problem, which we indicate it after abrief review on the successes of the standard inflation while clarifying the horizon problem.It is well–known that the most important problem of the standard cosmology, which is solved byproposition of an inflationary scenario, is the horizon problem. Of course, the inflation also solvesthe relic particle abundances (or the monopoles) and the flatness problems. However, it is generallybelieved that among these problems, the horizon problem is the most important one, for at leastthere are alternative scenarios that can resolve the other two problems in the same manner as theinflation does [35, 36]. The horizon problem arises when the universe is observed to be isotropicand homogeneous in the large scale structure. This requires that the initial conditions must be ina way which give such a universe. The problem with the standard cosmology is that although thematter fluctuations have been inhomogeneous at the initial level, but these fluctuations did not haveenough time for interactions and transforming information about their situations. Consequently,the inhomogeneous initial conditions should naturally result in inhomogeneous present large scalestructure which is in contradiction to the observations. The inflationary idea solves this problemby taking a homogeneous part of the initial condition, and inflates it to an appropriate size for thebeginning of the radiation dominated era. In our model, the horizon problem is solved in anotherway.Actually, the accelerating phase occurs during a initial −→ ϑ and a final = 2 ϑ , which is from t initial −→ −∞ to a finite final time, that is t final = ϑ (4 + ln ϑ ) / ˜ ℓ (assuming t = 0). This means thatthe accelerating phase takes infinite time, △ t = ( t final − t initial ) −→ ∞ , and during this phase, thematter fluctuations can interact with the other parts of initial conditions, exchange information abouttheir local structures, and hence, approach to an equilibrium state which is presented by a homoge-neous structure. Thus, although in our simple model, the accelerating phase cannot be interpretedas the standard inflationary era, but it can address the horizon problem of the standard cosmology. We have neglected the equality a = ϑ , for it makes t becomes −∞ . H t L
50 100 150 200 250 300 t - H t L Fig. 1 : The solid line shows the behavior of the scale factor for Case Ia with ℓ = C = − .
1. Below the dashed line(i.e. ϑ < a < ϑ = 2 C/ℓ = 2), one has an accelerating phase, and above it (i.e. a > ϑ = 2 C/ℓ = 2), a deceleratingphase. In the right figure, the dotted curve represents the ¨ a ( t ) which is negative for a > ϑ = 2. Note that, the ¨ a ( t )curve has been rescaled for a better clarification. This quasi static accelerating phase is very similar to the Hagedorn phase of string gas cosmology [37]. Region a > ϑ In this case, the expansion of universe is decelerating, and when a −→ ∞ , the first term inEq. (39) is the dominant one, hence in this limit, a ( t ) tends to t / that behaves as the radiation era.This phase occurs exactly after the above accelerating phase, and can be interpreted as the radiationdominated phase after the usual inflationary epoch in the standard cosmology. Indeed, this resultis completely in agreement with what is usually proposed for the universe in different cosmologicalmodels, the standard cosmology with or without an inflation.The behavior of the scale factor for Case Ia is plotted as a solid line in Fig. 1. ℓ & Positive C This case is very similar to the previous one, except the sign change in the acceleration. Hence,for the entire valid region of the scale factor, i.e. a >
0, the ˙ a > a < t → ∞ then a ( t ) tends to t / , exactly as the previous case, however here,there are not interesting properties. The behavior of the scale factor is depicted in Fig. 2(left). a − Cℓ < Once again, we investigate this case for different signs of the ℓ and C too. ℓ & C To describe this case, let us repeat Eq. (38) as12 a + ϑa + ϑ ln ( ϑ − a ) = ℓ ( − t + t ) , (42)where the valid domain of the scale factor is 0 ≤ a < ϑ , which the a ≥ a = 0 occurs at For it takes infinite time to double the value of the initial scale factor.
000 4000 6000 8000 10 000 t10203040a H t L
10 20 30 40 50 t0.20.40.60.81.0a H t L Fig. 2 : The solid line in the left figure shows the behavior of the scale factor for Case Ib with ℓ = − C = − .
1. As it isobvious, the scale factor asymptotically is proportional to t / (the dotted line). The solid line in the right figure showsthe behavior of the scale factor for Case IIa with ℓ = C = 0 .
1. When t → ∞ , the scale factor approaches ϑ = C/ℓ = 1. t = t − ( ϑ ln ϑ ) /ℓ . However, the upper bound of the scale factor approaches to a constant when t goes to infinity, i.e. for the late time, one gets a ( t → ∞ ) = ϑ = constant . (43)This approaching to a constant value is a direct consequence of the existence of a deformation pa-rameter, and more interestingly, it shows itself far from the big bang. As mentioned before, since thisdeformation parameter may be interpreted as a consequence of the quantum effects, then this featuremay also be viewed as a quantum gravity effect when the scale of the universe is significant. That is,this behavior can be a phenomenological property for the quantum gravity. However, these kind ofmodifications can be perceived as a semi–classical model or, as a model beyond the BD theory butstill in the classical regime. The scale factor decelerates in this choice, and its diagram is plotted inFig. 2(right). The graph shows that the behavior of the scale factor is in agreement with the resultsobtained by Ref. [32] in where a dynamical deformation between the lapse function and the scalefactor has been employed. ℓ & Negative C The case l >
C < ℓ Tends to Zero
As mentioned before, all equations reduce to their corresponding commutative ones when thedeformation parameter vanishes. Now, let us explicitly investigate it for Eq. (38). Hence, taking thelimit ℓ −→ C ℓ ln( Cℓ − a ) = C ℓ ln Cℓ − Caℓ − a − ℓa C − · · · , (44)where higher order terms in ℓ can be neglected. And obviously, the second and the third terms inthe above relation cancel the second and the first terms in Eq. (38), respectively. The first term inrelation (44) is a constant and can be absorbed by re–definition of t . Consequently, the scale factortends to t / which recovers the commutative solution (18) when ω goes to infinity, as expected.9 Conclusions
We have introduced a deformation in the phase space structure of the two existing fields of theBD theory in the spatially flat FRW metric. Also, we have traced the quantum footprints in thecosmological equations of motion in the comoving gauge. All the noncommutative equations areshown that do reduce to their corresponding counterparts when the deformation parameter tendsto zero, as expected. Then as proposed, we have investigated the effects when the BD couplingparameter goes to infinity. In this process, we have faced an integration constant that depends on theinitial conditions, however in order to be able to trace the effects, we assume it to be proportional tothe square root of the BD coupling parameter. We have also discussed our justifications for why wehave to fix it in this way and to render other side effects.Finally, different cosmological results have been deduced due to the different possible signs forthe two arbitrary parameters of the solutions, namely the deformation parameter, ℓ , and the anotherintegration constant (i.e. the C in Eq. (38)). For one class of the solutions, the result predicts aconstant value for the scale factor in the late time that is in agreement with the results obtained inRef. [32]. This feature may be interpreted as a quantum gravity footprint in the large scale.A more interesting result is achieved by the another class of the solutions. In this case, it is shownthat the existence of the deformation parameter (or equivalently, the quantum correction) removesthe big bang singularity by preventing the scale factor tends to zero. This also has an infinite temporalrange for an accelerating expansion region. However, this phase is not the standard inflationary phase,for its e–fold duration is a very small number, but it can appropriately overcome the horizon problemthat is the main one in the standard cosmology. Indeed, due to the very long time duration of thisphase, the matter fluctuations can transmit their information to the other parts of the universe, andconsequently become homogeneous. Implicitly, after this epoch, there is a graceful exiting and then,the universe enters in a radiation dominated era which is naturally in agreement with the standardcosmology, with or without an inflation. Note that, these consequences are held just by introducinga constant parameter without considering any potential in the model. It should also be mentionedthat the model just makes a (classical) background plausible to address the horizon problem similarto (classical) background of the inflationary models.However, one of the major success of the standard inflation is its prediction of (quantum) fluc-tuations’ behavior. The standard inflation anticipates a scale invariant spectral index which is ingood agreement with the observations. For our model, considering the fluctuation of dynamics isimportant, for not only to compare with the observational data but also, to test the stability of themodel. It should be checked whether inhomogeneous arbitrary initial conditions can have a significanteffect in the late time behavior or not. To overcome such general questions, one needs to performmore investigations, perhaps employing the perturbative analysis which in our model is still morecomplicated than in the standard inflation, due to the existence of the BD scalar field as well as thedeformation parameter. These are interesting investigations for further considerations, and are notin the scope of the current work.In fact the above achievement in solving the horizon problem can be viewed as a natural con-sequence of the noncommutativity approaches. That is, it is well–known that the varying speed oflight and noncommutative models are related to each other [38], where the first motivation for theformer models has been raised in order to achieve an alternative approach to the standard inflation.Indeed, the coordinate noncommutativity has been employed in the cosmological context for the samepurpose as well, see, e.g., Refs. [39, 40]. Appendix A: On Dynamical Deformation As Relation (19)
Let us first indicate the dimension of the Poisson bracket { P a , P φ } . In this work, we have employedthe units ¯ h = 1 = c , therefore, from the Plank length, l P = p ¯ hG/c , the dimensions of G and φ are [ G ] = L and [ φ ] = L − . The scale factor and the lapse function are dimensionless parameters,10he dimensions of coordinates and the BD Lagrangian are [ x µ ] = L and [ L ] = L − . Hence, one canconclude that [ P a ] = L − and [ P φ ] = L − , and consequently [ { P a , P φ } ] = L − . On the other hand,the dimension of the deformation parameter is [ l ] = L . Therefore, from dimensionality aspects ofview, the dynamical deformation (19) is a plausible choice. Besides, from simplicity point of view,with this choice, no other extra field has been introduced in the model.Of course, one may also propose other choices that still can satisfy the dimensionality of { P a , P φ } ,but the suggested relation (19) is a first order (linear) term in the deformation parameter as well.This suggestion is also a length indicator that can present and trace the quantum behaviors, and ifthe length indicator vanishes, one will recover the standard (classical) counterpart relations. On theother hand, in cosmological models a length parameter (e.g. the Planck length that is a function of¯ h ) is physically a more realistic and plausible choice. That is, among different choices that can beselected as a quantum indicator, a length scale is an appropriate one for cosmological models, thatcan compare different scales for the quantum or classical aspects as well. Appendix B: Discussion on Fixing Integration Constant φ To fix the inconvenience behavior of the φ when χ → ∞ , one may also apply the renormalizationprocedure alternatively to the matter field L matter . However in our model, there is no matter field and G (which is equal to 1 /φ in the BD theory) does not appear in the equations of motion and makesits value non–effective, but the above procedure can be applied as if the matter field is turned on.Hence, in this case, the value of Newtonian gravitational constant can be regularized by re–definitionof the matter field as well as the BD scalar field. To be more specific, in the presence of a matterfield, the BD Lagrangian is L = √− g (cid:18) φR − ωφ g µν φ ,µ φ ,ν + L barematter (cid:19) . (B.1)Now, as an overall constant has no role in the form of equations of motion, one can multiply theabove BD Lagrangian by a dimensionless parameter ( φ G bare ) − to get1 φ G bare √− g (cid:18) φR − ωφ g µν φ ,µ φ ,ν + L barematter (cid:19) = √− g " φφ G bare R − ω (cid:18) φφ G bare (cid:19) − g µν (cid:18) φφ G bare (cid:19) ,µ (cid:18) φφ G bare (cid:19) ,ν + 1 φ G bare L barematter . (B.2)The φ , that goes to infinity when ω −→ ∞ , can be absorbed in re–definition of φ and L barematter by arenormalization process such that L renormalized = √− g (cid:20) ¯ φR − ω ¯ φ g µν ¯ φ ,µ ¯ φ ,ν + L renormalizedmatter (cid:21) , (B.3)where ¯ φ = G − = φ renormalized ≡ φ/ ( φ G bare ) and L renormalizedmatter ≡ L barematter / ( φ G bare ). Thus,the difficulty of infinite value of φ can be solved, at least naively, by the renormalization procedure. Acknowledgement
We would like to thank H. Firouzjahi and M.M. Sheikh–Jabbari for fruitful discussions.
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