Loop quantum gravity and cosmological constant
aa r X i v : . [ g r- q c ] J a n Loop quantum gravity and cosmological constant
Xiangdong Zhang, ∗ Gaoping Long, † and Yongge Ma ‡ Department of Physics, South China University of Technology, Guangzhou 510641, China Department of Physics, Beijing Normal University, Beijing 100875, China
An one-parameter regularization freedom of the Hamiltonian constraint for loop quantum gravityis analyzed. The corresponding spatially flat, homogenous and isotropic model includes the twowell-known models of loop quantum cosmology as special cases. The quantum bounce nature istenable in the generalized cases. For positive value of the regularization parameter, the effectiveHamiltonian leads to an asymptotic de-Sitter branch of the Universe connecting to the standardFriedmann branch by the quantum bounce. Remarkably, by suitably choosing the value of theregularization parameter, the observational cosmological constant can emerge at large volume limitfrom the effect of quantum gravity, and the effective Newtonian constant satisfies the experimentalrestrictions in the meantime.
The origin of current cosmic acceleration is one of thebiggest challenges to modern physics, which is usuallycalled as the dark energy issue. Many possible mecha-nisms have been proposed to account for this issue, suchas the phenomenological models[1], modified gravity [2–5], higher dimensions [6] and so on. Among them, thecosmological constant is generally believed as the mostsimplest explanation [5, 7]. However, the nature of thecosmological constant is still mysterious. Whether it apurely classical effect or it has a quantum origin is a cru-cial open issue. It is well known that the awkward cos-mological constant problem would appear if one consid-ered quantum matter fields on a classical spacetime back-ground [5, 7]. A challenging question would be whether arealistic cosmological constant could emerge from certainquantum gravity theory.How to unify general relativity(GR) with quantummechanics remains as the biggest theoretical challengeto fundamental physics. Among various approaches toquantum gravity, loop quantum gravity (LQG) is no-table with its background independence [8–11]. The non-perturbative quantization procedure of LQG has beenapplied not only to GR, but also to the metric f ( R )theories[12, 13], scalar-tensor theories[14, 15], and higherdimensional gravity [16]. The idea and technique ofLQG have been successfully carried out in the symmetry-reduced models of loop quantum cosmology (LQC). Werefer to [17–20] for reviews on LQC.A remarkable result of LQC is that the classical bigbang singularity of the Universe can be avoided bya quantum bounce [17–22]. Moreover, LQC opens apromising avenue to relate quantum gravity effects to cos-mological observations of the very early Universe [23, 24].As in any quantization procedure of a classical the-ory, different regularization schemes exist also in LQCas well as in LQG [9, 10, 25, 26]. In particular, forthe LQC model of flat Friedmann-Lemaitre-Robertson- ∗ [email protected] † ‡ Corresponding author. [email protected]
Walker (FLRW) universe, alternative Hamiltonian con-straint operators were proposed [21, 27]. Note that, toinherit more features from LQG, the so-called Euclideanterm and Lorentzian term of the Hamiltonian constraintwere treated independently in the LQC model in [21].It was recently shown in [28, 29] that one of the quan-tum Hamiltonians proposed in [21] can lead to a newevolution scenario where the prebounce geometry couldbe described at the effective level by a de Sitter space-time. This raises the possibility to obtain a positivecosmological constant from a model of LQC. However,the cosmological constant obtained in [28] is very large(Λ ≈ . ℓ − P l ∼ m − ) and thus fails in fitting thecurrent observations which requires a very small cosmo-logical constant (Λ ob ∼ . × − m − ). Fortu-nately, this is not the end story. In this Letter, we willshow that there does exist certain regularization of theHamiltonian in LQC such that a small enough positivecosmological constant can emerge at the effective level.Moreover, the regularization choice inherits that of fullLQG. It is reasonable to infer that the effective Hamil-tonian could also be obtained by suitable semiclassicalanalysis of certain Hamiltonian of LQG.LQG is based on the connection-dynamical formu-lation of GR defined on a spacetime manifold M = R × Σ, where Σ denotes a three-dimensional spatialmanifold. The classical phase space consists of theAshtekar-Barbero variables ( A ia ( x ) , E ai ( x )), where A ia ( x )is a SU (2) connection and E ia ( x ) is a densitized triad[9–11]. The non-vanishing Poisson bracket is given by { A ia ( x ) , E bj ( y ) } = 8 πGγδ ba δ ij δ ( x, y ) (1)where G is the gravitational constant and γ is theBarbero-Immirzi parameter. The classical dynamics ofGR is thus obtained by imposing the Gaussian, dif-feomorphism and Hamiltonian constraints on the phasespace, where the latter represents the reparametrizationfreedom of time variable. In LQG, the notable Hamilto-nian constraint operator proposed in [30] and an alterna-tive one proposed in [25] are based on the regularizationschemes of the following expression of the Hamiltonianconstraint H g = 116 πG Z Σ d xN h F jab − ( γ + 1) ε jmn K ma K nb i ε jkl E ak E bl √ q , (2) where N is the lapse function, q denotes the determinantof the spatial metric, F iab is the curvature of connection A ia , and K ia represents the extrinsic curvature of Σ. Theso-called Euclidean term H E and Lorentzian term H L inEq. (2) are denoted respectively as H E = 116 πG Z Σ d xN F jab ε jkl E ak E bl √ q , (3)and H L = 116 πG Z Σ d xN ( ε jmn K ma K nb ) ε jkl E ak E bl √ q . (4)There is another alternative Hamiltonian constraint op-erator proposed in [26] for LQG, which is based on theregularization scheme of the following expression of theHamiltonian constraint, H g = − πGγ Z Σ d xN (cid:20) F jab ε jkl E ak E bl √ q + ( γ + 1) √ qR (cid:21) (5) where R is the 3-dimensional spatial curvature of Σ. It iseasy to see that expressions (2) and (5) are equivalent toeach other by using the classical identity (up to Gaussianconstraint)[9] H E = γ H L − Z Σ √ qR. (6)Here, we point out that there is an one-parameter free-dom to express the classical Hamiltonian constraint inthe connection formalism. The general expression can bewritten as H g = λH E − (1 + λγ ) H L + ( − λ ) Z Σ √ qR, (7)where λ is an arbitrary real number to represent the free-dom of choices. Clearly, the expression (2) corresponds tothe choice of λ = 1, while the expression (5) correspondsto the case of λ = − γ .Now we consider the spatially flat FLRW model. Onehas to introduce an “elemental cell” V on the spatial man-ifold R and restrict all integrals to this cell. Then onechooses a fiducial Euclidean metric o q ab on R , as well asthe orthonormal triad and co-triad ( o e ai ; o ω ia ) adapted to V such that o q ab = o ω iao ω ib . Via fixing the degrees of free-dom of local gauge and diffeomorphism transformations,one can obtain the reduced connection and densitizedtriad as [17] A ia = ˜ cV − o ω ia , E bj = pV − p det( q ) o e bj , where V o is the volume of V measured by o q ab , ˜ c, p areonly functions of the cosmological time t . To identify a dynamical matter field as an internal clock, we employ amassless scalar field φ with Hamiltonian H φ = p φ | p | , (8)where p φ is the momentum of φ . Hence the phase space ofthe cosmological model consists of conjugate pairs (˜ c, p )and ( φ, p φ ), with the following nontrivial Poisson brack-ets, { ˜ c, p } = 8 πG γ, { φ, p φ } = 1 . (9)Note that the gravitational variables are related to thescale factor a by | p | = a V and ˜ c = γ ˙ aV .In the physically reasonable ¯ µ scheme of LQC [20], itis convenient to introduce new conjugate variables forgravity by the canonical transformation v := 2 √ sgn ( p )¯ µ − , b := ¯ µ ˜ c, where ¯ µ = q ∆ | p | with ∆ = 4 √ πγG ~ being the minimumnonzero eigenvalue of the area operator [31]. The newvariables also form a pair of conjugate variables as { b, v } = 2 ~ . In LQC, the kinematical Hilbert space for the geometrypart is defined as H grkin := L ( R Bohr , dµ H ), where R Bohr and dµ H are respectively the Bohr compactification ofthe real line and Haar measure on it [17]. The kinematicalHilbert space for the scalar field part is defined as in usualSchrodinger representation by H sckin := L ( R, dµ ). Hencethe whole Hilbert space of our model is a direct product, H kin = H grkin ⊗ H sckin . In H grkin , there are two elementaryoperators, d e ib/ and ˆ v . It turns out that the eigenstates | v i of ˆ v contribute an orthonormal basis in H grkin . In the v -representation, the actions of these two operators onthe basis read c e ib | v i = | v + 1 i , ˆ v | v i = v | v i . (10)Let | φ ) be the generalized eigenstates of ˆ φ in H sckin . Wedenote | φ, v ) := | v i⊗| φ ) as the generalized basis for H kin .Notice that the spatial curvature R vanishes in thespatially flat FLRW model. Hence the general expression(7) reduces to H g = 116 πG Z d x h λF jab − ( λγ + 1) ε jmn K ma K nb i ε jkl E ak E bl √ q (11) By the regularization procedure mimicking that infull LQG, both the Euclidean term H E [20] and theLorentzian term H L [21] have been quantized as well-defined operators in H grkin . Therefore the operators ˆ H G corresponding to (11) is ready. Its action on a wave func-tion Ψ( v, φ ) in H kin is the following difference equation,ˆ H G Ψ( v, φ ) = f + ( v )Ψ( v + 4 , φ ) + ( f ( v ) + g ( v ))Ψ( v, φ )+ f − ( v )Ψ( v − , φ ) + g + ( v )Ψ( v + 8 , φ )+ g − ( v )Ψ( v − , φ ) , (12)where f + ( v ) = − λ q π √ G ~ πGγ (cid:12)(cid:12)(cid:12) | v + 2 | − | v | (cid:12)(cid:12)(cid:12) | v + 1 | ,f − ( v ) = f + ( v − , f ( v ) = − f + ( v ) − f − ( v ) .g + ( v ) = − (1+ λγ ) √ × γ / (8 πG ) / ~ / L ˜ g + ( v ) , ˜ g + ( v ) = h M v (1 , f + ( v + 1) − M v ( − , f + ( v − i × ( v + 4) M v (3 , × h M v (5 , f + ( v + 5) − M v (3 , f + ( v + 3) i ,g − ( v ) = − (1+ λγ ) √ × γ / (8 πG ) / ~ / L ˜ g − ( v ) , ˜ g − ( v ) = h M v (1 , − f − ( v + 1) − M v ( − , − f − ( v − i × ( v − M v ( − , − × (cid:2) M v ( − , − f − ( v − − M v ( − , − f − ( v − (cid:3) ,g o ( v ) = − (1+ λγ ) √ × γ / (8 πG ) / ~ / L ˜ g o ( v ) , ˜ g o ( v ) = h M v (1 , f + ( v + 1) − M v ( − , f + ( v − i × ( v + 4) M v (3 , × h M v (5 , f − ( v + 5) − M v (3 , − f − ( v + 3) i + h M v (1 , − f − ( v + 1) − M v ( − , − f − ( v − i × ( v − M v ( − , − × h M v ( − , f + ( v − − M v ( − , − f + ( v − (cid:3) , with M v ( a, b ) := | v + a |−| v + b | and L = q πγG ~ . Thus,the Hamiltonian constraint equation of our LQC modelcan be written as ˆ H G + √ p φ (∆) [ | v | − ! Ψ( φ, v ) = 0 , (13)where the action of the Hamiltonian of matter field reads √ p φ (∆) [ | v | − Ψ( v, φ ) = − √ ~ B ( v ) ∂ Ψ( v, φ ) ∂φ , (14)with B ( v ) = (cid:0) (cid:1) | v | (cid:12)(cid:12)(cid:12) | v + 1 | / − | v − | / (cid:12)(cid:12)(cid:12) [20]. Notethat we still have the freedom to choose a particular valueof the parameter λ . It is obvious that, if one set λ = − γ ,Eq.(13) would coincide with the quantum dynamics in[20], while by choosing λ = 1, one of the Hamiltonians in[21] would be obtained. Our viewpoint is that the valueof λ should be fixed by observations. To this aim, let usstudy the effective theory indicated by Eq.(13). It hasbeen showed in [21] that the expectation values of the Euclidean term H E and the Lorentzian term H L undersuitable semiclassical coherent states read respectively,to the leading order as h b H E i = β πG ∆ sin ( b ) , h b H L i = β πGγ ∆ sin (2 b ) , (15)where β = 2 πG ~ γ √ ∆. Thus the total effective Hamilto-nian constraint of the model reads H F = − β πGγ ∆ | v | sin b (cid:0) − (1 + λγ ) sin ( b ) (cid:1) + β | v | ρ, (16) where ρ = p φ V with V = | p | being the physical volumeof V .Now we consider the effective Hamiltonian (16) with λ >
0. At the kinematical level, the matter energy-density ρ can be solved by the effective Hamiltonian con-straint H F = 0 as ρ = 38 πG ∆ γ sin b (1 − (1 + λγ ) sin b ) . (17)This in turn implies two solutions b + and b − satisfyingsin ( b ± ) = 1 ± q − ρρ c λγ ) , (18)where ρ c = πG (1+ λγ ) γ ∆ . Takeing into account thefact that 0 < sin b ≤ λγ , it is easy to see that ρ isbounded by its maximum value ρ c . The effective equa-tions of motion of the model with respect to the cosmo-logical time t can be derived by the Hamiltonian con-straint (16). In particular, one can obtain˙ v = β πG ~ γ ∆ | v | sin(2 b )(1 − λγ ) sin b ) , (19)˙ b = − p φ ~ βv − β πG ~ γ ∆ sin b (1 − (1 + λγ ) sin b ) . (20)Hence we have ˙ v = 0 for b c satisfying sin ( b c ) = λγ ) .Moreover, the second order derivative of v can be calcu-lated at this point as ¨ v | b = b c = 24 πG (1 + β ) β | v | sin (2 b )(1 + λγ ) ρ | b = b c > Hence the matter density ρ takes its maximum at thebounce point. Therefore the point v | b = b c is the minimumwhere the quantum bounce of the Universe happens. Interms of the scale factor a , the Friedmann and Raychaud-huri equations of this model are derived as H = 1 γ ∆ sin ( b )(1 − sin ( b ))(1 − λγ ) sin b ) (21)¨ aa = ( H ) + 1 γ √ ∆ ˙ b (1 − ( b ) − λγ ) sin b (3 − b )) , (22) where H denotes the Hubble parameter and b has twosolutions as shown in (18). We are going to show thatthe two solutions correspond to the two periods of theevolution of the Universe, divided by the bounce point.The Hamiltonian (16) implies that p φ is constant ofmotion and hence φ is monotonic with respect to t . Byidentifying φ as a dynamical time, we obtain the followinganalytic solution of the effective equations of motion inthe case of λ > x ( φ ) = 11 + λγ cosh ( q β ~ πG ∆ γ ( φ − φ o )) , (23) ρ ( φ ) = 3 λ πG ∆ ( sinh( q β ~ πG ∆ γ ( φ − φ o ))1 + λγ cosh ( q β ~ πG ∆ γ ( φ − φ o )) ) (24) and v ( φ ) = s πG | p φ | ∆3 λβ λγ cosh ( q β ~ πG ∆ γ ( φ − φ o )) | sinh( q β ~ πG ∆ γ ( φ − φ o )) | , (25) where we defined x := sin b , and φ o is an integral con-stant. Eq.(23) indicates that the range of x in the physi-cal evolution covers the interval (0 , λγ ). This confirmsthat the bounce point of x = λγ ) does appear in thephysical evolution. Moreover, by using the Hamiltonian(16) and Eq.(25), the relation between φ and t can besolved as t ( φ ) − t = 2 πG ~ ∆ γ √ λ sgn[ p φ ( φ − φ )]3 β cosh( s β ~ πG ∆ γ ( φ − φ )) − λγ λγ ln | coth( s β πG ~ ∆ γ ( φ − φ )) | , where sgn[ p φ ( φ − φ )] denotes the sign of p φ ( φ − φ ).Therefore, either of the two domains φ > φ and φ < φ can cover the range of t . We thus consider the domain φ > φ without loss of generality. Then the infinite pastand infinite future of t correspond to φ → φ +0 and φ → + ∞ respectively. Hence Eq.(23) ensures that the twosolutions b − and b + in (18) correspond to the two periodsgiven by 0 < sin b − ≤ λγ ) and λγ ) ≤ sin b + < λγ of the evolution of the Universe, divided by thebounce point at b c . By Eqs. (25) and (23), it is obviousthat v → ∞ is achieved at ( φ − φ o ) → + ∞ or ( φ − φ o ) → + , and correspondingly the behaviour of b is given by b → φ − φ o ) → + ∞ ,b → b ≡ arcsin ( 1 p (1 + λγ ) ) if ( φ − φ o ) → + . (26)To study the asymptotics of the effective Friedmann andRaychaudhuri equations, we expand Eq.(21) and Eq.(22)by b and b − b respectively up to the second order termin the above large v limit and thus obtain H = 8 πG ρ, (for b →
0) (27) H = (cid:18) − λγ λγ (cid:19) πGρ , (for b → b )(28)and ¨ aa = − πG ρ + 3 P ) , (for b → , (29)¨ aa = − (cid:18) − λγ λγ (cid:19) πG ρ + 3 P ) + Λ3 , (for b → b ) , (30) where we defined the pressure of matter by P = − ∂H φ ∂V ,and an effective cosmological constantΛ ≡ λ (1 + λγ ) ∆ . (31)The asymptotic behaviors of the scalar curvature R =6( H + ¨ aa ) are given by R = − πGρ, (for b → , (32) R = − πGρ (cid:18) − λγ λγ (cid:19) + 4Λ , (for b → b ) . (33)Thus, if the Universe started to collapse from an initial
50 100 150 200 250 300 Λ LD FIG. 1: The Λ∆ as a function of λ . The γ = 0 . state of spatially flat FLRW configuration, it would un-dergo a quantum bounce and evolve into an asymptoticde-Sitter branch. A notable feature of this picture isthat an effective cosmological constant Λ emerges fromthe quantum gravity effect. Eq.(28) implies an effec-tive Newtonian constant G λ = − λγ λγ G in the de-Sitterepoch. For the special choice of λ = 1, G λ received alarge correction which is inconsistent with the currentexperiments [32]. However, the regularization freedomdoes admit us to choose certain sufficiently small λ suchthat G λ satisfies the experimental restrictions.The value of Λ is determined by the value of λ byEq.(31), and the function Λ∆( λ ) is plotted by Fig.1.To reproduce the current observed cosmological constantΛ ob , the corresponding λ has two solutions, λ ∼ γ ∆Λ ob and λ ∼ ∆Λ ob . It is obvious that for a reasonable choiceof γ ∼ . λ is too big to give an acceptable G λ .Moreover, in this case the critical density ρ c ∼ ob πG be-comes very small and thus conflicts with the experiments.However, λ is sufficiently small to give an acceptable G λ and in the meantime leads to a bouncing density ofPlanck order as ρ c = πG (1+ λγ ) γ ∆ ∼ γ πG ∆ . Forsuch a λ , Eqs. (28) and (30) reduce to H = 8 πGρ ob , (34)¨ aa = − πG ρ + 3 P ) + Λ ob . (35)They are nothing but the standard Friedmann and Ray-chaudhuri equations with the observational cosmologicalconstant!To summarize, an one-parameter regularization free- dom of the Hamiltonian constraint for LQG is introducedby Eq.(7). The corresponding spatially flat FLRW modelis studied. The quantum difference equation represent-ing the evolution of the universe and its effective Hamil-tonian are obtained. The general expression (7) includesthe Euclidean term quantum dynamics [20] by choosing λ = − γ and the Euclidean-Lorentzian term dynamics[21] by choosing λ = 1. The quantum bounce natureof LQC is tenable in the general case when the matterdensity approaches ρ c = πG (1+ λγ ) γ ∆ . For a chosen λ >
0, the effective Hamiltonian of our LQC model canlead to a branch of Universe with an asympotic positivecosmological constant connecting to the FLRW branchthrough the quantum bounce. Remarkably, by choos-ing λ = ∆Λ ob , the standard Friedmann equation withthe observational cosmological constant Λ ob can be ob-tained at large volume limit of the asymptotic de-Sitterbranch. In the meantime, unlike the case of λ = 1, theeffective Newtonian constant G λ also satisfies the exper-imental restrictions. Therefore, there does exist suitableregularization scheme for LQC, such that the cosmologi-cal constant emerges from the effect of quantum gravity.Moreover, since the effective Hamiltonian of LQC givenin [21] can be derived from full LQG by suitable semi-classical analysis [33], it is reasonable to infer that thecosmological constant derived in this Letter be originatedfrom the effect of LQG.This work is supported by NSFC with Grants No.11775082, No. 12047519, No. 11961131013 and No.11875006. [1] J. Friemann, M. Turner, D. Huterer, Dark energy and theaccelerating Universe,
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