The cosmological perturbation theory in loop cosmology with holonomy corrections
aa r X i v : . [ h e p - t h ] M a y The cosmological perturbation theory in loop cosmology withholonomy corrections
Jian-Pin Wu ∗ Department of Physics, Beijing Normal University, Beijing 100875, China
Yi Ling † Center for Relativistic Astrophysics and High Energy Physics,Department of Physics, Nanchang University, 330031, China
Abstract
In this paper we investigate the scalar mode of first-order metric perturbations over spatiallyflat FRW spacetime when the holonomy correction is taken into account in the semi-classicalframework of loop quantum cosmology. By means of the Hamiltonian derivation, the cosmologicalperturbation equations is obtained in longitudinal gauge. It turns out that in the presence of metricperturbation the holonomy effects influence both background and perturbations, and contributethe non-trivial terms S h and S h in the cosmological perturbation equations. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION In loop quantum cosmology two main quantum gravity effects lead to remarkable modifi-cations to the standard description of the early universe(for a detailed review, see Ref. [1]).One is due to the holonomy correction and the other is due to the inverse volume correction.Such modifications can successfully avoid the Big Bang singularity [2–5], and replace it bythe Big Bounce even at the semi-classical level [6, 7]. In addition, it is very interesting tonotice that quantum gravity effects may lead to the occurrence of the super-inflationaryphase [8]. As shown in Ref. [9], such a super-inflationary phase can also resolve the horizonproblem with only a few number of e-foldings. Therefore, it is possible to construct a phaseof inflation or an alternative to inflation in the framework of loop quantum cosmology.As we all known, the inflationary phase is crucial for understanding the structure forma-tion and anisotropies of the CMB. In order to address these issues in the framework of loopquantum cosmology, we must consider the cosmological perturbation theory with modifica-tions due to quantum gravity effects. In the earlier work by Bojowald et al. [10, 11], by meansof the Hamiltonian derivation they have obtained the cosmological perturbation equationwith inverse volume corrections for scalar modes in longitudinal gauge. They show thatsuper-horizon curvature perturbations are not preserved. Recently, they have also derivedthe gauge-invariant quantities and the corresponding gauge-invariant cosmological pertur-bation equations with inverse volume corrections for scalar modes [12, 13]. In addition,the vector modes and tensor modes with corrections from loop quantum gravity have beeninvestigated [14, 15].At the same time, some pioneer work have already been devoted to understanding theprimordial power spectrum in the perturbation theory of LQC [9, 16–22]. First of all, inRef. [9, 18], it is shown that a scale invariant spectrum can be obtained. More importantlythese attempts imply that the quantum gravity effects may leave an imprints on the powerspectrum which can be potentially detected in the future experiments such as the Plancksatellite. However,above considerations are restricted to the scalar field perturbations withfixed background. To provide a complete and more precise understanding on the perturba-tion theory in loop cosmology, it is essential to take the metric perturbation into account.Along this direction it is worthwhile to point out that another potential observables, pri-mordial gravitational waves have already been investigated intensively in LQC [23].2lthough, in Ref. [10–13] the cosmological perturbation equations with inverse volumecorrections have been derived in longitudinal gauge and gauge-invariant manner respectively,the metric perturbations with holonomy corrections is still absent. In the present paper, bymeans of the Hamiltonian derivation, we will derive the cosmological perturbation equationswith holonomy corrections in longitudinal gauge.The outline of our paper is the following. For comparison, we firstly present a briefreview on the perturbation equations in standard classical cosmology in section II. Afterintroducing the basic variables in loop cosmology in section III, we will demonstrate adetailed derivation on the cosmological perturbation equation with holonomy corrections insection IV. The discussion is given in section VI.
II. THE CLASSICAL COSMOLOGICAL PERTURBATION EQUATIONS
Before proceeding to the effective loop quantum cosmology with holonomy corrections,we first briefly review the classical perturbation equations in standard cosmology. A detailedderivation can be found in Ref. [24]. Let us now consider a spatially flat background metricof FRW type ds = a ( η )( − dη + δ ab dx a dx b ) . (1)where η is the conformal time. The spatial part of the metric describes isotropic and homo-geneous 3-surfaces. Then one can perturb the background metric ds = a ( η ) (cid:2) − (1 + 2Φ) dη + (1 − δ ab dx a dx b (cid:3) . (2)Here we only consider the scalar modes in longitudinal gauge, which is thus diagonal.Through this paper, we will consider the scalar field ϕ as the matter source. Expanding theEinstein’s equation linearly, one can obtain the cosmological perturbation equation ∇ Φ − H ˙Φ − ( ˙ H + 2 H )Φ = 4 πG ( ˙¯ ϕ ˙ δϕ + ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ) , (3)¨Φ + 3 H ˙Φ + ( ˙ H + 2 H )Φ = 4 πG ( ˙¯ ϕ ˙ δϕ − ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ) , (4) ∂ a ( ˙Φ + H Φ) = 4 πG ˙¯ ϕδϕ ,a , (5)where a dot denotes a derivative with respect to the conformal time η . H is the Hubbleexpansion rate in the conformal time, and for later convenience, we have identified a with3 p which is introduced in (10). Note that in the case of vanishing anisotropic stresses, twoscalar functions Φ and Ψ coincide, Φ = Ψ. Therefore, in above equations we have set Φ = Ψ,which simplifies the equations considerably . Moreover, among these equations above onlytwo of them are independent. Combining these equations, one can obtain the followingsecond order differential equation for Φ¨Φ − ∇ Φ + (6 H + 2¯ p V , ¯ ϕ ( ¯ ϕ )˙¯ ϕ ) ˙Φ + (2 ˙ H + 4 H + V , ¯ ϕ ( ¯ ϕ )˙¯ ϕ H )Φ = 0 . (6)In addition, the background and the perturbed Klein-Gordon equation can respectivelyexpressed as ¨¯ ϕ + 2 H ˙¯ ϕ + ¯ pV ,ϕ ( ¯ ϕ ) = 0 , (7)¨ δϕ + 2 H ˙ δϕ − ▽ δϕ + ¯ pV , ¯ ϕ ¯ ϕ ( ¯ ϕ ) + 2¯ pV , ¯ ϕ ( ¯ ϕ )Φ − ϕ ˙Φ = 0 . (8) III. THE BASIC VARIABLES
Now we intend to study the scalar mode of first-order metric perturbations around spa-tially flat FRW spacetime when the holonomy corrections is taken into account in the semi-classical framework of loop quantum cosmology. To derive the cosmological perturbationequations we adopt the Hamiltonian approach which has been developed in the effectiveloop quantum cosmology with inverse triad corrections [10, 12]. We summarize the basicidea and steps as follows.In loop quantum gravity, instead of the spatial metric q ab , a densitized triad E ai is pri-marily used, which satisfies E ai E bi = q ab detq . Moreover, in the canonical formulation thespace-time metric is given by ds = − N dη + q ab ( dx a + N a dη )( dx b + N b dη ) , (9)where N and N a are lapse function and shift vector respectively.By comparing the above equation with the FRW metric (1), the background variables,¯ N , ¯ N a and ¯ E ai , can be expressed as respectively¯ N = √ ¯ p ; ¯ N a = 0; ¯ E ai = ¯ pδ ai , (10) However we must point out that, as a matter of fact, Φ = Ψ is a consequence of equations of motion,which can also be seen in this paper K ia ,can be derived from the following relation¯ K ab = 12 ¯ N ( ˙¯ q ab − D ( a ¯ N b ) ) = ˙ aδ ab . (11)where D is the covariant spatial derivation. Thus, the extrinsic curvature can be expressedas ¯ K ia = ¯ E bi q | det ( ¯ E cj ) | ¯ K ab = ˙¯ p p δ ia =: ¯ kδ ia . (12)In equation (12), we have defined the background extrinsic curvature as ¯ k =: ˙¯ p p = ˙ aa ,which can also be obtained from the background equations of motion [12]. Therefore, inclassical FRW background, the extrinsic curvature is nothing but the conformal Hubbleparameter H . However, in the effective loop quantum cosmology, the relation between theextrinsic curvature and the conformal Hubble parameter will change due to quantum gravitycorrections, which we will see in the next section.The canonical perturbed variables can be related to the perturbed metric variables bycomparing the perturbed metric (2) with the canonical one(9). It turns out that the per-turbed triad is given by δE ai = − p Ψ δ ai , (13)and the perturbed lapse function is δN = ¯ N Φ . (14)As shown in the above, the extrinsic curvature components can be diagonal, thus it canbe expanded as K ia = ¯ K ia + δK ia = ¯ kδ ia + δK ia . (15)The perturbed extrinsic curvature will be derived from the equation of motion in thefollowing. We can assume that δE ai and δK ia do not have homogeneous modes, namely Z Σ δE ai δ ia d x = 0 , Z Σ δK ia δ ai d x = 0 . (16)And the homogeneous mode is defined by¯ p = 13 V Z Σ E ai δ ia d x, ¯ k = 13 V Z Σ K ia δ ai d x , (17)5here we integrate over a bounded region of coordinate size V = R Σ d x . Then we canconstruct the Poisson brackets of the background and perturbed variables [13], { ¯ k, ¯ p } = 8 πG V , { δK ia ( x ) , δE bj ( y ) } = 8 πGδ ij δ ba δ ( x − y ) . (18)In addition, we point out that the similar conditions will be required in the perturbedlapse δN , the scalar field δϕ and conjugate momentum δπ such that Z Σ δN d x = 0 , Z Σ δϕd x = 0 , Z Σ δπd x = 0 , (19)which is used in expanding the Hamiltonian constraint. While the homogeneous mode ofthe scalar field and its conjugate momentum is¯ ϕ = 1 V Z Σ ϕd x, ¯ π = 1 V Z Σ πd x . (20)Therefore, the Poisson brackets of the background and perturbed variables of scalar fieldis { ¯ ϕ, ¯ π } = 13 V , { δϕ ( x ) , δπ ( y ) } = δ ( x − y ) . (21) IV. THE COSMOLOGICAL PERTURBATION THEORY WITH HOLONOMYCORRECTIONS
Now we turn to the derivation of the cosmological perturbation theory in the effectiveloop quantum cosmology with holonomy corrections. For more details on the Hamiltoncosmological perturbation theory, we refer to Ref.[10, 12].Thanks to the holonomy corrections, in the isotropic and homogeneous models, the ef-fective Hamiltonian can be obtained at the phenomenological level by simply replacing thebackground Ashtekar connection ¯ k by sin ¯ µγ ¯ k ¯ µγ , where γ is the Barbero-Immirzi parameter.The parameter ¯ µ depends on the quantization scheme and may be a function of ¯ p . Morediscussions about the parameter ¯ µ , we can refer to Ref. [4, 25].However, when the inhomogeneities are taken into account, it is no longer true. To studythe effects of holonomy corrections on inhomogeneous perturbations, we similarly substitutethe appearance of ¯ k in the classical Hamiltonian by a general form sin m ¯ µγ ¯ km ¯ µγ where m is aninteger. In the context of vector modes [14] and tensor modes [15], due to the requirementof the anomaly cancellation, we can fix the parameter m . Since the evolution of all modes6hould be generated by one general Hamiltonian constraint, it would be reasonable to use thevalues found for vector modes and tensor modes also for scalars. However, it must been alsopointed out that the restrictions of anomaly cancellation from the vector modes and tensormodes has not been checked for scalars in the presence of holonomy corrections. Completeconsistency is realized only if all modes can be anomaly-free with holonomy corrections forspecific parameter values.Subsequently, one can write down the expressions for the gravitational Hamiltonian den-sity in a similar manner H hG = H h (0) G + H h (1) G + H h (2) G with H h (0) G = −
6( sin ¯ µγ ¯ k ¯ µγ ) √ ¯ p , H h (1) G = −
4( sin 2¯ µγ ¯ k µγ ) √ ¯ pδ cj δK jc − √ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) δ jc δE cj + 2 √ ¯ p ∂ c ∂ j δE cj , H h (2) G = √ ¯ pδK jc δK kd δ ck δ dj − √ ¯ p ( δK jc δ cj ) − √ ¯ p ( sin 2¯ µγ ¯ k µγ ) δE cj δK jc − p / ( sin ¯ µγ ¯ k ¯ µγ ) δE cj δE dk δ kc δ jd + 14¯ p / ( sin ¯ µγ ¯ k ¯ µγ ) ( δE cj δ jc ) − δ jk p / ( ∂ c δE cj )( ∂ d δE dk ) , (22)where the superscript “ h ” represents the holonomy corrections and the corresponding clas-sical expressions can be found in Ref.[12, 13]. We now only consider the scalar field as thematter source. Its Hamiltonian density expands as H M = H (0) M + H (1) M + H (2) M . Since thematter is free from the holonomy corrections, the expressions of scalar field Hamiltoniandensity, H M = H π + H ∇ + H ϕ , expanding up to the second order, are as the classical cases[12, 13], H (0) π = ¯ π p / , H (0) ∇ = 0 , H (0) ϕ = ¯ p / V ( ¯ ϕ ) , (23) H (1) π = ¯ πδπ ¯ p / − ¯ π p / δ jc δE cj p , H (1) ∇ = 0 , H (1) ϕ = ¯ p / ( V , ¯ ϕ ( ¯ ϕ ) δϕ + V ( ¯ ϕ ) δ jc δE cj p ) , (24)and H (2) π = 12 δπ ¯ p / − ¯ πδπ ¯ p / δ jc δE cj p + 12 ¯ π ¯ p / ( δ jc δE cj ) p + δ kc δ jd δE cj δE dk p ! , H (2) ∇ = 12 √ ¯ pδ ab ∂ a δϕ∂ b δϕ , H (2) ϕ = 12 ¯ p / V , ¯ ϕ ¯ ϕ ( ¯ ϕ ) δϕ + ¯ p / V , ¯ ϕ ( ¯ ϕ ) δϕ δ jc δE cj p +¯ p / V ( ¯ ϕ ) ( δ jc δE cj ) p − δ kc δ jd δE cj δE dk p ! . (25)7 . The background equations In the isotropic and homogeneous FRW background, the diffeomorphism constraint van-ishes. Therefore background equations are generated only by the background Hamiltonianconstraint, which can be expressed as H h (0) [ ¯ N ] = 116 πG Z Σ d x ¯ N [ H (0) G + 16 πG ( H (0) π + H (0) ϕ )] . (26)Thus the explicit expression for the background Hamiltonian constraint is − πG √ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) + ¯ π p / + ¯ p / V ( ¯ ϕ ) = 0 . (27)Then, by means of Poisson bracket, we can derive the equation of motion for the gravi-tational variables ¯ k and ¯ p .˙¯ k = { ¯ k, H h (0) [ ¯ N ] } = − [ 12 ( sin ¯ µγ ¯ k ¯ µγ ) + ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ] + 4 πG [ − ¯ π p + ¯ pV ( ¯ ϕ )] . (28)˙¯ p = { ¯ p, H h (0) [ ¯ N ] } = 2¯ p ( sin 2¯ µγ ¯ k µγ ) . (29)Similarly, the equation of motion for scalar field ¯ ϕ and its conjugate momentum field ¯ π can also be derived as ˙¯ ϕ = { ¯ ϕ, H h (0) [ ¯ N ] } = ¯ π ¯ p . (30)˙¯ π = { ¯ π, H h (0) [ ¯ N ] } = − ¯ p V , ¯ ϕ ( ¯ ϕ ) . (31)Note that in above Poisson brackets, we have used the relation ¯ N = √ ¯ p . Substitutingthe relation (30) into the constraint equation (27) gives rise to the corrected Friedmannequation ( sin ¯ µγ ¯ k ¯ µγ ) = 8 πG ϕ + ¯ pV ( ϕ )] . (32)At the same time, equation (28) is just the corrected Raychaudhuri equation˙¯ k + 12 ( sin ¯ µγ ¯ k ¯ µγ ) + ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) = 4 πG [ − ˙¯ ϕ pV ( ¯ ϕ )] . (33)In the classical limit, ¯ µ →
0, above two equations can be reduced to the Friedmann andRaychaudhuri equation in the standard cosmology. Finally, the Klein-Gordon equation canbe derived from Eqs. (30), (31) and (29)¨¯ ϕ + 2( sin 2¯ µγ ¯ k µγ ) ˙¯ ϕ + ¯ pV , ¯ ϕ ( ¯ ϕ ) = 0 . (34)8n addition, from the equation of motion (29), one can find that the extrinsic curvature¯ k is related to the conformal Hubble parameter H bysin 2¯ µγ ¯ k µγ = ˙¯ p p =: H . (35)Therefore, due to the holonomy corrections, the conformal Hubble parameter H is notsimply equal to the extrinsic curvature ¯ k but receives corrections. For consistency, in ournext derivation we will continuously use the extrinsic curvature ¯ k rather than the conformalHubble parameter. Only at the end, we will use the conformal Hubble parameter H insteadof sin 2¯ µγ ¯ k µγ in the perturbation equations. B. The perturbed equations
In this subsection, we will derive the cosmological perturbation equation with holonomycorrections. Firstly we will derive the equations of motion of perturbed variables. In thecanonical formulation, the equation of motion of any phase space function f is determinedby Poisson bracket, ˙ f = { f, H } . Here H is the total Hamiltonian, which is a sum of theHamiltonian constraint H [ N ] and the diffeomorphism constraint D [ N a ], H = H [ N ]+ D [ N a ].Since the zero-order and first-order shift vectors vanish, the diffeomorphism constraints isidentically satisfied up to the second-order. Thus, the equations of motion of the perturbedvariables are only generated by the Hamiltonian constraint. The perturbed Hamiltonianconstraint up to the second-order is written as ˜ H h [ N ] = ˜ H h [ ¯ N ] + ˜ H h [ δN ] with˜ H h [ ¯ N ] = 116 πG Z Σ d x ¯ N [ H (2) G + 16 πG ( H (2) π + H (2) ∇ + H (2) ϕ )] , ˜ H h [ δN ] = 116 πG Z Σ d xδN [ H (1) G + 16 πG ( H (1) π + H (1) ϕ )] . (36)Note that we have used the conditions that the perturbed variables do not have homo-geneous modes as described in Eq.(16) and (19). As well, we input the boundary conditionrequiring that the integration over the boundary vanishes, namely Z Σ ¯ N [ H h G + 16 πG ( H π + H ϕ )] = 0 , Z Σ δN [ H h G + 16 πG ( H π + H ϕ )] = 0 . (37)Therefore, the equations of motion of perturbed variables are generated only by thesecond order part of Hamiltonian constraints. Thus, we can arrive at the equation of motion9f the perturbed variables by means of the Poisson bracket δ ˙ K ia ≡ { δK ia , ˜ H h [ ¯ N ] + ˜ H h [ δN ] } = ¯ N ¯ p / [ − ¯ p ( sin 2¯ µγ ¯ k µγ ) δK ia −
12 ( sin ¯ µγ ¯ k ¯ µγ ) δE dk δ ka δ id + 14 ( sin ¯ µγ ¯ k ¯ µγ ) ( δE dk δ kd ) δ ia + 12 δ ik ∂ a ∂ d δE dk ] − δN √ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) δ ia + 1 √ ¯ p ∂ a ∂ i δN +4 πG ¯ N ¯ p / [ − ¯ πδπ ¯ p δ ia + 12 ¯ π ¯ p ( 12 δE dk δ kd δ ia + δE dk δ ka δ id ) + ¯ p V , ¯ ϕ ( ¯ ϕ ) δϕδ ia +¯ pV ( ϕ )( 12 δE dk δ kd δ ia − δE dk δ ka δ id )] + 4 πGδN [ −
12 ¯ π ¯ p / + √ ¯ pV ( ¯ ϕ )] δ ia , (38)˙ δE ai ≡ { δE ai , ˜ H h [ ¯ N ] + ˜ H h [ δN ] } = ¯ N √ ¯ p [ − ¯ pδK jc δ ci δ aj + ¯ p ( δK jc δ cj ) δ ai + ( sin 2¯ µγ ¯ k µγ ) δE ai ] + 2 δN ( sin 2¯ µγ ¯ k µγ ) √ ¯ pδ ai , (39) δ ˙ ϕ ≡ { δϕ, ˜ H h [ ¯ N ] + ˜ H h [ δN ] } = ¯ N ¯ p / ( δπ − ¯ π δE cj δ jc p ) + δN ¯ p / ¯ π , (40) δ ˙ π ≡ { δπ, ˜ H h [ ¯ N ] + ˜ H h [ δN ] } = ¯ N ¯ p / [¯ p ∇ δϕ − ¯ p V , ¯ ϕ ¯ ϕ δϕ −
12 ¯ p V , ¯ ϕ ¯ ϕ δE cj δ jc ] . (41)Furthermore, using Eqs.(13) and (29), we can obtain the perturbed extrinsic curvature δK ia from equation (39), δK ia = − δ ia [ ˙Ψ + ( sin 2¯ µγ ¯ k µγ )(Ψ + Φ)] . (42)Similarly, using Eq.(13), equations (40) and (41) can be respectively reexpressed as δ ˙ ϕ = δπ ¯ p + ¯ π ¯ p (3Ψ + Φ) . (43) δ ˙ π = ¯ p ∇ δϕ − ¯ p V , ¯ ϕ ¯ ϕ δϕ + 3¯ p V , ¯ ϕ Ψ . (44)Now, we derive the Hamiltonian’s equation using the equation of motion of δK ia . Col-lecting the expressions δE ai (13), δK ia (42) , δN (14), and equations (30), (43), one canobtain { ¨Ψ + ( sin 2¯ µγ ¯ k µγ )(2 ˙Ψ + ˙Φ) + [(cos 2¯ µγ ¯ k −
12 ) ˙¯ k + ( sin 2¯ µγ ¯ k µγ ) + ˙¯ µ ¯ µ (¯ k cos 2¯ µγ ¯ k − sin 2¯ µγ ¯ k µγ ) −
12 ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ](Ψ + Φ) + ¯ pV ( ¯ ϕ )(Φ − Ψ) } δ ia + ∂ a ∂ i (Φ − Ψ)= 4 πG ( ˙¯ ϕ ˙ δϕ − ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ) . (45)10hen deriving this equation, we have used the relation4 πG ˙¯ ϕ = ( sin ¯ µγ ¯ k ¯ µγ ) − ˙¯ k − ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) , (46)which can be obtained from the corrected Friedmann equation (32) and Raychaudhuri equa-tion (33). From equation (45), we can read the off-diagonal equation ∂ a ∂ i [Φ − Ψ] = 0 , (47)which implies Φ = Ψ. Therefore, in the following derivation, we will identify Φ with Ψ.Then the diagonal equation gives¨Φ + 3( sin 2¯ µγ ¯ k µγ ) ˙Φ + [(2 cos 2¯ µγ ¯ k −
1) ˙¯ k + 2( sin 2¯ µγ ¯ k µγ ) + 2 ˙¯ µ ¯ µ (¯ k cos 2¯ µγ ¯ k − sin 2¯ µγ ¯ k µγ ) − ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ]Φ = 4 πG ( ˙¯ ϕ ˙ δϕ − ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ) . (48)Subsequently, we will consider the diffeomorphism constraint equation. The perturbeddiffeomophism constraint with holonomy corrections is D [ N c ] = 18 πG Z Σ d xδN c [¯ p∂ c ( δ dk δK kd ) − ¯ p ( ∂ k δK kc ) − ( sin 2¯ µγ ¯ k µγ ) δ kc ( ∂ d δE dk )+8 πG ¯ π∂ c δϕ ] . (49)The diffeomorphism constraint equation can be obtained by varying the diffeomorphismconstraint with respect to the shift perturbation:8 πG δD [ δN c ] δ ( δN c ) = ¯ p∂ c ( δ dk δK kd ) − ¯ p ( ∂ k δK kc ) − ( sin 2¯ µγ ¯ k µγ ) δ kc ( ∂ d δE dk ) + 8 πG ¯ π∂ c δϕ = 0 . (50)Using the expressions δE ai (13), δK ia (42) and equation (30), the above equation reducesto ∂ c [ ˙Φ + ( sin 2¯ µγ ¯ k µγ )Φ] = 4 πG ˙¯ ϕ∂ c δϕ . (51)Finally, we will derive the Hamiltonian constraint equation. We note that after thevariation with respect to the background lapse ¯ N , the constraint equation will be second-order and can be neglected. So one can obtain the Hamiltonian constraint equation by onlyvarying the perturbed lapse δNδ ˜ H h [ N ] δ ( δN ) = 116 πG [ − µγ ¯ k µγ ¯ k √ ¯ pδK ia δ ai − ( sin ¯ µγ ¯ k ¯ µγ ¯ k ) √ ¯ p δE ai δ ia + 2 √ ¯ p ∂ a ∂ i δE ai ]+ ¯ πδπ ¯ p / − ( ¯ π p / − ¯ p / V ( ¯ ϕ )) δE ai δ ia p + ¯ p / V , ¯ ϕ ( ¯ ϕ ) δϕ = 0 . (52)11ubstituting the expressions δE ai (13), δK ia (42) and equation (30) into the above equationyields the Hamilton constraint equation ∇ Φ −
3( sin 2¯ µγ ¯ k µγ ) ˙Φ − [ ˙¯ k +6( sin 2¯ µγ ¯ k µγ ) −
4( sin ¯ µγ ¯ k ¯ µγ ) +¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ]Φ = 4 πG [ ˙¯ ϕ ˙ δϕ +¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ] . (53)In addition, using Eqs.(43) and (44), with the help of the background equations (29),(30) and (31), the perturbed Klein-Gordon equation can be expressed as δ ¨ ϕ + 2( sin 2¯ µγ ¯ k µγ ) δ ˙ ϕ − ∇ δϕ + ¯ pV , ¯ ϕ ¯ ϕ δϕ + 2¯ pV , ¯ ϕ Φ − ϕ ˙Φ = 0 . (54)Now, we replace sin 2¯ µγ ¯ k µγ by Hubble parameter H in the perturbation equations (53), (48)and (51) such that these equations can be reexpressed as ∇ Φ − H ˙Φ − [ ˙¯ k + 6 H −
4( sin ¯ µγ ¯ k ¯ µγ ) + ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ]Φ = 4 πG ( ˙¯ ϕ ˙ δϕ + ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ) , (55)¨Φ + 3 H ˙Φ + [2 ˙ H − ˙¯ k + 2 H − ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ]Φ = 4 πG ( ˙¯ ϕ ˙ δϕ − ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ) , (56) ∂ a ( ˙Φ + H Φ) = 4 πG ˙¯ ϕδϕ ,a . (57)As we have emphasized in the introduction, among the three classical perturbationsequations (3), (4) and (5) only two are independent. However, when the gauge-fixing hasbeen done before deriving equations of motion in the presence of quantum corrections, it maynot produce all terms correctly such that this consistency can not be maintained. In order topreserve the consistency for quantum corrected equations (55), (56) and (57), the additionalcorrection terms must be required. The simplest way is only to modify the equations (56)as follow by introducing some additional correction terms,¨Φ+ { H + 1 H [ ˙ H − ˙¯ k − ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ] } ˙Φ+[2 H +4 ˙ H − k − p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ]Φ = 4 πG ( ˙¯ ϕ ˙ δϕ − ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ) . (58)The proof of consistency of these equations (55), (58) and (57) has been given in theappendix. Obviously, in the classical limit, ¯ µ →
0, the equations (55), (58) and (57) reduceto the classical cosmological perturbations (3), (4) and (5) respectively. In addition, wemust also point out that since the gauge-fixing has been done before deriving equations ofmotion, the introduce of the additional correction terms is not unique. In order to obtain themore complete and unambiguous quantum corrected cosmological perturbations equation,we must consider the gauge invariant variables and derive these perturbations equations in a12auge invariant manner, which is under progress. Combing these equations, one can obtainthe following second order differential equation for Φ¨Φ − ∇ Φ + { H + 2¯ p V , ¯ ϕ ( ¯ ϕ )˙¯ ϕ + 1 H [ ˙ H − ˙¯ k − ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) ] } ˙Φ+[8 H + 4 ˙ H − k −
4( sin ¯ µγ ¯ k ¯ µγ ) − p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) + 2¯ p V , ¯ ϕ ( ¯ ϕ )˙¯ ϕ H ]Φ = 0 . (59)In addition, using the relation between the extrinsic curvature ¯ k and the conformal Hubbleparameter H (35), one can obtain( sin ¯ µγ ¯ k ¯ µγ ) = 1 ± p − µγ ) H µγ ) . (60)If we denote S h = H − ( sin ¯ µγ ¯ k ¯ µγ ) = H − ± √ − µγ ) H µγ ) , which results from the holonomycorrections in the presence of the metric perturbation, and S h = ˙ H − ˙¯ k − ¯ p ∂∂ ¯ p ( sin ¯ µγ ¯ k ¯ µγ ) , whichbe introduced by the requirement of the consistency, then the above second order differentialequation can be further rewritten as¨Φ − ∇ Φ + [6 H + 2¯ p V , ¯ ϕ ( ¯ ϕ )˙¯ ϕ + S h H ] ˙Φ + 2[2 H + ˙ H + ¯ p V , ¯ ϕ ( ¯ ϕ )˙¯ ϕ H − S h + 2 S h ]Φ = 0 . (61)Up to now, we have completed the derivation of the cosmological perturbation equationsin the effective loop quantum cosmology with holonomy corrections. V. DISCUSSION
The effects of quantum gravity on structure formation, generally called trans-Planckianissues, have been investigated intensively (for example, we can refer to [26]). In loop quantumcosmology, the analogous issues have also been investigated in Ref.[9, 16–20]. However, inRef.[19], they assume that after a super-inflation phase, the universe underwent a normalinflation stage. Then they find that the loop quantum effects can hardly lead to any imprintin the primordial power spectrum. Although in Ref.[9] the scale invariant spectrum wasobtained and the holonomy effects also leave their imprint on the power spectrum, onlythe holonomy effects from a fixed background were taken into account. In Ref.[20], thecosmological perturbation equations with holonomy corrections were derived in longitudinalgauge. But they consider only the cases of large scale metric perturbations. In this paper,along the Hamiltonian approach we have derived the cosmological perturbation equation13or scalar modes in longitudinal gauge in the presence of holonomy corrections. In thepresence of metric perturbation, we find that holonomy effects influence both backgroundand perturbations, which contribute the non-trivial terms S h and S h . Therefore, theholonomy effects will affect the power spectrum such that it is possible that the quantumgravity effects will leave their imprint on the cosmic microwave background observed today.In the future work, we will investigate analytically and numerically the characters of powerspectrum in the presence of holonomy corrections, which might open a window to test theloop quantum gravity effects.In addition, when ignoring the additional corrections term S h , which be introduced bythe requirement of the consistency, the second order differential equation (61) become¨Φ − ∇ Φ + 2( H − ¨¯ ϕ ˙¯ ϕ ) ˙Φ + 2[ ˙ H − H ¨¯ ϕ ˙¯ ϕ + 2 H + 2 S h ]Φ = 0 . (62)here we have used the Klein-Gordon equation (34).In this case, we can furthermore introduce the Mukhanov-Sasaki variable υ = a ˙¯ ϕ Φ. Thenthe cosmological perturbation equation (62) reduces to¨ υ − ∇ υ + [( ¨¯ ϕ ˙¯ ϕ ) . − ( ¨¯ ϕ ˙¯ ϕ ) + ˙ H − H + 4 S h ] υ = 0 , (63)In momentum space, the cosmological perturbation equation (63) can be written as¨ υ − [ κ − S h − m eff ] υ = 0 , (64)where κ denotes the momentum and m eff = ( ¨¯ ϕ ˙¯ ϕ ) . − ( ¨¯ ϕ ˙¯ ϕ ) + ˙ H − H . Therefore, the cosmologicalperturbation equation (64) can be effectively viewed as imposing such a modified dispersionrelation at quantum gravity phenomenological level. Obviously, in such a modified dispersionrelation, both the energy and momentum are bounded. Here, we point out that, in Ref.[27], Y. Ling et. al have also proposed a bounded modified dispersion relation, motivatedby the isotropic homogenous effective loop quantum cosmology with holonomy corrections.Although both are bounded, they are also very different, implying we can not simply usethe background corrections instead of perturbation corrections. In the future work, we willfurthermore discuss the implications of such two modified dispersion relations.Our present paper is the first step towards studying the holonomy corrected cosmologicalperturbation equations in the presence of metric perturbation. Since constraints are modi-fied, the form of gauge invariant variables should change as well. Therefore, it is necessary14o study the perturbations with different gauges or in a gauge invariant manner in thisformalism, which is under progress. Acknowledgement
J. P. Wu is grateful to Prof. Yongge Ma and Wei-Jia Li for helpful discussion. J.P. Wu is partly supported by NSFC(No.10975017). Y. Ling is partly supported byNSFC(No.10875057), Fok Ying Tung Education Foundation(No. 111008), the key projectof Chinese Ministry of Education(No.208072) and Jiangxi young scientists(JingGang Star)program. He also acknowledges the support by the Program for Innovative Research Teamof Nanchang University.
Appendix A: The proof of consistency of these equations (55), (58) and (57)
In this appendix, we will give a proof of the consistency of these equations (55), (58) and(57). Without loss of generality, we will only derive the Eq. (58) from the Eqs. (55) and(57). From the perturbation equation (57), by taking the spatial derivation, we can obtain ddη ( ∇ Φ) + H ∇ Φ − πG ˙¯ ϕ ∇ δϕ = 0 . (A1)In addition, using the corrected Raychaudhuri equation (33) and the perturbation equa-tion (55), the term ∇ Φ can be expressed as ∇ Φ = 3 H ˙Φ + (cid:2) H − πG ( ˙¯ ϕ + ¯ pV ( ¯ ϕ )) (cid:3) Φ + 4 πG ( ˙¯ ϕ ˙ δϕ + ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ) . (A2)Therefore, we can obtain the following expressions: ddη ( ∇ Φ) = 3 H ¨Φ + [3 ˙ H + 6 H − πG ( ˙¯ ϕ + ¯ pV ( ¯ ϕ ))] ˙Φ+[12 H ˙ H − πG (2 ˙¯ ϕ ¨¯ ϕ + ˙¯ pV ( ¯ ϕ ) + ¯ p ˙¯ ϕV , ¯ ϕ ( ¯ ϕ ))]Φ+4 πG [ ¨¯ ϕ ˙ δϕ + ˙¯ ϕ ¨ δϕ + ˙¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ + ¯ pV , ¯ ϕ ( ¯ ϕ ) ˙ δϕ + ¯ p ˙¯ ϕV , ¯ ϕ ¯ ϕ ( ¯ ϕ ) δϕ ] , (A3) H ∇ Φ = 3 H ˙Φ + [6 H − πG H ( ˙¯ ϕ + ¯ pV ( ¯ ϕ ))]Φ + 4 πG H [ ˙¯ ϕ ˙¯ ϕ + ¯ pV , ¯ ϕ ( ¯ ϕ ) δϕ ] . (A4)In addition, we can also expressed the term 4 πG ˙¯ ϕ ∇ δϕ as following with the help of (54)4 πG ˙¯ ϕ ∇ δϕ = 4 πG ˙¯ ϕ [ δ ¨ ϕ + 2 H ˙¯ ϕ + ¯ pV , ¯ ϕ ¯ ϕ ( ¯ ϕ ) δϕ + 2¯ pV , ¯ ϕ ( ¯ ϕ )Φ − ϕ ˙Φ] . (A5)15ollecting all the above expressions (A3), (A4) and (A5), we can obtain the perturbationequation (58) by straightly calculating. Similarly, we can also derive equation (55) or (57)from the remaining two equations. Therefore, among these equations above only two ofthem are independent. [1] M. Bojowald, Loop quantum cosmology, Living Rev. Relativity 8, 11, (2005) [gr-qc/0601085].[2] M. Bojowald, Absence of Singularity in Loop Quantum Cosmology, Phys. Rev. Lett. 86 (2001)5227 [gr-qc/0102069].[3] A. Ashtekar, T. Pawlowski, and P. Singh, Quantum nature of the big bang: an analytical andnumerical investigation, Phys. Rev. D 73:124038, (2006) [gr-qc/0604013].[4] A. Ashtekar, T. Pawlowski, and P. Singh, Quantum nature of the big bang: improved dynam-ics, Phys.Rev.D74:084003,2006, [gr-qc/0607039].[5] A. Ashtekar, M. Bojowald and J. Lewandowski, Mathematical structure of loop quantumcosmology, Adv. Theor. Math. Phys. 7, 233 (2003) [gr-qc/0304074];M. Bojowald, G. Date and K. Vandersloot, Homogeneous loop quantum cosmology: The roleof the spin connection, Class. Quant. Grav. 21, 1253 (2004) [gr-qc/0311004];P. Singh and A. Toporensky, Big crunch avoidance in k = 1 loop quantum cosmology, Phys.Rev. D 69, 104008 (2004) [gr-qc/0312110];G. V. Vereshchagin, Qualitative approach to semi-classical loop quantum cosmology, JCAP0407, 013 (2004) [gr-qc/0406108];G. Date, Absence of the Kasner singularity in the effective dynamics from loop quantumcosmology, Phys. Rev. D 71, 127502 (2005) [gr-qc/0505002];G. Date, Absence of the Kasner singularity in the effective dynamics from loop quantumcosmology, Phys. Rev. D 71, 127502 (2005) [gr-qc/0505002];G. Date and G. M. Hossain, Genericity of big bounce in isotropic loop quantum cosmology,Phys. Rev. Lett. 94, 011302 (2005) [gr-qc/0407074];R. Goswami, P. S. Joshi and P. Singh, Quantum evaporation of a naked singularity, Phys.Rev. Lett. 96, 031302 (2006) [gr-qc/0506129].[6] M. Bojowald, The Early Universe in Loop Quantum Cosmology, J. Phys. Conf. Ser. 24 (2005)77, [gr-qc/0503020].
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