Under an appropriate change of the perturbation variable Lifshitz-Khalatnikov
propagation equations for the scalar perturbation reduce to d'Alembert
equation. The change of variables is based on the Darboux transform.Comment: LaTeX, 9 page

Golda, Zdzislaw A.

Woszczyna, Andrzej

Zawada, Karolina

English

arXiv

Under an appropriate change of the perturbation variable Lifshitz-Khalatnikov propagation equations for the scalar perturbation reduce to d'Alembert equation. The change of variables is based on the Darboux transform

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arXiv:gr-qc/0511040v1  8 Nov 2005Canonical gauge-invariant variables for scalarperturbations in synchronous coordinatesZdzis law A. Golda, Andrzej Woszczyna and Karolina ZawadaAstronomical Observatory, Jagellonian Universityul. Orla 171, 30–244 Krako´w, PolandAbstractUnder an appropriate change of the perturbation variable Lifshitz-Khalatnikov propagation equations for the scalar perturbation reduce tod’Alembert equation. The change of variables is based on the Darbouxtransform.1 IntroductionThe gauge-invariant perturbation theories eﬃciently eliminate non-physical perturbations, yet they provide the propagation equations in noncanon-ical form — diﬀerent for each theory. In this case, pure artefacts of the choiceof the reference system, gauge or the perturbation variable, are likely to beconfused with new dynamical phenomena yet unknown in the laboratory-scalephysics. This particularly concerns the large scale limit, where some “non-oscillatory behaviour” outside the particle horizon is commonly expected.In this paper we show that the deﬁnition of the gauge-invariant variables stillpossesses a freedom to chose the canonical variables, i.e. variables which satisfyd’Alembert equation in its standard form. To show that we have used the classi-cal Lifshitz-Khalatnikov formalism, which after being appropriately extended isequivalent to other “manifestly” gauge invariant descriptions. A systematic con-struction of the gauge-invariant canonical variables is provided. Consequently,the scalar perturbation propagates like a massless scalar ﬁeld in the Roberson-Walker space-time, and therefore there is a close analogy between the pertur-bation theory (the acoustics of the expanding universes) and the ﬁeld theory inthe curved space-time. Propagation of the sound waves in the early universemay be also considered as an example of the acoustic geometry in the sense ofUnruh [1].The paper is organized as follows: In the second Section we remind themethod of elimination of spurious modes in the Lifshitz-Khalatnikov theory [2].This Section is a generalization of the procedure given in [3] to the case ofarbitrary space curvature and arbitrary density-dependent pressure p = p(ǫ).1In Section 3 we present the general method of construction of gauge-invariantquantities in the synchronous system of reference. We show that the procedureis not unique, and therefore, there exist vast classes of physically relevant gauge-invariant variables. The remaining freedom to choose between them may be usedto better describe the perturbation dynamical properties, or to construct moreadequate observables. A speciﬁc choice depends on the researcher motivation,and on the character of the problem to be solved. Finally in Section 4 we adoptthe previously deﬁned methods to obtain a canonical form of the perturbationequations — the d’Alembert equation.2 Synchronous system of reference — techniquesof reduction of the gauge freedomConsider the Robertson-Walker universe of arbitrary space curvature (K =−1, 0,+1)gµν = a2(η) diag[−1, 1, sin2(√Kχ)K,sin2(√Kχ)Ksin2 θ], (1)with the hydrodynamic energy-momentum tensorT µν = (ǫ+ p)uµuν + p gµν (2)and the barotropic equation of state p = p(ǫ). We investigate small perturba-tions δgµν of the metric (1). We limit ourselves to scalar (density) perturbations,therefore, δgµν is deﬁned by two scalar functions λ(k)(η) and µ(k)(η)δgµ0 = 0, (3)δg nm =∑k(λ(k)(η)Pnm + µ(k)(η)Qnm)+ c.c., (4)where P nm and Qnm stand for scalar harmonics [2],Q nm =13Q(k)(x)δnm , (5)P nm =1k2 −KQ;n;m +Qnm , (6)Q(k)(x) is the complex solution of the Helmholtz equation Q(k)(x);m;m = −(k2−K)Q(k)(x), x = {χ, θ, ϕ}, and the amplitudes λ(k)(η) and µ(k)(η) satisfy thesystem of two second order equations convoluted to each otherλ′′(k)(η) + 2a′(η)a(η)λ′(k)(η) −k2 −K3[λ(k)(η) + µ(k)(η)]= 0, (7)µ′′(k)(η) +[2 + 3c2s(η)] a′(η)a(η)µ′(k)(η)+k2 − 4K3[1 + 3c2s(η)] [λ(k)(η) + µ(k)(η)]= 0. (8)2cs(η) denotes the sound velocity: cs(η) =√p′(η)/ǫ′(η) 6= 0. Then the densitycontrast1 δ = δǫ/ǫ is a linear combination [2]δ(η,x) =∑kδ(k)(η)Q(k)(x) + c.c., (9)andδ(k)(η) =A(k)3ǫ(η)a2(η)[(k2 − 4K)(λ(k)(η) + µ(k)(η)) + 3a′(η)a(η)µ′(k)(η)], (10)where A(k) are arbitrary complex numbers. The system (7–8) deﬁnes the fourdimensional phase-space, which means that the system possesses two physi-cal and two gauge degrees of freedom. Pure gauge solutions are known, andfor λ(k)(η) they respectively read [2]G1(η) = 1, (11)G2(η) = −(k2 −K)∫1a(η)dη. (12)Diﬃculties with ﬁxing the gauge freedom have stimulated elaboration of thegauge-invariant theories [4, 5, 6, 7, 8, 9]. After the famous Bardeen’s paper [10]these theories were intensively developed for almost two decades. Yet, the orig-inal Lifshitz-Khalatnikov formalism, when appropriately extended, provides anequally good description of inhomogeneities. The procedure is as follows: wereduce (7–8) system to the single fourth-order equation for λ(k)(η).λ(k)(4)(η) + a(η)H(η)[4 + 3c2s(η)]λ(k)(3)(η)+{−5K + a2(η)[2(ǫ(η)3− p(η))+(9H2(η) − ǫ(η)) c2s (η)]+(k2 −K) c2s (η)}λ′′(k)(η) + a(η)H(η) [−a2(η) [ǫ(η) + 3p(η)]+ 2a2(η)H2(η)[1 + 3c2s (η)]+(k2 −K) c2s (η)]λ′(k)(η) = 0, (13)where H(η) stands for the Hubble parameter H(η) = a′(η)/a2(η). The knowl-edge of the gauge solutions (11–12) enables one to extract the gauge space fromthe space of all solutions. First, we write the solutions in the formλ(k)(η) = f1λ1(k)(η) + f2λ2(k)(η) + g1G1(k)(η) + g2G2(k)(η) (14)with explicitly separated linear subspace g1G1(k)(η) + g2G2(k)(η) carrying allthe gauge freedom (g1, g2 two arbitrary coeﬃcients). Subsequently, we adoptthe Darboux transform2 [12] of λ(k)(η) to express the two remaining (physical)1In more rigorous notation one should write δ = δǫ/ǫ0, where ǫ0 denotes background,unperturbed energy density. For the sake of simplicity we skip the subscript “0” in formulas.2Darboux transform of a function f(x) is defined as f̂(x) = A(x)f(x)+B(x)f ′(x) (comparealso formula (18) in section 3) with arbitrary x-dependent coefficients A and B. The sametransform is extensively exploited in the soliton theory [11].3degrees of freedom. A new perturbation variable B(k)(η) appearsB(k)(η) = a(η)dd η[(dd ηλ(k)(η)G1(k)(η))(dd ηG2(k)(η)G1(k)(η))−1]= a(η)dd ηdd ηλ(k)(η)dd ηG2(k)(η).(15)On strength of (13) the variable B(k)(η) satisﬁes the second order equationB′′(k)(η)−[1− 3c2s(η)]a(η)H(η)B′(k)(η)−[(13+ c2s (η))a2(η)ǫ(η)− (k2 −K) c2s (η)]B(k)(η) = 0. (16)Equation (16) does not contain gauge the coeﬃcients, therefore, the variableB(k)(η) is gauge-invariant. Now, the function λ(k)(η) can be found by the inverseDarboux transform of the two linearly independent solutions to equation (16)3λ(k)(η) = f1(K − k2)∫1a(η)[∫B1(k)(η)a(η)dη]dη+f2(K − k2)∫1a(η)[∫B2(k)(η)a(η)dη]dη + g1G1(k)(η) + g2G2(k)(η). (17)On can insert λ(k)(η) to (7) and solve it algebraically to obtain the secondunknown function µ(k)(η). As a result, by ﬁxing four numbers {f1, f2, g1, g2} oneuniquely determines the metric correction δgµν . The metric correction is notgauge-invariant and cannot be such in any perturbation formalism. Quantitiescontaining time integrals of B(k)(η) would exhibit similar properties, while thosecontaining B(k)(η) and derivatives are obviously gauge-invariant.3 Families of gauge-invariant variables in thesynchronous reference systemThe metric perturbation can be written in the form (17) if the solutions of theequation (16) are explicitly known. However, even in the case when these so-lutions are not known one may employ the equation (16) to “convert” somenon-gauge-invariant perturbation variables into gauge-invariant ones. This par-ticularly refers to perturbation measures based on hydrodynamic quantities,since they involve the metric derivatives and are free of the metric elementsthemselves. The procedure is following:1) For the non-gauge-invariant perturbation variable X we take the linear com-bination of X and its time derivative X ′ (the Darboux transform)X̂(η,x) = c1(η)[∂∂ηX(η,x) + c2(η)X(η,x)], (18)3The integration constants in the inverse Darboux transform can be set arbitrary, becausethe gauge freedom is already guaranteed by free choice of the coefficients g1 and g2.4which means thatX̂(η,x) =∑kX̂(k)(η)Q(k)(x) + c.c., (19)andX̂(k)(η) = c1(η)[dd ηX(k)(η) + c2(η)X(k)(η)]. (20)2) We apply the equation (16) to formally express the coeﬃcients X̂(k)(η) in theformX̂(k)(η) = F1(η)dd ηB(k)(η) + F2(η)B(k)(η) + F3(η)∫B(k)(η)a(η)dη. (21)3) We look for c1(η) and c2(η) such that X̂(k)(η) contains solely function B(k)(η)and its derivatives but not integrals. (Integral over η in (21) introduces arbi-trary constant c(k), which depends on the wave vector k. The variable X̂(η,x)(formula (19)) would contain then terms being arbitrary functions of the spacecoordinate x. In this way integration constant c(k) restores one degree of thegauge freedom.) The relevant condition F3(η) = 0 leads to a linear equationfor c2(η), and leaves c1(η) free. Therefore, the new gauge-invariant variableX̂(η,x) is actually a c1(η)-dependent family of variables, where c1(η) is an ar-bitrary function of time. Below we show some examples.3.1 The density contrastOn strength of (9), (7–8) and (17) the density contrast δ(k)(η) can be expressedby use of the gauge invariant function B(k)(η).δ(k)(η) = − 3H(η)a3(η)ǫ(η)B′(k)(η) −1a2(η)[k2 −Ka2(η)ǫ(η)− 1]B(k)(η)+32[p(η) + ǫ(η)]H(η)ǫ(η)∫B(k)(η)a(η)dη. (22)The contrast δ(k)(η) is not gauge-invariant, as it is well known [10]. The gaugefreedom is caused by the last term in (22) containing the integral of B(k)(η).Let us introduce a new variable — the “modiﬁed density contrast”δ̂(k)(η) = c1(η)[dd ηδ(k)(η) + c2(η)δ(k)(η)]. (23)With the aid of (16) one obtainsδ̂(k)(η) = F1(η)dd ηB(k)(η) + F2(η)B(k)(η) + F3(η)∫B(k)(η)a(η)dη, (24)5where the function F3(η) reads4F3(η) =32c1(η)H(η)[1 +p(η)ǫ(η)] [dd ηln(1a(η)ǫ′(η)ǫ(η))+ c2(η)]. (25)By settingc2(η) = − dd ηln(1a(η)ǫ′(η)ǫ(η)), (26)one eliminates the integral of B(k)(η) (F3(η) = 0), hence δ̂(k)(η) becomes a gaugeindependent quantityδ̂(k)(η) = c1(η)ǫ′(η)a(η)ǫ(η)dd η[a(η)ǫ(η)ǫ′(η)δ(k)(η)]. (27)This variable is still deﬁned up to the factor c1(η) being an arbitrary functionof time.3.2 The expansion rate contrastIdentically we proceed with inhomogeneities in the expansion rate5 δϑ(η,x).δϑ(η,x) =∑kδϑ(k)(η)Q(k)(x) + c.c., (28)whereδϑ(k)(η) =1a3(η)[k2 −Ka2(η)(p(η) + ǫ(η))− 32]B′(k)(η)− H(η)a2(η)[k2 −Ka2(η)(p(η) + ǫ(η))− 32]B(k)(η)+32[12(p(η) + ǫ(η)) − Ka2(η)]∫B(k)(η)a(η)dη . (29)We introduce δ̂ϑ(k)(η) deﬁned as a linear combinationδ̂ϑ(k)(η) = c1(η)[dd ηδϑ(k)(η) + c2(η)δϑ(k)(η)], (30)and eliminate the integral of B(k)(η). We ﬁnally obtainδ̂ϑ(k)(η) = c1(η)[dd ηδϑ(k)(η) −dd ηln[H ′(η)a(η)]δϑ(k)(η)]. (31)4F1(η) and F2(η) are irrelevant to the problem discussed here, therefore we skip writingthem explicitly.5ϑ ≡ uµ;µ is the expansion [13].63.3 Other variablesBy employing the same procedure one may deﬁne gauge-invariant variableswhich combine inhomogeneities in Hubble ﬂow with inhomogeneities of the en-ergy densityr̂(η,x) = c1(η)[δǫ(η,x)− 13ǫ′(η)H ′(η)δϑ(η,x)]. (32)It is clear that functions F (δ̂, δ̂ϑ, r̂, . . . , ǫ,H, . . .) of the gauge-invariant variables{δ̂, δ̂ϑ, r̂, . . .} and the parameters of the background universe {ǫ,H, . . .} are alsogauge-invariant. None of these variables is preferred by the perturbation theoryitself. The choice of the variable depends on its physical meaning and usefulness,its relation to other physical concepts and theories, and eventually on the abilityto construct relevant observables.4 Perturbation equations in the canonical formIn all the examples above we constructed the gauge-invariant variables which re-fer to the synchronous system of reference. The function c2(η) was uniquely de-termined for each δ̂(η), ϑ̂(η) and X̂(η) by the demand of their gauge-invariance.The freedom to make an arbitrary choice of the time-dependent factor c1(η) stillremains, and therefore, we are now able to construct gauge-invariant variableswith particular dynamical properties. Consider the variable δ̂(η) with c1(η)chosen asc1(η) = a2(η)H2(η)ǫ(η)ǫ′(η). (33)We obtain the gauge-invariant variable6δ̂(k)(η) = a(η)H2(η)dd η[a(η)ǫ(η)ǫ′(η)δ(k)(η)], (34)which satisﬁes the propagation equation of the formδ̂′′(k)(η) +[2a′(η)a(η)− c′s(η)cs(η)]δ̂′(k)(η) + c2s (η)(k2 −K)δ̂(k)(η) = 0, (35)with the function a(η) deﬁned asa(η) = a(η)√1cs(η)ǫ(η) + p(η)3H2(η). (36)Now we introduce the new time parameter dξ = cs(η)dη (acoustic conformaltime [14]). After this reparametrization the propagation equation reduces tod’Alembert equation∇µ∇µ δ̂(ξ,x) = 0 (37)6Please note the misprint in paper [14] formula (7), already corrected in gr-qc/0302091.7in the Robertson-Walker space-time with coordinates {ξ, χ, θ, ϕ} and the metricformgµν = a2(ξ) diag[−1, 1, sin2(√Kχ)K,sin2(√Kχ)Ksin2 θ], (38)Therefore, the modiﬁed density contrast δ̂(ξ,x) propagates in the Robertson-Walker space-time with the scale factor a(ξ) in the same manner as minimallycoupled scalar ﬁeld in the Robertson-Walker space-time with the scale factora(η) [15]. All the classical results obtained in the ﬁeld theory on the curvedspace-times apply to the density perturbations in the expanding universe. Onecan introduce the LagrangianL = 12gµν δ̂,µδ̂,ν (39)for ﬁeld δ̂(ξ,x) and derive the equation of motion (37) by use of the Euler-Lagrange equations∇µ ∂∂δ̂,µL− ∂∂δ̂L = 0. (40)An appropriate change of the perturbation variables [14] reduces the propa-gation equation of any gauge-invariant theory to the form of the equation (16).By use of the procedure discussed above one can construct canonical variables,and as a consequence, each of these formalism can be expressed in the Lagrange-Hamilton language.The authors thank L.M. Soko lowski for reading the manuscript and for con-structive critics.References[1] W. Unruh, Phys. Rev. Lett. 46, 1351 (1981),W. Unruh, Phys. Rev. D 51, 2827 (1995),M. Visser, Class. Quantum Grav. 15, 1765 (1998),N. Bilic, Class. Quantum Grav. 16, 3953 (1999).[2] E.M. Lifshitz, I.M. Khalatnikov, Adv. Phys. 12, 185 (1963).[3] Z.A. Golda, A. Woszczyna, J. Math. Phys. 42, 456 (2001).[4] D.W. Olson, Phys. Rev. D 14, 327 (1976).[5] A. Woszczyna, A. Kulak, Class. Quantum Grav. 6, 1665 (1989).[6] M. Bruni, P.K.S. Dunsby, G.F.R. Ellis, Astrophys. J. 395, 34 (1992).[7] D. Lyth, E. Stewart, Astrophys. J. 361, 343 (1990).8[8] R. Brandenberger, R. Kahn, and W. Press, Phys. Rev. D 28, 1809 (1983).[9] V. F. Mukhanov, H.A. Feldman, R.H. Brandenberger, Phys. Rep. 215, 203(1992).[10] J.M. Bardeen, Phys. Rev. D 22, 1882 (1980).[11] G.L. Lamb, Elements of Soliton Theory, John Wiley & Sons, New York1980.[12] D. Zwillinger, Handbook of Differential Equations, Academic Press, Boston1989.[13] S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time,Cambridge University Press, London 1973.[14] Z.A. Golda, A. Woszczyna, Phys. Lett. A 310, 357 (2003).[15] N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space, CambridgeUniversity Press, Cambridge 1982.9

Elements of Soliton Theory,

Handbook of Differential Equations,

Quantum Fields in Curved Space,

The Large Scale Structure of Space-Time,

Canonical gauge-invariant variables for scalar perturbations in synchronous coordinates

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Canonical gauge-invariant variables for scalar perturbations in synchronous coordinates

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