Quantum-to-classical transition of primordial cosmological perturbationsin de Broglie-Bohm quantum theory: The bouncing scenario

Nelson Pinto-Neto and Grasiele SantosICRA-Centro Brasileiro de Pesquisas Físicas–CBPF, rua Xavier Sigaud, 150, Urca,

CEP22290-180 Rio de Janeiro, Brazil

Ward StruyveDepartments of Mathematics and Philosophy, Rutgers University, Hill Center, 110 Frelinghuysen Road,

Piscataway, New Jersey 08854-8019, USA(Received 16 October 2013; published 14 January 2014)

In a previous work we have exhibited a clear description of the quantum-to-classical transition ofcosmological quantum fluctuations in the inflationary scenario using the de Broglie–Bohm quantumtheory. These fluctuations are believed to seed the small inhomogeneities, which are then responsible forthe formation of large scale structures. In this work we show that using the de Broglie–Bohm theory, it isalso possible to describe the quantum-to-classical transition of primordial perturbations which takes placearound a bouncing phase, even if the latter is caused by quantum effects due to the quantization of thebackground geometry.

DOI: 10.1103/PhysRevD.89.023517 PACS numbers: 98.80.Qc, 03.65.Ta, 04.60.Ds

I. INTRODUCTION

According to the usual accounts of structure formation, aperiod of inflation drove the early universe to a veryhomogeneous and isotropic state in which ordinary matterhas been dispersed away, with only quantum vacuumfluctuations remaining [1–5]. These fluctuations thenresulted into classical perturbations of the energy densityand then started to form structures (such as stars, galaxiesand clusters of galaxies) through gravitational instabilities.(There are also models, known as warm inflation models,according to which those classical inhomogeneities origi-nate from thermal fluctuations [6,7]). The same picture isadvocated in bouncing models [8]. In these models, theinitial singularity is absent: the universe goes through acollapsing phase from a very large volume up to aminimum size (at which point some new physics takesplace, like quantum effects, nonminimal couplings, and soon [8]), and then starts to expand, reaching the standardradiation-dominated expansion phase before nucleosynthe-sis. In the far past of the contracting phase, the universe isassumed to be very big and dust dominated, with a verysmall dust energy density. Inhomogeneities in the energydensity are expected to dissipate so that only quantumvacuum fluctuations around a uniform distribution remain.Similarly as in the inflationary scenario, structures in theUniverse are thought to originate from those quantumfluctuations.There has been a lot of discussion in the literature about

the possible mechanisms behind the transition fromcosmological quantum fluctuations to the small classicalinhomogeneities which seed the structure formation [9–21].Asuccessful explanationneeds todealwith themeasurement

problem [1,2,4]. Namely, the initial quantum state is homo-geneous and isotropic and is supposed to result in classicalfluctuations that are not symmetric. The Schrödinger evolu-tion preserves the translational and rotational symmetry.Hence, according to standardquantum theory, this symmetrycan only be broken through collapse of the wave function.This collapse is supposed to happen uponmeasurement. Butin the early universe there is no measurement device orobserver that could cause sucha collapse.Evenworse,wearedealing with a description of the whole universe and hencethere is not even place for an externalmeasurement device orobserver. All structures, includingmeasurement devices andobservers, are supposed to emerge from the primordialfluctuations themselves. As such, to address the problemof the quantum-to-classical transition of fluctuations, weneed to use an alternative to quantum theory that is free of themeasurement problem.In a recent paper [22], we have considered the de

Broglie–Bohm theory [23–25] to study the quantum-to-classical transition of these fluctuations for the case ofinflationary theory (see [26] for an earlier study). We haveshown that this transition can be explained very easily andnaturally. The de Broglie–Bohm theory solves the meas-urement problem by postulating an actual field configura-tion, which is guided in its motion by the wave function. Inthe cosmological scenario, this actual field configuration isneither homogeneous nor isotropic, so it does not share thesymmetries of the quantum state, and it behaves classically(i.e., obeys the classical field equations) when expected. Inour analysis we did not appeal to decoherence. If there issuitable decoherence, then our results still hold. In otherapproaches that solve the measurement problem, such asthe many worlds theory, an explanation of the transition

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requires suitable decoherence. Much work has been per-formed on studying possible sources for the decoherence(it could come, for instance, from the coupling betweendifferent modes, or from interactions with other matterfields), as well as possible time scales at which it couldoccur, see e.g. [15] and references therein. However, itseems fair to say that there are no conclusive resultsyet concerning the source and time scales of thedecoherence [3].Other approaches to the measurement problem worth

mentioning are dynamical collapse theories. Various col-lapse models are being developed in order to account forthe quantum-to-classical transition of primordial fluctua-tions (see e.g. [16–18,20,21] and references therein).In this paper, we address the problem of the quantum-

to-classical transition of cosmological perturbations forbouncing scenarios [27]. In inflationary models, fromthe perspective of the de Broglie–Bohm quantum theory,the fast suppression of the decaying mode and the con-sequent dominance of the growing mode ensured thequantum-to-classical transition of the perturbations.However, in bouncing models growing and decayingmodes are interchanged at the bounce. Hence, a priori,it is not clear when the dominance of one mode over theother becomes really effective, and it is not trivial to knowwhen the perturbations will begin to behave classically.Moreover, bounces can happen due to quantum gravita-tional effects [28], and this could also disturb theclassicality of the perturbations. We will show that, inspite of these facts, in general bouncing models, theperturbation modes begin to behave classically beforethe bounce takes place when the physical scale of theperturbations has become larger than the curvature scale ofthe background and the contraction has lasted a sufficientlylong period of time. In fact, this is the case for allperturbations of cosmological interest. The calculationswere done, again, in the context of the de Broglie–Bohmquantum theory.In the next section we will summarize the classical

dynamics of cosmological perturbations in general bounc-ing models. In Sec. III, we will consider the quantumdescription of the perturbations, in the context of the deBroglie-Bohm theory, and show how the quantum-to-classical transition takes place. In Sec. IV a particularexample of a quantum bouncing model is presented. Weend up with the conclusions in Sec. V.

II. LINEAR COSMOLOGICAL PERTURBATIONSIN GENERAL BOUNCING MODELS

Let us first consider the classical description of cosmo-logical perturbations. In the next section we will turn to thequantum description.The perturbations are considered in a background

Friedmann-Robertson-Walker model, with scale factor aand uniform total matter distribution with density ρ and

pressure p, and are described by the Mukhanov-Sasakivariable vðη;xÞ, which combines both fluctuationsof the matter and metric. The parameter η is conformaltime defined by adη ¼ dt, with t being cosmic time.The Lagrangian for the Mukhanov-Sasaki variable, whichcan be derived from the Einstein-Hilbert action, is given by

Lv ¼Z

dx1

2

�v02 þ

�z0

z

�2

v2 − c2sδij∂iv∂jv − 2z0

zvv0

�

(1)

(assuming natural units c ¼ ℏ ¼ 1), where the primesdenote derivatives with respect to conformal time and

z ¼ffiffiffiβ

pxHcs

; β ¼ 3

2

8πG3

a2ðρþ pÞ; c2s ¼dpdρ

:

(2)

H ¼ a0=a is the conformal Hubble function, which relatesto the Hubble function H ¼ a−1da=dt through H ¼ aH,and x ¼ a0=a is the redshift function. Subscripts 0 refer topresent-day values. The Lagrangian yields the followingequations of motion for the Fourier modes vkðηÞ:

v00k þ�c2sk2 − z00

z

�vk ¼ 0: (3)

Defining Ω ¼ ρ=ρc, where ρc is the critical densitytoday, and using the energy conservation equation

dρdt

þ 3Hðρþ pÞ ¼ 0 ⇒dρdx

¼ 3ðρþ pÞx

; (4)

we obtain

β ¼ 1

2xR2H

dΩdx

; z2 ¼ 1

2c2sxwdΩdx

;

c2s ¼x3

ddx

ln

�1

x2dΩdx

�; (5)

where RH ¼ 1=ða0H0Þ is the present value of the comovingHubble radius.In the case where the background matter is a single fluid

with p ¼ wρ; we have Ω ¼ Ω0x3ð1þwÞ and

c2s ¼ w; z2 ¼ 3ð1þ wÞ2wx2

; (6)

and Eq. (3) reduces to

v00k þ�wk2 − a00

a

�vk ¼ 0: (7)

The general solution of the mode equation (3) can beformally expanded in powers of k2 as [29]

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vkz ¼ A1;k

�1 − c2sk2

Rηηi

dη̄z2ðη̄Þ

Rη̄ηiz2ð ¯̄ηÞd ¯̄ηþ � � �

�þ

A2;k

Rηηi

dη̄z2ðη̄Þ

�1 − c2sk2

Rη̄ηiz2ð ¯̄ηÞd ¯̄η R ¯̄η

ηid ¯̄̄η

z2ð ¯̄̄ηÞ þ � � ��;

(8)

where we have presented the terms up to order Oðk2Þ. ηirefers to some initial time. The coefficients A1;k and A2;kare two constants, determined by the field vk and its firstderivative at time ηi, namely, A1;k ¼ vkðηiÞ=zðηiÞ andA2;k ¼ v0kðηiÞzðηiÞ. They are roughly of the same orderof magnitude [30]. We will be interested in the situationwhere c2sk2 ≪ z00=z, because the modes of physical interesttoday, even starting with wavelengths much smaller thanthe curvature scale in the far past of the contracting phase,will certainly become bigger than the curvature scale whenthe universe was still very large before the bounce. Hence,in this regime, we will take only the first terms of theseries above.This description of the perturbations is valid, in the case

where entropy perturbations are negligible, in the con-tracting and expanding phases when the dynamics is givenby the General Relativity Einstein’s equations, and throughthe bounce itself in the case of the quantum bouncepresented in [28], which will be considered here inSec. IV. For general bounces, it is not clear that solution(8) is valid through the bounce; neither do we have anyparticular analytic solution in order to evaluate it away fromthe bounce, as was done in the case of the quantum bounceof [28]. However, if the bounce is short enough, an estimateof Eq. (8) away from the bounce, where General Relativityis valid, will be sufficient to evaluate the orders ofmagnitude of the amplitudes, assuming that a short bouncedoes not change the mode evolution too much.In order to analyze the quantum-to-classical transition,

we focus on the term

A2;k

Zη

−∞dη̄z̄2

; (9)

which appears in the solution (8) (we are assuming that thecontracting phase begins at a very large negative conformaltime, which we take to be ηi → −∞, and that the bounceoccurs for η near zero). This term grows with time. We canwrite

Zη

−∞dη̄

z2ðη̄Þ ¼�B −

Z∞

η

dη̄z2ðη̄Þ

�; (10)

where

B ¼Z

∞

−∞dηz−2 (11)

is a constant. For the case where the bounce is dominatedby a single fluid with equation of state parameter wq of the

order of one [31], this constant was evaluated in Ref. [27]and reads

B ≈4xb

3ð1 − wqÞEðxbÞz2ðxbÞ; (12)

where E ¼ H=H0 ¼ffiffiffiffiffiffiffiffiffiffiΩðxÞp

, and a subscript b refers to thevalues of the physical quantities at the bounce. It must beunderstood that, although evaluated at xb, the functionsEðxÞ and z2ðxÞ in Eq. (12) are the usual general relativisticexpressions for them which are valid just before the bouncebut maybe not through the bounce itself.Let us now provide an estimate for the constant B.

For realistic bounces, occurring at energy scales biggerthan the nucleosynthesis energy scale, we have thatxb ¼ a0=ab ≫ 1010. As wq is of order 1, and assumingthat zðxbÞ can be approximated by the value it takes in asingle fluid dominated bounce, i.e., that it is given by theexpression (6), then B will be very big: B ≫ 1010. Hence,the solution for the mode functions vk near the bounce(where η is near zero) is given by

vk ≈ ½A1;k þ A2;kB�zðηÞ − A2;kzðηÞZ

∞

η

dη̄z2ðη̄Þ

≈ A2;kzðηÞ�B −

Z∞

η

dη̄z2ðη̄Þ

�: (13)

In the last approximation we used that A1 and A2 areassumed to be roughly of the same order, and that theintegral is roughly equal to B=2.Equation (13) is the main result of this section and will

be used in the next one to achieve the classical limit.Remember that it is valid for perturbation modes for whichc2sk2 ≤ z00=z (i.e., when their physical wavelengths aremuch larger than the curvature scale of the contractingbackground) and in case the background space hasalready contracted enough [which was important inderiving (12)].

III. THE DE BROGLIE-BOHM–APPROACHTO PERTURBATIONS

We now consider a quantum mechanical treatment of theMukhanov-Sasaki variable, keeping the backgroundclassical. The classical equation looks very similar to thatof inflationary scenario. Therefore we can proceed in asimilar way as in [22].The Lagrangian (1), which formally looks like that of a

free scalar field with a time-dependent mass, can bestraightforwardly quantized. Assuming a product wavefunctional Ψ ¼ Πk∈R3þΨkðvk; v�k; ηÞ, the Schrödingerequation for each wave function Ψk satisfies

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i∂Ψk

∂η ¼�− ∂2

∂v�k∂vkþc2sk2v�kvk−iz0

z

� ∂∂v�kv

�kþvk

∂∂vk

��Ψk:

(14)

In the de Broglie–Bohm approach, there is also an actualfield vðη;xÞ whose Fourier modes satisfy the guidanceequations

v0k ¼ ∂Sk∂v�k þ z0

zvk; v�0k ¼ ∂Sk

∂vk þ z0

zv�k; (15)

where Ψk ¼ jΨkj exp ðiSkÞ.Whenever z00=z is negligible with respect to c2sk2, which

happens in the far past in the contracting phasefor the modes of physical interest, vacuum initial conditionscan be imposed on the wave function. This yields thesolution

Ψk ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2π

p jfkðηÞjq exp

�− 1

2jfkðηÞj2jvkj2 þ i

��jfkðηÞj0jfkðηÞj

− z0

z

�jvkj2 −

Zη d~η2jfkð~ηÞj2

��; (16)

with fk a solution to the classical mode equation (3) that isisotropic (so that A1 and A2 in (8) depend only on k) andthat satisfies fkðηiÞ ¼ 1=

ffiffiffiffiffi2k

pwith jηij ≫ 1 (so that

A1;k ≡ A1;k ¼ 1=ffiffiffiffiffi2k

p). The quantum state Ψ is homo-

geneous and isotropic.For this quantum state, the solutions to the guidance

equations read

vkðηÞ ¼ vkðηiÞjfkðηÞjjfkðηiÞj

: (17)

Note that this result is independent of the precise form offkðηÞ. Since fk is a solution of the classical equation ofmotion (3), the classical limit will be achieved wheneverjfkðηÞj, as a function of time, is proportional to fkðηÞ.Around the bouncewe have that, for large wavelengths, i.e.,c2sk2 ≪ z″=z,

fkðηÞ ≈ A2;kzðηÞ�B −

Z∞

η

dη̄z2ðη̄Þ

�; (18)

[see Eq. (13)] and thus,

fkðηÞ ∝ jfkðηÞj:

Hence,

vkðηÞ ∝ fkðηÞ; (19)

which means that the perturbations (17) are evolvingclassically.

IV. EXAMPLE OF A QUANTUM BOUNCE

As an example of a bounce solution, we present a modelin which a bouncing phase occurs due to quantum effectsoriginating from a quantized background geometry. Themodel, described in detail in [28], describes a perfect fluidwith equation of state p ¼ wρ modeling the matter source,in a flat background, quantized following the Wheeler-DeWitt prescription. Considering only scalar perturbations,the Wheeler-DeWitt equation for Ψ½a; v; T� reads

i∂Ψ∂T ¼ 1

4að3w−1Þ=2 ∂

∂a�að3w−1Þ=2 ∂

∂a�Ψ

þ −a3w−12

Zdx

δ2Ψδv2

þ a3wþ1w2

Zdxv;iv;iΨ;

(20)

where the scale factor a is the only remaining gravitationaldegree of freedom of the background, and T is the fluiddegree of freedom used to define time. The perturbationsare described by the variable v, which is the gauge-invariantMukhanov-Sasaki variable.We assume a product wave function of the form

Ψ½a; v; T� ¼ Ψð0Þ½a; T�Ψð2Þ½a; v; T�; (21)

where Ψð0Þ satisfies

i∂Ψð0Þ∂T ¼ að3w−1Þ=2

4

∂∂a

�að3w−1Þ=2 ∂

∂a�Ψð0Þ: (22)

From the above equation we get the following continuityequation for ρða; TÞ ¼ að1−3wÞ=2jΨð0Þða; TÞj2:

∂ρ∂T − ∂

∂a�a3w−12

∂S∂a ρ

�¼ 0; (23)

where S is the phase of Ψð0Þ. Using the de Broglie–Bohmapproach, we postulate an actual scale factor that obeys theguidance equation

dadT

¼ − a3w−12

∂S∂a : (24)

Using a Gaussian with widthffiffiffiffiffiffiffiffiffiffiTc=2

pas the initial wave

function (see Ref. [28]), integration of this equation yields

aðTÞ ¼ ab

�1þ

�TTc

�2� 1

3ð1−wÞ; (25)

where ab is an integration constant.

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Changing the description to conformal time according tothe relation dη ¼ ½aðTÞ�3w−1dT, we obtain the followingequation for the wave functional of the perturbations

i∂ ~Ψð2Þ∂η ¼ 1

2

Zdx

�− δ2

δv2þ wv;iv;i

− ia0

a

�δ

δvvþ v

δ

δv

��~Ψð2Þ; (26)

where, as before, the primes denote derivative with respectto η, aðηÞ is the solution of the guidance equation (24), andthe wave functional ~Ψð2Þðv; ηÞ is related to Ψð2Þða; v; TÞthrough the conditional wave function Ψð2ÞðaðηÞ; v; TðηÞÞ(see again [28] for details). If we decompose vðη;xÞ in itsnormal modes and assume a product wave functional~Ψð2Þ ¼ Πk∈R3þΨkðvk; v�k; ηÞ, we get the same equationas (14), with zðηÞ replaced by aðηÞ.In this case, the equation of motion for the variable fkðηÞ

defined in Eq. (16) (with z replaced by a) is identical toEq. (7), except for the fact that the quantity a is now a givenfunction of time, the Bohmian trajectory shown in (25).Therefore, solution (8), with z replaced by a and c2s by w, isvalid through the bounce itself and the constant B is givenexactly by

B ¼ 8w9ð1 − w2Þ ffiffiffiffiffiffi

Ω0

p x3ð1−wÞ=2b : (27)

This exact calculation is in accordance with the estimate(12). As before, with w of the order one, B is very large andEqs. (13) and (19) are valid, and the quantum-to-classicaltransition takes place. (Note that in this case, the equation

of state parameter wq of the fluid dominating the bounce isjust w.)

V. CONCLUSIONS

Using the de Broglie–Bohm theory, we have establishedthe quantum-to-classical transition of primordial perturba-tions for a large class of bouncing models. We alsoconsidered the particular example of a quantum bouncedescribed in [28], for which there is an analytical solutionfor perturbations during the bounce.While the wave function of the perturbations is homo-

geneous and isotropic, the actual perturbation vðx; ηÞ is notsymmetric. It is a superposition of A1;k and A2;k modes. Aswe showed in Sec. III, during the bounce the A1;k modebecomes negligible compared to the A2;k mode [seeEq. (13)], so that the mode vkðηÞ and hence the fieldvðx; ηÞ behave classically.As in the inflationary case, we did not invoke

decoherence. If at some stage in this transition there isdecoherence in the field basis, this will not destroy theclassical behavior of the fields.

ACKNOWLEDGMENTS

G. B. Santos would like to thank Faperj of Brazil forfinancial support. N. Pinto-Neto would like to thank CNPqof Brazil for financial support. W. Struyve acknowledgesthe support of a grant from the John Templeton Foundation.The opinions expressed in this publication are those of theauthors and do not necessarily reflect the views of the JohnTempleton Foundation.

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[30] In fact, as A1 and A2 depend on k, this assertion depends onthe scale we are talking about. For an almost scale-invariantspectrum of cosmological perturbations, the A2 term islarger than the A1 term for all scales of cosmological interest(see Ref. [28]), which enforces the argumentation describedbelow for the transition of quantum-to-classical behavior inbouncing models. Only for very short wavelengths can theA1 term be bigger than A2.

[31] We added the subscript q in the equation of stateparameter wq of the fluid dominating the bounce todistinguish it from the equation of state parameter w ofthe dust fluid (like dark matter) which dominated theUniverse at the beginning of the contracting phase,when the Universe was very large. While w ≈ 0 (whichleads to an almost scale-invariant spectrum—which isobserved—for wavelengths that become longer than thecurvature scale during the contracting phase [28]), wehave that wq is of the order of one. For example, in thecase of radiation wq ≈ 1=3 or in the case of stiff matterwq ≈ 1.

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