Bouncing models and inflation

Astron. Nachr. /AN 335, No. 6/7, 727 – 732 (2014) / DOI 10.1002/asna.201412100

Bouncing models and inflation

N. Pinto-Neto�

ICRA - Centro Brasileiro de Pesquisas Fısicas – CBPF, rua Xavier Sigaud, 150, Urca, CEP22290-180, Rio de Janeiro,Brazil

Received 2014 Apr 30, accepted 2014 May 13Published online 2014 Aug 01

Key words cosmology: theory – early universe – inflation

In this contribution I will make a critical comparison among viable inflationary, and bouncing models for the primordialUniverse which I will describe, and I will discuss how the usual problems and features of the standard cosmological modelare addressed by them in order to investigate if they can be distinguished by future observations.

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1 Introduction

The release of the Planck satellite data (Planck Collabo-ration 2013) has set the inflation paradigm (Linde 1982;Mukhanov & Chibsov 1981; Starobinsly 1979) on a firmerground. Apart from solving some of the standard cosmo-logical puzzles (horizon, flatness, isotropy, and absence ofmonopoles), the simplest models predict an almost scaleinvariant spectrum of long wavelength scalar perturbationswith a low amplitude of tensor perturbations, as observed.

However, there are some weak points and unansweredissues in the inflationary paradigm: The type of scalar fieldthat drives inflation has never been observed and it is stillan open question whether its theoretical properties can befound in some well motivated fundamental particle physicstheory (Lyth 2007); the presence of trans-Planckian wave-lengths, which casts doubts on the validity of the cosmolog-ical perturbation predictions of inflation (Martin & Bran-denberger 2001); it does not really solve the homogeneityproblem as long as one still needs special initial conditionsin a patch to begin inflation (Hollands & Wald 2002); fi-nally, one of the most important problems of the standardcosmological model, the existence of an initial singularity(a point where no physics is possible), is not addressed byinflation (Borde & Vilenkin 1997).

Concerning this last issue, any mechanism which elimi-nates the initial singularity yields at most two possible sce-narios: either there is a quantum creation of a small but finiteUniverse and hence a beginning of time (Hartle & Hawking1983; Vilenkin 1986), or the Universe is eternal, with no be-ginning of time. This last possibility can be divided in twoother categories: the Universe has always expanded, with along accelerated phase before the usual decelerated expan-sion of the standard model from either an asymptoticallyzero volume flat spacetime (Gasperini & Veneziano 1993)or from a finite but small compact region (Mulryne at al.

� Corresponding author: [email protected]

2005; Falciano, Pinto-Neto & Santini 2007); the Universehad a contracting phase before the usual expanding phase,hence performing a bounce (Acacio, Pinto-Neto & Sagioro-Leal 1998; Colistete, Fabris & Pinto-Neto 2000; Novello &Salim 1979).

The transition from accelerated to decelerated expan-sion or from contraction to expansion demands non stan-dard or non classical physics, and/or non standard matter inorder to avoid the singularity in between. Up to now, there isnot a single observation which favorizes one of these threescenarios (time creation, eternal expansion, bounce) with re-spect to the other, rendering these three possibilities suscep-tible of physical investigation, without prior preferences.

Bouncing models differentiates, however, very stronglyfrom the other two scenarios above in one important as-pect: initial conditions may not be anymore put in a verysmall and tiny region, perhaps with Planckian size, but in avery large and almost flat Universe. In this framework, theflatness and the homogeneity problems are viewed withina very different perspective. Hence, bouncing models notonly solve, by construction, the singularity problem, butthey may also possibly solve other important puzzles of thestandard model without the need of an inflationary phase.From now on, we turn to the way a bounce addresses theusual puzzles, before presenting an actual model in whichthese features can be readily implemented.

2 Standard problems and comparison withinflation

We will now make a list of some standard problems in Cos-mology and how bouncing models and inflation deal withthem.

1) The singularity problem: of course bouncing modelssolve the problem by construction while inflation does notaddress this question.

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728 N. Pinto-Neto: Bouncing models and inflation

2) The horizon problem: both types of approach solvethis problem. In bouncing models, as the horizon is definedby d

H≡ a(t)

∫ t

tia−1(t)dt, with ti some initial time, then,

considering that the contracting phase is led by a perfectfluid with equation of state − 1

3 < ω < 1 as in this contribu-tion, then the scale factor behaves as a(t) ∝ (−t)2/[3(1+ω)],so that we obtain the horizon as

dH

=3 (1 + ω)

1 + 3ω

{|ti|(1+3ω)/[3(1+ω)] |−t|2/[3(1+ω)]

+ t}

.

(1)

If ω > − 13 , then clearly, as ti → −∞ (bouncing case),

then dH

diverges. At any finite time before or after thebounce, the horizon is infinite and remains so for all sub-sequent times. Note that in the presence of dark energy(−1 < ω < − 1

3 ) the situation is different and the hori-zon has a finite size depending on the time on which thedark fluid dominates over ordinary fluids in the contractingphase.

3) The flatness problem: This issue is related with theclassical equation

˙|Ω− 1| = −2a

a3.

As Ω is close to 1 now, it must have been arbitrarily closerin the past in a Big-Bang model with decelerated expansionbecause the RHS of the above equation is positive. To solvethis problem, one must have had a long period where theRHS of the above equation is negative in order for Ω getsnear 1 as much as necessary before it begins its deceleratedexpansion. This can be accomplished either through an in-flationary expansion phase (a > 0 and a > 0), or through along decelerated contracting phase (a < 0 and a < 0), thecase of the type of bounces considered here. In this frame-work, we would say that the Universe is almost flat nowbecause it has expanded much less than all it has contractedbefore. It is a nice solution to this problem, within a verydifferent perspective on how we view initial conditions.

4) Initial conditions for structure formation: The mainachievement of inflation, the prediction that the spectral in-dex of scalar perturbations is close to one, can also be ob-tained in bouncing models as it will be shown in this contri-bution. In the simple model presented here, nS can be veryclose to 1 but not less, contrary to the simplest inflationarymodels, where it must be less than one. However, there isa possibility to get nS < 1 also in such bouncing models,as we will see later on. The physical evolution of the per-turbations is qualitative the same in both approaches: onegives vacuum initial conditions to the perturbations in thefar past, when the Universe is almost flat and the curvaturescale is much bigger than the wavelength of the perturba-tion, the Universe contracts, the curvature scale becomessmaller than the wavelength of the perturbation and the am-plification of perturbations begin to take place, it makes asmooth bounce to the expanding phase, and afterwards thecurvature scale becomes again much bigger than the wave-length of the perturbation, which returns to its oscillatory

regime. The constraints on the parameters due to the nor-malization conditions and the compromise that the modelshould describe our real Universe impose that the scale fac-tor at the bounce must be large in Planck units. Once this isdone, there are no trans-Plankian problems, as it can happenin inflation, and no depart from linearity.

5) The homogeneity problem: This is one of the deep-est problems of the Standard Model. There are mainlytwo approaches to this issue: some theory of initial con-ditions, perhaps coming from quantum cosmology, wouldexplain the outcome of the very special homogeneous andisotropic Friedmann solution, or some dynamical process inthe early Universe would erase all initial inhomogeneitiesand anisotropies. Inflation is an example of the second ap-proach. It is unquestionable that it alleviates the problemsubstantially (Goldwirth & Piran 1992), but it is not clear,specially concerning homogeneity, whether inflation itselfprecludes special initial conditions to occur (Hollands &Wald 2002). An example of the first approach is the Weylcurvature hypothesis (Pelavas & Lake 2000; Penrose 1979),based on thermodynamical considerations. The arrow oftime seems to imply that the Universe had initially a verylow total entropy, whose gravitational part, the dominantone, should depend on the Weyl tensor, implying that thispart of the curvature tensor should be negligible initially.Note however that this particular hypothesis is not sufficientto guarantee homogeneity as long as the conformal factor ofthe metric may have non negligible spatial gradients at thebeginning.

In bouncing models, one may open a complete new per-spective and solve this problem using a mixture of thesetwo approaches. In a very large initial Universe, the energy-momentum tensor of matter should be very small, andhence, due to Einstein’s equations, the Ricci tensor. How-ever this does not assure that the geometry is almost flatbecause Einstein’s equations do not fix the Weyl tensor.Supposing, however, the Weyl curvature hypothesis, impliesthat geometry must indeed be almost flat at that time. Whathappens with the inhomogeneities which should be presentthen?

In a first approach, as the Universe was very big, rar-efied and almost flat, the self gravitation of any inhomo-geneity with δρ/ρ � 1 is negligible, as long as ρ is verysmall (which is not true if we take initial conditions whenthe Universe is very small and ρ is big). Hence, any inho-mogeneity gets dissipated in the same way as the air in aroom gets rapidly homogeneous if perturbed. Gravity andthe entropy of the gravitational field are not important inthis regime, only the entropy of matter, which has all timedisposable to reach its maximum value.

One can go one step forward and assume a slow cosmo-logical contraction. In that case, the dust field velocity satis-fies v ∝ 1/a, its number density is n ∝ 1/a3, and its meanfree path is, consequently, λMFP = 1/(nσ) ∝ a3, where σis the dust cross section.

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Astron. Nachr. /AN 335, No. 6/7 (2014) 729

The dissipation time of an inhomogeneity of wavelengthλ reads

td =λ

v

(1 +

λ

λMFP

).

As we have a dust contraction, a ∝ R2/3H , where RH =

yRH0 is the value of the Hubble radius at any time, and RH0

is the value of the Hubble radius today. Writing λ = xRH0,one gets

td ∝ xy2/3

(1 + const

x

y3

)TH0,

where TH0 is the value of the Hubble time today. Comparingwith a typical Hubble time TH = yTH0 at that time one getsthat

tdtH∝ x

y1/3

(1 + const

x

y3

).

The dependence on y obtained on the simple calculationabove shows that, for a sufficiently large RH = yRH0, anyscale up to the size of our Universe today becomes ho-mogeneous, being dissipated before gravity can play anyrole. In fact, depending on the amount of time spent in thislow contraction regime, the dissipation is so effective thatonly quantum fluctuations given by the uncertainty princi-ple would remain, justifying vacuum initial conditions forthe perturbations.

6) Dark energy: This problem is not addressed neitherby inflation nor by the bouncing models we investigatedhere. However, the existence of dark energy is harmless forinflation but may be important for bouncing models just be-cause their effects become important only when the Uni-verse is large. If one assumes that the field or cosmologicalconstant which models dark energy was always there, that itwas not created at the bounce, then one must take care aboutthe horizon problem, as discussed above, but one could alsoenvisage a mechanism for having a spectral index less thenone as long as one may have an effective equation of stateparameter smaller than zero due to the presence of such afield. This is something to be investigated in more detail inthe future.

3 An example of a bouncing model comingfrom quantum cosmology

The action we shall begin with is that of General Relativity(GR) with a perfect fluid (Mukhanov, Feldman & Branden-berger 1992)

S = SGR + Sfluid = − 1

6�2Pl

∫ √−gR d4x−∫ √−gε d4x,

(2)

where �Pl

= (8πGN/3)1/2 is the Planck length in natu-

ral units (� = c = 1), ε is the perfect fluid energy density

whose pressure p is provided by the relation p = ωε, and ωis a non-vanishing constant.

Let the geometry of space-time be given by

gμν = g(0)μν + hμν , (3)

where g(0)μν represents a homogeneous and isotropic cosmo-

logical background,

ds2 = g(0)μν dxμdxν = N2(t)dt2 − a2(t)δijdxidxj , (4)

where we are restricted to a flat spatial metric, and the hμν

represents linear scalar perturbations around it which we de-compose into

h00 = 2N2φ

h0i = −NaB,i (5)

hij = 2a2(ψγij − E,ij).

Substituting Eqs. (5) and (4) into the Einstein-Hilbertaction (2), performing Legendre and canonical transforma-tions, redefining N with terms which do not alter the equa-tions of motion up to first order, all this without ever us-ing the background equations of motion, the Hamiltonianup to second order is simplified to (Peter, Pinho & Pinto-Neto 2007)

H =N[H

(0)0 + H

(2)0

]+ΛNPN +

∫d3xφπψ +

∫d3xΛφπφ,

(6)

where

H(0)0 ≡ −�2

PlP 2

a

4aV+

PT

a3ω, (7)

and

H(2)0 ≡ 1

2a3

∫d3xπ2 +

aω

2

∫d3xv,iv,i. (8)

The quantities N , φ, ΛN , and Λφ play the role of Lagrange

multipliers of the constraints H(0)0 + H

(2)0 ≈ 0, πψ ≈ 0,

PN ≈ 0, and πφ ≈ 0, respectively. The momenta Pa, πφ,πψ , PN and PT are conjugate respectively to a, φ, ψ, N, T ,this last variable playing the role of time.

The variable v is related with the gauge invariantBardeen potential Φ (Peter, Pinho & Pinto-Neto 2007)through

Φ,i,i = −3�2

Pl

√(ω + 1)ε0

2√

ωa

(v

a

)′

, (9)

which coincides with Eq. (12.8) of Mukhanov, Feldman &Brandenberger (1992), relating v and Φ:

Φ,i,i = −

√3

2

�PlHγ

c2s

(v

z

)′

, (10)

where

γ ≡ 1− H′H2

, z ≡ a

cs

√γ, (11)

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and cs (c2s ≡ dp0/dε0 = p′0/ε′0) is the velocity at which den-

sity perturbations propagates, and if the classical equationsof motion, that can be recast in the form

ε0 + p0 = ε0(1 + ω) =2H2γ

3�2Pl

a2,

are used.The above quantities have been redefined in order to be

dimensionless. For instance, the physical scale factor aphys

can be obtained from the dimensionless a present in Eq. (7)through aphys = �

Pla/√

V , where V is the co-moving vol-ume of the background space-like hyper-surfaces, which wesuppose to be compact. The constraint H

(0)0 + H

(2)0 is the

one which generates the dynamics, yielding the correct Ein-stein equations both at zeroth and first order in the pertur-bations, as can be checked explicitly. The others imply thatN , φ, and ψ are not relevant. The unique perturbed degreeof freedom is v, as obtained in Mukhanov, Feldman & Bran-denberger (1992). We would like to emphasize again that inorder to obtain the above results, no assumption has beenmade about the background dynamics: The Hamiltonian (6)is ready to be applied in the quantization procedure.

In the Dirac quantization procedure, the first class con-straints must annihilate the wave functional

Ψ[N, a, φ(xi), ψ(xi), v(xi), T

],

yielding

∂

∂NΨ =

δ

δφΨ =

δ

δψΨ = HΨ = 0. (12)

The first three equations impose that the wave functionaldoes not depend on N , φ and ψ: as mentioned above, Nand φ are, respectively, the homogeneous and inhomoge-neous parts of the total lapse function, which are just La-grange multipliers of constraints, and ψ has been substitutedby v(xi), the unique degree of freedom of scalar perturba-tions, as expected.

As PT appears linearly in H , and making the gaugechoice N = a3ω, one can interpret the T variable as a timeparameter. Hence, the equation

HΨ = 0 (13)

assumes the Schrodinger form

i∂

∂TΨ =

1

4

{a(3ω−1)/2 ∂

∂a

[a(3ω−1)/2 ∂

∂a

]}Ψ (14)

− a3ω−1

2

∫d3x

δ2

δv2Ψ +

a3ω+1ω

2

∫d3xv,iv,iΨ,

where we have chosen the factor ordering in a in order toyield a covariant Schrodinger equation under field redefini-tions.

If one makes the ansatz

Ψ[a, v, T ] = Ψ(0)(a, T )Ψ(2)[a, v, T ], (15)

where Ψ(0)(a, T ) satisfies the equation

i∂

∂TΨ(0)(a, T ) =

a(3ω−1)/2

4

∂

∂a

[a(3ω−1)/2 ∂

∂a

]Ψ(0)(a, T ),

(16)

then we obtain for Ψ(2)[a, v, T ] the equation.

i∂

∂TΨ(2)(a, v, T ) = −a(3ω−1)

2

∫d3x

δ2

δv2Ψ(2)(a, v, T )

+ωa(3ω+1)

2

∫d3xv,iv,iΨ(2)(a, v, T ).

(17)

Let us use the ontological Bohm-de Broglie interpretationof quantum mechanics (Holland 1993), where quantum tra-jectories can be defined, and let us study the background.We change variables to

χ =2

3(1− ω)−1a3(1−ω)/2,

obtaining the simple equation

i∂Ψ(0)(a, T )

∂T=

1

4

∂2Ψ(0)(a, T )

∂χ2. (18)

This is just the time reversed Schrodinger equation for a onedimensional free particle constrained to the positive axis.

We will choose the initial normalized wave function

Ψ(init)(0) (χ) =

(8

T0π

)1/4

exp

(−χ2

T0

), (19)

where T0 is an arbitrary constant. The wave solution for alltimes in terms of a reads

Ψ(0)(a, T ) =

[8T0

π(T 2+T 20 )

]1/4

× exp

[−4T0a3(1−ω)

9(T 2+T 20 )(1−ω)2

]

× exp

{−i

[4Ta3(1−ω)

9(T 2+T 20 )(1−ω)2

+ 12 arctan

(T0

T

)− π

4

]}.

(20)

Due to the chosen factor ordering, the probability den-sity ρ(a, T ) has a non trivial measure and it is given by

ρ(a, T ) = a(1−3ω)/2∣∣Ψ(0)(a, T )

∣∣2. Its continuity equationcoming from Eq. (18) reads

∂ρ

∂T− ∂

∂a

[a(3ω−1)

2

∂S

∂aρ

]= 0, (21)

which implies in the Bohm interpretation that

a = −a(3ω−1)

2

∂S

∂a, (22)

in accordance with the classical relations a = {a, H} =−a(3ω−1)Pa/2 and Pa = ∂S/∂a.

Inserting the phase of (20) into Eq. (22), we obtain theBohmian quantum trajectory for the scale factor:

a(T ) = a0

[1 +

(T

T0

)2] 1

3(1−ω)

. (23)


Astron. Nachr. /AN 335, No. 6/7 (2014) 731

Note that this solution has no singularities and tends to theclassical solution when T → ±∞. Remember that we arein the gauge N = a3ω, and T is related to conformal timethrough

NdT = adη =⇒ dη = [a(T )]3ω−1 dT. (24)

The Bohmian quantum trajectory a(T ) can be used inEq. (17). Indeed, since one can view a(T ) as a function ofT , it is possible to implement a time dependent canonicaltransformation, yielding

i∂Ψ(2)

∂η=

∫d3x

(−1

2

δ2

δv2+

ω

2v,iv

,i − a′′

2av2

)Ψ(2). (25)

This is the most simple form of the Schrodinger equa-tion which governs scalar perturbations in a quantum mini-superspace model with fluid matter source. The correspond-ing time evolution equation for the operator v in the Heisen-berg picture is given by

v′′ − ωv,i,i −

a′′

av = 0, (26)

where a prime means derivative with respect to conformaltime. In terms of the normal modes vk, the above equationreads

v′′k +

(ωk2 − a′′

a

)vk = 0. (27)

With this simple equation, we can evaluate the spectral in-dex and amplitude of scalar cosmological perturbations.

Hence, using quantum cosmology and the de Broglie-Bohm interpretation (Holland 1993), thanks to which onecan define trajectories and a scale factor evolution with time,we have obtained a simple model whose scalar and tensorperturbations can be made arbitrary close to scale invariance(Peter, Pinho & Pinto-Neto 2007). The model consists ofa classically contracting single dust perfect fluid in whichthe big crunch is avoided through quantum effects in thegeometry described by the Wheeler-DeWitt equation, turn-ing the Universe evolution to an expanding phase, whichsoon becomes classical again. This transition is smoothlydescribed by a Bohmian quantum trajectory containing abounce. Perturbations begin in a vacuum state during thecontraction epoch, when the Universe was very large andalmost flat, and are subsequently evolved in a fully quan-tum way through its all history. Hence, we have presenteda non-singular model without horizons where perturbationsof quantum mechanical origin can be described all alongand generate structures in the Universe. Note that if one hadused other interpretations of quantum mechanics instead ofde Broglie-Bohm’s, e.g. the many world, where the notionof trajectories is not immediate, the implementation of thecalculations for the perturbations could have been muchmore involved, if possible.

In order to match the CMB normalization, we find thatthe characteristic length at the bounce must be of the or-der of a few thousands Planck lengths, thereby making the

model fully consistent. Indeed, we should like to emphasizethat one would expect precisely the Wheeler-de Witt equa-tion to be valid and important in this regime (a sufficientlylow scale for quantum gravity effects be important, but notso low in order for not being affected by string and/or loopquantum gravity effects). Moreover, it predicts a slightlyblue spectrum, which may be seen as a drawback of themodel. However, in more realistic and elaborated models,other fluids must be considered, like radiation and dark en-ergy. It seems that adding these fluids will not spoil our re-sults as long as only a single fluid dominates at the bounce,and that dust dominates when the scales of cosmologicalinterest become greater than the curvature scale in the clas-sical contracting phase. If in that period dark energy hassome effect imposing a slightly negative effective w, thenone could even obtain a slightly red spectrum instead of theblue one derived here.

Finally, we have obtained that once the constraint on thescalar index is taken into account, the tensor-to-scalar ra-tio which follows is predicted to be small and varying withthe square root of the spectral index. Thus, this category ofmodel is falsifiable and comparable with inflationary pre-dictions for which the relation is linear.

4 Conclusions

The theory of linear quantum perturbations has been suc-cessfully applied in the framework of a classical inflationarybackground: only the perturbations were quantized, leadingto a sort of semi-classical approximation to quantum gravity(Mukhanov, Feldman & Brandenberger 1992). We have de-veloped a Hamiltonian formalism where not only the pertur-bations but also the background could be quantized (Peter,Pinho & Pinto-Neto 2005, 2006, 2007; Pinho & Pinto-Neto2007). This led to a picture of quantum perturbations evolv-ing in a non singular bouncing background space-time froma vacuum state yielding spectral indices and amplitudes thatcan be made to agree with observations provided the dom-inant fluid in the background when the perturbation scalebecomes smaller than the curvature radius is dust. The cur-vature scale at the bounce can always be set larger than thePlanck length, and hence the calculations are not spoiledby higher order quantum gravity effects. Finally, such amodel can be extended to include a radiation dominateddecelerating phase before nucleosynthesis without corrupt-ing its main features properties. This thus provides a sim-ple theoretical framework where only the basic principlesof general relativity and quantum mechanics, together withthe assumption of the existence of a dust like fluid (darkmatter?), yield what can be argued to be a sensible model.Furthermore, such behaviors can also be obtained in othernon quantum bouncing models (Finelli, Peter & Pinto-Neto2008), indicating that these are not particular properties ofthe specific models here discussed.

We have also argued that general bounces provide dif-ferent perspectives on old issues such as flatness and homo-

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geneity. In fact, these problems may be alleviated or solvedusing simple physical arguments which can only be appliedin this context.

There are however many open questions left to be ad-dressed and some weak points. Let us list them below.

– Baryogenesis and dark energy are not addressed, but thelatter could actually provide a means of obtaining a red-der spectrum.

– Was radiation always there or it was produced at thebounce, e.g. through the evaporation of mini black holesor monopolonium bound states? Does its presence altersthe amplitude of the perturbations, and if so, how?

– As primordial perturbations are enhanced at the bounce,similarly one could think that it also might lead to largeamounts of particle production. The relic density of thethese particles needs be evaluated for each model.

– Although spatial curvature is expected to be negligibleduring most of the evolution, particularly in the expand-ing phase, it may be quite important at the bounce itself,and modify the amplitude of the perturbations.

All the properties of bouncing models and their open is-sues show that they seem to provide a robust alternative toinflation. A less ambitious role, although still very impor-tant, should be that they can complement inflation by solv-ing the singularity problem, ease the homogeneity problemand yield appropriate initial conditions for it (Falciano &Lilley & Peter 2008). In any case, bounce cosmology leadsto numerous new, hopefully measurable, ideas and effects,yet to be investigated. The tensor-to-scalar prediction is al-ready an example of such an effect rendering the paradigmtestable.

As a final remark, we would like to stress that, in con-tradistinction with models in which time begins, there isno point on asking what is the probability of appearanceof some particular eternal model out of nothing. Contraryto usual perspectives, one can as well assume existence tobe conceptually prior to non-existence, i.e. existence itselfmay not be deserving explanation. This is the idea underly-ing our category of models: the Universe always existed andits “appearance” is thus a non question.

Acknowledgements. I would like to thank CNPq of Brazil for fi-nancial support.

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