A covariant and gauge invariant formulation of the cosmological "backreaction"
aa r X i v : . [ g r- q c ] F e b BA-TH/623-09CERN-PH-TH/2009-249arXiv:0912.3244
A covariant and gauge invariant formulationof the cosmological “backreaction”
M. Gasperini , , G. Marozzi and G. Veneziano , Dipartimento di Fisica, Universit`a di Bari,Via G. Amendola 173, 70126 Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari,Via G. Amendola 173, 70126 Bari, Italy GR ε CO – Institut d’Astrophysique de Paris, UMR7095, CNRS,Universit´e Pierre & Marie Curie, 98 bis boulevard Arago, 75014 Paris, France CERN, Theory Unit, Physics Department,CH-1211 Geneva 23, Switzerland Coll`ege de France, 11 Place M. Berthelot, 75005 Paris, France
Abstract
Using our recent proposal for defining gauge invariant averages we give a general-covariant formulation of the so-called cosmological “backreaction”. Our effective covari-ant equations allow to describe in an explicitly gauge invariant form the way classicalor quantum inhomogeneities affect the average evolution of our Universe.
Introduction
It is well known that the homogeneous and isotropic Friedman-Lemaitre-Robertson-Walker(FLRW) metric describes the geometric properties of our Universe only on sufficiently largescales of distance, and has to be interpreted as the “averaged” cosmological metric: namely,the metric emerging from an appropriate smoothing-out of the local inhomogeneities andanisotropies. The same interpretation of averaged variables has to be assigned to the matterenergy and momentum density, sourcing the FLRW metric in the cosmological Einsteinequations.A problem (a rather old one, see for instance [1]) thus appears due to the fact thatthe Einstein equations for the averaged geometry are different, in general, from the aver-aged Einstein equations . This is because the averaging procedure does not commute, ingeneral, with the non-linear differential operators appearing in Einstein’s equations. As aconsequence, the dynamics of the averaged geometry is affected by so-called “backreaction”terms, originating from the contribution of the inhomogeneities present in the metric andmatter sectors.Interest in these themes has considerably risen after the suggestion that the cosmicacceleration, recently observed on large scales, could be unrelated to phantomatic dark-energy sources, but – perhaps more simply – to the dynamical effects of the backreaction(see e.g. [2]) thus solving the well-known“coincidence problem”. This possibility has focusedcurrent researches on a new problem: the gauge invariance of the averaging procedure.Indeed, in the absence of gauge invariance, the computed backreaction effects depend notonly on the hypersurface chosen to compute the average integrals, but also on the chosencoordinate frame (see e.g. [3]-[6] for recent discussions).At present, the most commonly used averaging procedure in a cosmological context isbased on the foliation of space-time into three-dimensional hypersurfaces comoving withthe matter sources, and on the volume integration over such spacelike hypersurfaces [7](see [8] for a recent review, and [6, 9] for an extension to more general hypersurfaces).This procedure is applied, in particular, to the scalar part of the cosmological Einsteinequations, i.e. to the so-called Hamiltonian constraint and Raychaudhuri’s equation. Byspatially averaging such equations on a comoving domain D (and considering, for simplicity,dust fluid sources) one obtains [7]: (cid:18) ˙ a D a D (cid:19) = 8 πG h ρ i D −
16 ( h Q i D + hRi D ) , (1.1) − ¨ a D a D = 4 πG h ρ i D − h Q i D . (1.2)1ere the dot denotes derivatives with respect to the cosmic time of the comoving syn-chronous coordinates, a D is an effective scale factor, related to the volume V D of theintegration domain (normalized by a reference volume scale V D ) by a D = ( V D /V D ) / ,where V D = Z D d x q | det g ij | , (1.3)and g ij is the intrinsic metric of the comoving hypersurfaces. Also, the brackets denotespatial average over D , namely h· · ·i D = 1 V D Z D d x q | det g ij | ( · · · ) , (1.4) ρ is the energy density of the dust sources, and R is the scalar intrinsic curvature associatedwith the spatial metric g ij . Finally, h Q i is a correction called “kinematical” backreaction,arising in a particular gauge (see section 2) from the averages of two scalar quantities: thetrace Θ of the expansion tensor and the scalar shear σ : h Q i D = 23 (cid:16) h Θ i D − h Θ i D (cid:17) − h σ i D . (1.5)The above averaged equations are obtained within a spatial slicing of the space-timemanifold induced by the flow lines of the matter sources. In this paper we present acovariant version of the effective equations for the averaged cosmological quantities (calledhereafter, for simplicity, averaged cosmological equations), based on a recently proposedgauge invariant averaging prescription [4]: the backreaction effects we obtain depend on thehypersurface chosen to define the physical observer, but do not depend on the particularchoice of coordinates. In the appropriate class of gauges we recover the recent results givenin [6, 9]. In addition, for an observer at rest with respect to the matter sources, we recoverthe same results as in [7, 8].The paper is organized as follows. In Sect. 2 we present an explicitly covariant versionof the Hamiltonian constraint, Raychaudhuri’s equation, and of the projected conservationequation for a generic foliation of spacetime and energy-momentum tensor. In Sect. 3 webriefly summarize the main aspects of our gauge-invariant averaging prescription, and wederive the corresponding general-covariant version of the averaged cosmological equations.Our conclusive remarks are briefly presented in Sect. 4. In order to introduce spatial averages of physical quantities we consider a general class offoliations of spacetime by spacelike hypersurfaces Σ( A ) over which a scalar field A ( x ) takes2onstant values (the so-called level-sets of A ). Let n µ be the future-directed unit normal toΣ( A ), defined by n µ = − ∂ µ A ( − ∂ µ A∂ ν A g µν ) / , n µ n µ = − − , + , + , +)). Let us also introduce the projector h µν into the hypersurfaces by: h µν = g µν + n µ n ν , h µρ h ρν = h µν , h µν n µ = 0 . (2.2)The Einstein equations G µν = T µν (we use units in which 8 πG = 1) can then be projectedalong n µ and h µν , and give rise to three (sets of) equations that can be chosen in thefollowing form: G µν n µ n ν = T µν n µ n ν ≡ ε , (2.3) G µν n µ h νρ = T µν n µ h νρ ≡ J ρ , (2.4) R µν h µρ h νσ = T µν h µρ h νσ − h ρσ T ≡ S ρσ − h ρσ T . (2.5)They correspond to an explicitly covariant version of the so-called Arnowitt-Deser-Misner(ADM) equations.It is always possible to make contact with the more conventional ADM formalism bychoosing a class of gauges in which the scalar field A ( x ) is homogeneous. We shall call itthe ADM gauges. In such a gauge the normal vector n µ takes the form: n µ = N ( − , , , , n µ = 1 N (1 , − N i ) , (2.6)where N and N i are, respectively, the so-called lapse function and shift vector. In thisgauge: h µi = δ µi , h = 0 , h ij = g ij , (2.7)where g ij is the induced 3–metric (or first fundamental form) on Σ. The spacetime metricof the foliated spacetime is given by: ds = g µν dx µ dx ν = − N dt + g ij ( dx i + N i dt )( dx j + N j dt ) , (2.8)and its inverse by: ✷ ≡ g µν ∇ µ ∇ ν = − N − ( ∇ − N i ∇ i ) + (3) g ij ∇ i ∇ j . (2.9)where (3) g ij is the inverse of the 3 × g ij .In this class of ADM gauges Eqs. (2.3) and (2.4) reduce to the convential form ofthe Hamiltonian and momentum constraints, respectively, while Eq. (2.5) generates the3econd order evolution equations. In the ADM context such evolution equations are splittedinto twice as many equations, those defining the second fundamental form (or extrinsiccurvature) K ji , and those describing the evolution of K ji itself (see, e.g. [8]).In order to give a covariant and gauge invariant formulation of the cosmological back-reaction we shall make use of the covariant Eqs. (2.3)–(2.5), which lead to an explicitlyscalar form of the Hamiltonian constraint and Rachayduri’s equation. To this purpose, letus first consider the spacetime flow generated by the timelike vector field n µ , and define theprojected expansion tensor of the flow worldlines asΘ µν ≡ h αµ h βν ∇ α n β = 13 h µν Θ + σ µν + ω µν , (2.10)whereΘ ≡ ∇ µ n µ , σ µν ≡ h αµ h βν (cid:18) ∇ ( α n β ) − h αβ ∇ τ n τ (cid:19) , ω µν ≡ h αµ h βν ∇ [ α n β ] , (2.11)are the expansion scalar, the shear tensor and the rotation tensor, respectively. In our case,with n µ given by Eq. (2.1), we have a zero rotation tensor and we can write σ µν = Θ µν − h µν Θ , σ ≡ σ µν σ νµ = 12 (cid:18) Θ µν Θ νµ −
13 Θ (cid:19) , (2.12)where σ is the shear scalar.Consider first the Hamiltonian constraint (2.3), whose left hand side can be decomposedas follows: R µν n µ n ν + 12 R = 12 (cid:16) Θ − Θ µν Θ νµ (cid:17) + 12 R s . (2.13)In the ADM gauge the term in brackets on the right hand side can be expressed in termsof the extrinsic curvature as: Θ − Θ µν Θ νµ = K − K ij K ji , (2.14)while R s is a scalar which, in the ADM gauge, goes over to the intrinsic scalar curvature R associated with the induced metric g ij . We can thus rewrite the Hamiltonian constraintcompletely in terms of scalar quantities as: R s + Θ − Θ µν Θ νµ = R s + 23 Θ − σ = 2 T µν n µ n ν = 2 ε . (2.15)Coming now to the explicitly scalar form of Rachayduri’s equation, we note that itcorresponds to the linear combination of Eq. (2.3) and of the trace of Eq. (2.5), leading to: R µν n µ n ν = T µν h µν − T. (2.16)4fter some straightforward calculation the above equation takes the form: − n µ ∇ µ Θ = 2 σ + 13 Θ − ∇ ν ( n µ ∇ µ n ν ) + (cid:18) T µν − g µν T (cid:19) n µ n ν = 2 σ + 13 Θ − ∇ ν ( n µ ∇ µ n ν ) + ε + 12 T . (2.17)So far we have made no assumptions on the form of T µν . Let us now restrict ourselvesto the case of a perfect fluid with: T µν = ( ρ + p ) u µ u ν + pg µν , (2.18)where u µ is the 4-velocity comoving with the fluid, and ρ and p are, respectively, the(scalar) energy density and pressure in the fluid’s rest frame (the generalization to severalnon-interacting fluids is straightforward). We stress that u µ and n µ are in general distinct:the former depends on the properties of the matter sources, the latter depends on thechoice of the hypersurfaces on which we want to average, hence should be determined bythe particular problem at hand. With this model of sources we can then express the basicquantities entering our equations (2.15) and (2.17) as: ε = T µν n µ n ν = ( ρ + p )( u µ n µ ) − p , T = T µµ = − ρ + 3 p . (2.19)For later use, we note that the trace T of the energy momentum tensor is unaffectedby the possible “tilt” (misalignment) between u µ and n µ , while the sources of the covariantADM Eqs. (2.3)–(2.5) – namely the objects that we may call the ADM energy density ε ,the ADM pressure π = S ρρ /
3, and the ADM current J ν – become: ε = ρ − ( ρ + p ) (cid:16) − ( u µ n µ ) (cid:17) , (2.20) π = p −
13 ( ρ + p ) (cid:16) − ( u µ n µ ) (cid:17) , (2.21) J ν = ( ρ + p ) u β n β (1 + u µ n µ ) u ν . (2.22)On the other hand, a straightforward calculation in the ADM gauge leads to:( u µ n µ ) = 1 + (3) g ij u i u j ≥ , (2.23)meaning that this quantity is always larger than 1. We can thus introduce a “tilt angle” α T such that sinh α T = ( u µ n µ ) −
1, and we can rewrite ǫ and π in the more convenient form ε = ρ + ( ρ + p ) sinh α T , π = p + 13 ( ρ + p ) sinh α T . (2.24)Let us finally consider the condition following from the projected energy-momentumconservation law, n µ ∇ µ T µν = 0. It is known that such condition is not implied by the5wo projected scalar Einstein equations considered above, and has to be added to the setof averaged cosmological equations as an additional independent constraint [7, 8]. In thegeneral context we are considering (with n µ = u µ ), the projected conservation equation canbe written explicitly as follows: u µ ∂ µ [( ρ + p ) u ρ n ρ ] + n µ ∂ µ p + ( ρ + p ) [ ∇ µ u µ u ρ n ρ − Θ µν u µ u ν ] = 0 . (2.25) Let us begin from the four dimensional integral of a scalar S ( x ) as defined in [4]: I ( S, Ω) = Z Ω( x ) d x q − g ( x ) S ( x ) ≡ Z M d x q − g ( x ) S ( x ) W Ω ( x ) . (3.1)The integration region Ω ⊆ M is defined in terms of a suitable scalar window function W Ω , selecting a region with temporal boundaries determined by the space-like hypersurfacesΣ( A ) (defined in Sect. 2), and with spatial boundary determined by the coordinate condition B < r , where B is a (positive) function of the coordinates with space-like gradient ∂ µ B ,and r is a positive constant. As we are interested in the variation of the volume averagesalong the flow lines normal to Σ( A ), we choose in particular the following window function: W Ω ( x ) = n µ ∇ µ θ ( A ( x ) − A ) θ ( r − B ( x )) , (3.2)where θ is the Heaviside step function, and n µ is defined in Eq. (2.1).As discussed in [4], if B ( x ) is a scalar function the integral (3.1) is not only a scalarunder general coordinate transformations but is also gauge invariant (to all orders): namely,it is invariant under the local field reparametrizations induced by any coordinate changewhen old and new fields are evaluated at the same space-time position. If B is not a scalar,the spatial boundary can be a source of breaking of covariance and gauge invariance. In [10]we will discuss in more detail such a breaking, and confirm that it goes away in the limit oflarge spatial volumes (with respect to the typical scale of inhomogeneities) [4]. Using thewindow function (3.2), the integral (3.1) becomes: I ( S, A ) = Z M d x q − g ( x ) δ ( A ( x ) − A )( − ∂ µ A ∂ µ A ) / θ ( r − B ( x )) S ( x ) . (3.3)Let us now consider the derivative of I ( S, A ) with respect to A , a quantity that, like I itself, is covariant and gauge invariant (apart from a possible gauge dependence inducedby the spatial boundary): ∂I ( S, A ) ∂A = − Z d x q − g ( x ) δ ′ ( A ( x ) − A ) ( − ∂ µ A ∂ µ A ) / θ ( r − B ( x )) S ( x )6 − Z d x q − g ( x ) ∂ δ ( A ( x ) − A ) [ ∂ A ( x )] − ( − ∂ µ A ∂ µ A ) / θ ( r − B ( x )) S ( x ) . (3.4)Within adapted ADM coordinates, where A is homogeneous, we can always choose thosewith vanishing shift (i.e. with g = − N and g i = 0). In such coordinates we have: ∂I ( S, A ) ∂A = − Z d x √− g ∂ δ ( A ( t ) − A ) ( − g ) / θ ( r − B ( x )) S ( x )= Z d x δ ( A ( t ) − A ) ∂ h √− g ( − g ) / θ ( r − B ( x )) S ( x ) i = Z d x q | γ | δ ( A ( t ) − A ) h θ ( r − B ( x )) ( N Θ S + ∂ S ) − δ ( r − B ( x )) S∂ B i , (3.5)where γ = det g ij , and we have used that, in these coordinates,Θ = N − ∂ log √ γ. (3.6)The above equation can be easily recast into the following covariant and gauge invariantform, ∂I ( S, A ) ∂A = I ∂ µ A∂ µ S∂ µ A∂ µ A , A ! + I S Θ( − ∂ µ A∂ µ A ) / , A ! − I ∂ µ A∂ µ B∂ µ A∂ µ A Sδ ( r − B ) , A ! , (3.7)that reduces to Eq. (3.5) in the special coordinates we have been using. Note that the lastfactor of the above equation is absent if n µ is orthogonal to the gradient of B ( n µ ∂ µ B = 0),namely if B does not depend on the time coordinate of the gauge used in Eq. (3.5). Weshall restrict ourselves to this case hereafter.Let us define now the covariant averaging prescription for a scalar S ( x ) on the hyper-surfaces of constant A , following [4], as: h S i A = I ( S, A ) I (1 , A ) . (3.8)Using Eq. (3.5) we can easily obtain the derivative of the averaged scalar h S i A as: ∂ h S i A ∂A = h ∂ A S i A + (cid:28) S N Θ ∂ A (cid:29) A − h S i A (cid:28) N Θ ∂ A (cid:29) A , (3.9)and Eq. (3.7) immediately gives us the generally covariant version of the above equation: ∂ h S i A ∂A = * ∂ µ A∂ µ S∂ µ A∂ µ A + A + * S Θ( − ∂ µ A∂ µ A ) / + A − h S i A * Θ( − ∂ µ A∂ µ A ) / + A . (3.10)This is the covariant and gauge invariant generalization of the Buchert-Ehlers commutationrule [11], and will be the starting point for our generalization of the averaged cosmological7quations. However, let us first illustrate the precise connection to the special version ofthis rule obtained in [11].In ADM coordinates, with A homogeneous, Eq. (3.3) reads: I ( S, A ) = Z Σ A d x q | γ ( t , ~x ) | S ( t , ~x ) θ ( r − B ( t , ~x )) , (3.11)where we have called t the time when A ( t ) takes the constant values A , and the averagesare referred to a section of the three-dimensional hypersurface Σ A where A ( x ) = A . Thisis exactly the type of spatial integrals used in [7] (with a domain D determined by thecondition B ( x ) < r ), and thus leads to the same definition of averages (see Eq. (1.4)). Letus compute, in this case, the corresponding version of Eq. (3.10). Multiplying both sidesby ∂ t A we obtain the equation ∂ t h S i A = D ∂ t S − N i ∂ i S E A + h SN Θ i A − h S i A h N Θ i A = h N n µ ∂ µ S i A + h SN Θ i A − h S i A h N Θ i A , (3.12)which generalises the commutation rule given in [11] to the case of non-vanishing shiftvector. For N i = 0 the standard result is recovered.Let us come now to the covariant formulation of the averaged cosmological equations.Starting from the generally covariant volume integral I (1 , A ) = Z d x √− g δ ( A ( x ) − A ) q − ∂ µ A∂ µ A θ ( r − B ( x )) (3.13)we define, along the lines of the previous approach [7], an effective scale factor e a such that1 e a ∂ e a∂A = 13 I (1 , A ) ∂ I (1 , A ) ∂A . (3.14)We then find, using Eqs. (3.7), (3.8),1 e a ∂ e a∂A = 13 * Θ( − ∂ µ A∂ µ A ) / + A . (3.15)We are now in the position of presenting the covariant generalization of the averaged equa-tion (1.1). Taking the square of the previous equation, using the Hamiltonian constraint inthe form of Eq. (2.15), and explicitly reintroducing Newton’s constant in the formulae, weeasily obtain: (cid:18) e a ∂ e a∂A (cid:19) = 8 πG * ε ( − ∂ µ A∂ µ A ) + A − * R s ( − ∂ µ A∂ µ A ) + A − * Θ ( − ∂ µ A∂ µ A ) + A − * Θ( − ∂ µ A∂ µ A ) / + A + 13 * σ ( − ∂ µ A∂ µ A ) + A . (3.16)8n order to arrive at the covariant generalization of the second Buchert’s equation (1.2)we start with the simple relation1 e a ∂ e a∂A = ∂∂A (cid:18) e a ∂ e a∂A (cid:19) + (cid:18) e a ∂ e a∂A (cid:19) . (3.17)Using Eq. (3.15), and the general commutation rule (3.10) for the first term on the righthand side, we obtain − e a ∂ e a∂A = 29 * Θ( − ∂ µ A∂ µ A ) / + A + 16 * ∂ µ A∂ µ ( ∂ ν A∂ ν A )( − ∂ µ A∂ µ A ) / Θ + A − * Θ ( − ∂ µ A∂ µ A ) + A + 13 * ∂ µ A∂ µ Θ( − ∂ µ A∂ µ A ) / + A . (3.18)Inserting then the covariant Raychaudhuri’s equation (2.17) in the last term of the aboveequation we are lead to the final result, − e a ∂ e a∂A = 4 πG * ε + 3 π ( − ∂ µ A∂ µ A ) + A − * ∇ ν ( n µ ∇ µ n ν )( − ∂ µ A∂ µ A ) + A + 16 * ∂ µ A∂ µ ( ∂ ν A∂ ν A )( − ∂ µ A∂ µ A ) / Θ + A − * Θ ( − ∂ µ A∂ µ A ) + A − * Θ( − ∂ µ A∂ µ A ) / + A + 23 * σ ( − ∂ µ A∂ µ A ) + A , (3.19)where we have used the relation ρ − p = ε − π (see Eq. (2.24)). Equations (3.16) and(3.19), together with the averaged conservation equation discussed below, are the mainresults of this paper.Let us now observe that Eq. (3.16), written in the ADM gauge, and multiplied by( ∂ t A ) , reduces to the equation recently presented in [6, 9]: (cid:18) e a ∂ e a∂t (cid:19) = 8 πG h N ρ i A + 8 πG D N ( ρ + p ) sinh α T E A − h N Ri A − (cid:16) h N Θ i A − h N Θ i A (cid:17) + 13 h N σ i A . (3.20)We have used Eq. (2.24) to replace the ADM parameter ε with the fluid proper energyand pressure. When sinh α T = 0 (namely when the averaging hypersurfaces coincide withthose orthogonal to the fluid velocity, n µ = u µ ), we exactly recover the correspondingBuchert’s equation (see the second paper of [7]). In the synchronous gauge ( N = 1) andfor dust sources ( p = 0) we recover instead Buchert’s Eq. (1.1), apart from the additionalcontribution arising from a nonvaninshing tilt angle α T .Consider now the second equation (3.19), and let us first note that1 e a ∂ e a∂t = (cid:18) ∂A ∂t (cid:19) e a ∂ e a∂A + 1 e a ∂ A ∂t ∂ e a∂A . (3.21)9e go then to the ADM coordinates, as before, imposing in addition the convenient gaugechoice N i = 0, and we apply Eq. (2.24) to express ε and π in terms of ρ and p . By usingEq. (3.19) to eliminate the first term on the right hand side of Eq. (3.21) we obtain − e a ∂ e a∂t = 4 πG D N ( ρ + 3 p ) E A + 8 πG D N ( ρ + p ) sinh α T E A − (cid:16) h N Θ i A − h N Θ i A (cid:17) + 23 h N σ i A − (cid:28) Θ ∂N∂t (cid:29) A − D N (3) g ij ∇ i ∇ j N E A , (3.22)in agreement with [6]. Again, for sinh α T = 0, we also recover the corresponding Buchert’sequation (see for instance [8]). We may note that, when sinh α T = 0, the “tilt effects” givea negative contribution (assuming ρ + p >
0) to the average cosmic acceleration.Let us conclude this section by noting that, as anticipated in Sect. 2, the set of aver-aged cosmological equations has to be complemented by the general-covariant average ofthe conservation equation (2.25). Such an operation can be performed straightforwardly,according to the general procedure outlined above. We shall present here, for simplicity, thecovariant version of the averaged conservation equation in the particular case in which thespace-time foliation is referred to the fluid comoving frame. In such a case, setting n µ = u µ in Eq. (2.25), and applying the general-covariant commutation rule (3.10), we obtain ∂∂A h ρ i A = − * Θ p ( − ∂ µ A∂ µ A ) / + A − h ρ i * Θ( − ∂ µ A∂ µ A ) / + A (3.23)(note that, in this case, the scalar A corresponds to the velocity potential of the fluidsources). Going, as before, to the ADM coordinates, and multiplying by ∂ t A , we recoverthe known result ∂∂t h ρ i D + h N Θ p i D − h ρ i h N Θ i D = 0 , (3.24)already presented in various papers [7, 8]. The main results of this paper are the covariant and gauge invariant formulation of theBuchert-Ehlers commutation rule, Eq. (3.10), and of the effective equations for the averagedevolution of a perfect fluid-dominated Universe, Eqs. (3.16), (3.19),(3.23). The average isperformed over a generic class of hypersurfaces, not necessarily orthogonal to the fluid flowlines. We stress that our results allow to compute averaged quantities in a completelyarbitrary coordinate system. 10he results obtained in this paper can be directly applied to the case of the quantum cos-mological backreaction, by using the correspondence between quantum expectation valuesand classical averages performed over all three-dimensional space, as illustrated in detailsin [4]. Hence, in particular, can be applied to study the effect of the quantum fluctuationswithin the canonical formalism of cosmological perturbation theory.At the same time, the classical averaged equations (possibly extended to light-like hy-persurfaces) may provide a covariant starting point for determining whether present inho-mogeneities can significantly contribute to the observed cosmic acceleration.
Acknowledgements
One of us (MG) is very grateful to the Coll`ege de France for its warm hospitality andsupport. One of us (GM) wishes to thank Syksy Rasanen for useful discussions. GM wassupported by the GIS “Physique des Deux Infinis”.
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