Xavier Roy
email: roy@obs.univ-lyon1.fr

Thomas Buchert
email: buchert@obs.univ-lyon1.fr

Sante Carloni
email: sante.carloni@esa.int

Nathaniel Obadia
email: nathaniel.obadia@ens-lyon.fr

Global gravitational instability of FLRW backgrounds -Interpreting the dark sectors

Keywords: numbers: 04.20.-q, 04.25.-D, 04.40.-b, 95.35.+d, 95.36.+x, 98.80.Es, 98.80.Cq

The standard model of cosmology is based on homogeneous-isotropic solutions of Einstein's equations. These solutions are known to be gravitationally unstable to local inhomogeneous perturbations, commonly described as evolving on a background given by the same solutions. In this picture, the FLRW backgrounds are taken to describe the average over inhomogeneous perturbations for all times. We study in the present article the (in)stability of FLRW dust backgrounds within a class of averaged inhomogeneous cosmologies. We examine the phase portraits of these latter, discuss their fixed points and orbital structure, and we provide detailed illustrations. We show that FLRW cosmologies are unstable in some relevant cases: averaged models are driven away from them through structure formation and accelerated expansion. We find support for the proposal that the dark components of the FLRW framework may be associated to these instability sectors. Our conclusion is that FLRW cosmologies have to be considered critically as for their role to serve as reliable models for the physical background.

The problem of a physical background in cosmology

The standard description of the global evolution of the universe relies on the class of homogeneous-isotropic Friedmannian-Lemaître-Robertson-Walker (FLRW) solutions of Einstein's equations. By construction all inhomogeneities in matter and geometry, responsible for the formation of structures, average out on this given background. The conjecture is held that the FLRW background is the actual physical background.

This widely adopted framework needs to be challenged for several reasons. First, apart from the question of how to technically implement an average (a non-trivial subject for tensors), the result will depend in any case on the spatial domain over Confidential: not for distribution. Submitted to IOP Publishing for peer review 16 May 2011 which inhomogeneities are considered: a scale-dependence of the averaged variables must generically result, casting strong doubts on the scale-independent averaged values issued from the FLRW solutions (see [START_REF] Ellis | The universe seen at different scales[END_REF] for a summary of these thoughts). Secondly, it is actually naive to expect that a strictly homogeneous model provides the average, given the non-linearity of Einstein's equations and its implications (e.g. the noncommutativity of the time-evolution and averaging). This background issue may also be identified as the crucial question in the discussion of whether or not the need for dark components can be replaced by employing inhomogeneous models [START_REF] Kolb | Cosmological background solutions and cosmological backreactions[END_REF].

Looking at current cosmological structure formation models based on perturbation theories or N-body simulations, we find that the extra (backreaction) terms due to the averaging procedure vanish on the background of a homogeneous-isotropic solution. A closer look reveals that this property results by construction rather than by derivation, and it is a consequence of employing Newtonian or quasi-Newtonian schemes with periodic boundary conditions and a flat background geometry (see [START_REF] Buchert | Averaging inhomogeneous Newtonian cosmologies[END_REF] for the proof). This construction is not expected to work in the framework of general relativity because of: (i) the relevance of the spatial intrinsic curvature (the second derivatives of the metric may be significant even if the metric perturbations are negligible [START_REF] Buchert | Geometrical order-of-magnitude estimates for spatial curvature in realistic models of the universe[END_REF]), together with (ii) the fact that inhomogeneities are coupled to the spatial curvature evolution [START_REF] Buchert | On average properties of inhomogeneous fluids in general relativity: dust cosmologies[END_REF], and finally (iii) the absence of a conservation law for the averaged intrinsic curvature [START_REF] Buchert | On the curvature of the present day universe[END_REF].

It is of course conceivable that the homogeneous solutions provide in some spatial and temporal regimes a good approximation for the evolution of the averaged distribution. A systematic approach to address this assumption is to analyze the stability of the FLRW solutions in the space of averaged cosmologies. Although FLRW solutions are known to be unstable to local perturbations, we may be able to find that they enjoy stability if subjected to perturbations on average and on some large scale. In the present paper we follow this approach, and we find that the FLRW solutions can also be globally unstable within a class of averaged models. Structure formation and accelerated expansion imply that the physical background is driven away from the FLRW background in relevant cases. Our investigation is independent on the choice of scale, and it therefore contains all the dynamical situations a domain can undergo.

For the class of inhomogeneous cosmologies we consider in this work, we shall employ a framework that averages the scalar parts of Einstein's equations, in which case the averaged model is unambiguously defined (reviewed in [START_REF] Buchert | Dark energy from structure -a status report[END_REF][START_REF] Buchert | Towards physical cosmology: focus on inhomogeneous geometry and its nonperturbative effects[END_REF]) and, due to its covariant implementation, is under control [START_REF] Buchert | On average properties of inhomogeneous fluids in general relativity: perfect fluid cosmologies[END_REF][START_REF] Gasperini | A covariant and gauge invariant formulation of the cosmological "backreaction[END_REF][START_REF] Li | On the onset of cosmological backreaction[END_REF][START_REF] Marozzi | The cosmological backreaction: gauge (in)dependence, observers and scalars[END_REF]. We are going to explore the phase space of averaged cosmologies by working within a class of exact scaling solutions, generalizing previous results [START_REF] Buchert | Correspondence between kinematical backreaction and scalar field cosmologies -the 'morphon field[END_REF] ‡.

The study of the phase space of dynamical systems has a long tradition. The ‡ In [START_REF] Buchert | Correspondence between kinematical backreaction and scalar field cosmologies -the 'morphon field[END_REF] the scaling law was applied to the total averaged curvature, and we choose here to apply it to the averaged curvature deviation from a constant (Friedmannian) curvature. Taking a null constant curvature in our work leads back to the results of [START_REF] Buchert | Correspondence between kinematical backreaction and scalar field cosmologies -the 'morphon field[END_REF]. This difference of approach allows to study in this paper the stability of all FLRW backgrounds, not fully addressed in [START_REF] Buchert | Correspondence between kinematical backreaction and scalar field cosmologies -the 'morphon field[END_REF]. Note already that the phase portraits drawn in [START_REF] Buchert | Correspondence between kinematical backreaction and scalar field cosmologies -the 'morphon field[END_REF] correspond to the one-dimensional locus Ω D k = 0 in the plots of figure [START_REF] Arrowsmith | An introduction to dynamical systems[END_REF].

most comprehensive study of FLRW cosmologies that include radiation, dust and a cosmological constant has been provided by Ehlers and Rindler [START_REF] Ehlers | A phase-space representation of Friedmann-Lemaître universes containing both dust and radiation and the inevitability of a big-bang[END_REF]. Other studies, including Bianchi models, were treated by Wainwright and Ellis [START_REF] Wainwright | Dynamical System in Cosmology[END_REF] who proposed the so-called dynamical system approach, which we shall use in this paper. The same technique can be followed to investigate the dynamics of minimally coupled scalar fields in the FLRW framework [START_REF] Copeland | Exponential potentials and cosmological scaling solutions[END_REF]. Note that the backreaction fluid we study in this paper can also be interpreted as an effective scalar field [START_REF] Buchert | Correspondence between kinematical backreaction and scalar field cosmologies -the 'morphon field[END_REF] that allows comparison of this work with our analysis. More recently, dynamical system analysis was used for modified theories of gravity (fourth order [START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF][START_REF] Carloni | A dynamical system approach to higher order gravity[END_REF][START_REF] Carloni | Some remarks on the dynamical systems approach to fourth order gravity[END_REF][START_REF] Leach | Shear dynamics in Bianchi I cosmologies with R n -gravity[END_REF], scalar tensor [START_REF] Carloni | Cosmological dynamics of scalar tensor gravity[END_REF] and Hořava-Lifshitz [START_REF] Carloni | An analysis of the phase space of Hořava-Lifshitz cosmologies[END_REF]) leading to a greater understanding of the cosmology of these theories. Also, results of a recent investigation of LTB models [START_REF] Sussman | A dynamical systems study of the inhomogeneous ΛCDM model[END_REF] are in accord with our findings §.

In the next section we introduce the basic framework of averaged inhomogeneous cosmologies, choosing a vanishing cosmological constant for simplicity. We consider in section 3 the backreaction terms to obey scaling evolution laws, and we present and analyze the corresponding autonomous dynamical system. In section 4 we summarize and discuss the results, and we conclude in section 5.

Effective description of inhomogeneous cosmologies

Let us consider an inhomogeneous, irrotational and pressureless fluid (dust). The spatial average over a compact, restmass preserving domain D of the Raychaudhuri equation, the Hamilton constraint and the dust continuity equation reads in its rest frame (see [START_REF] Buchert | On average properties of inhomogeneous fluids in general relativity: dust cosmologies[END_REF][START_REF] Buchert | On average properties of inhomogeneous fluids in general relativity: perfect fluid cosmologies[END_REF][START_REF] Buchert | Dark energy from structure -a status report[END_REF] for details):

äD a D + 4πG 3 D = Q D 3 , (1)
ȧD a D 2 - 8πG 3 D + k D i a 2 D = - W D + Q D 6 , (2)
. D + 3 ȧD a D D = 0 , (3)
where the angle brackets stand for the Riemannian spatial volume average, and the overdot is the partial time-derivative, here identical to the covariant time-derivative. a D is the effective scale factor, D is the averaged energy density of the dust, Q D is the kinematical backreaction term, and W D is the averaged curvature deviation from a constant curvature, defined as:

a D := (V D /V D i ) 1/3 , D = D i a -3 D , (4)
Q D := 2 3 (θ -θ D ) 2 D -2 σ 2 D , W D := R D -6k D i a -2 D , (5)
with V D the volume of the domain, R its three-Ricci scalar curvature and θ and σ the expansion and shear rates of the dust. (Here and in the sequel, the subscript i stands § In [START_REF] Sussman | A dynamical systems study of the inhomogeneous ΛCDM model[END_REF] Sussman uses quasi-local variables instead of averaged ones. The relation he gives between them allows to establish common findings.

for the initial value.) Equation ( 2) is an integral of equation ( 1) under the conservation law (3), if and only if Q D and W D obey the integrability condition

a -6 D Q D a 6 D . + a -2 D W D a 2 D . = 0 . (6)
The set of equations (1-3) formally resembles, from a kinematical point of view, a Friedmannian cosmology sourced by two fluids, but there are some fundamental differences between the two frameworks. First, the system is valid for any metric, whereas the FLRW equations only hold for maximally symmetric metrics. Second, the second fluid (the backreaction fluid) emerges from the averaging process, it is not added in our model contrary to a two-fluid FLRW cosmology. This fluid represents the difference between the dynamics of a Friedmannian dust background and of a physical dust background obtained through the averaging procedure . Third, the dynamics of the averaged geometrical variables of our model generally differs from that of the FLRW model: R D might evolve differently from the Friedmannian curvature k D i a -2 D restricted to the same domain. Fourth and last, the averaging procedure is performed over a chosen domain, and it implies a scale-dependence of all the variables. In particular, one expects the backreaction fluid to exhibit rather distinct behaviors according to the scale [START_REF] Wiegand | Multiscale cosmology and structure-emerging dark energy: a plausibility analysis[END_REF].

For void domains ( = 0) the backreaction terms, encoded through geometrical invariants, correspond to inhomogeneities of the spatial geometry of the vacuum. For domains over which Q D = 0, inhomogeneities could still be locally present even though the kinematical backreaction identically vanishes on D due to an exact compensation between the expansion fluctuations and the shear ¶. In the same spirit W D = 0 only states that the averaged curvature deviation vanishes on D, but nothing can be inferred about the local curvature deviations within the domain. Homogeneity is only achieved, if the backreaction variables are required to vanish on all scales.

Since we aim at studying the instability of Friedmannian dust backgrounds, we shall consider for our analysis the quantity X D := Q D + W D , which corresponds to the whole averaged departure from the Friedmannian framework. From the constraint (6), a null X D implies vanishing Q D and W D , in which case we recover a FLRW background on average. Also, the dark components can be thought of as the manifestations of the deviation of the physical background from a FLRW one: a positive Q D contributes to accelerate the expansion of the domain and plays against gravity, whereas a negative Q D participates in the deceleration of the domain's expansion and adds to gravity (see equation ( 1)). In this sense, the instability sectors, defined through X D , represent the dark components of the concordance model.

Dynamical system for the scaling solutions

The scaling solutions

We need one additional relation to solve the system (1-3), and this closure relation expresses the freedom of the choice for the local structure of the inhomogeneities, encoded on average by the dynamics of the backreaction fluid + . A natural choice is given by establishing a formal analogy between the backreaction terms and a fluid, whose energy density and pressure depend on the volume of the domain [START_REF] Buchert | Correspondence between kinematical backreaction and scalar field cosmologies -the 'morphon field[END_REF]:

Q D ∝ V α D , W D ∝ V β D ⇒ Q D = Q D i a n D , W D = W D i a p D , (7)
where the scaling parameters n and p are real constants. Expressions [START_REF] Buchert | Towards physical cosmology: focus on inhomogeneous geometry and its nonperturbative effects[END_REF] together with (6) result in the following constraint the dynamical system has to satisfy:

(n + 6) Q D i a n D + (p + 2) W D i a p D = 0 . (8)
In what follows we shall concentrate on the class of scaling solutions n = p * , for which relation ( 8) implies

(n + 2)W D = -(n + 6)Q D ⇒ (n + 2)X D = -4Q D . (9)

The autonomous system

Let us introduce the Hubble functional H D := ȧD /a D and the volume deceleration functional q D := -1-ḢD /H 2 D . We define domain-dependent dimensionless cosmological characteristics as

Ω D m := 8πG 3H 2 D D , Ω D k := - k D i a 2 D H 2 D , Ω D X := - X D 6H 2 D , (10)
which by construction add up to 1 according to the Hamilton constraint (2). Upon using the expressions (9,10) we replace the system [START_REF] Arrowsmith | An introduction to dynamical systems[END_REF][START_REF] Barrow | On the stability of the Einstein static universe[END_REF][START_REF] Buchert | On average properties of inhomogeneous fluids in general relativity: dust cosmologies[END_REF][START_REF] Buchert | Dark energy from structure -a status report[END_REF] by

Ω D m -(n + 2)Ω D X = 2q D , (11)
Ω D m + Ω D k + Ω D X = 1 , (12)
Ω D m = Ω D m Ω D m -(n + 2) Ω D X -1 , (13)
Ω D X = Ω D X Ω D m -(n + 2) Ω D X + n + 2 , ( 14
)
where the prime denotes the derivative with respect to the evolution parameter N D := ln a D . The two last equations are easily derived noticing that (1/H 2 D ) = 2(q D + 1)/H 2 D . The autonomous system [START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF][START_REF] Carloni | A dynamical system approach to higher order gravity[END_REF] determines the orbit of a cosmological state, defined by the quartet (Ω D m , Ω D X , Ω D k , n), in the corresponding phase space, and its fixed points are provided in table 1. The phase space (Ω D m , Ω D X ) presents three invariant lines under the phase flow:

Ω D m = 0, Ω D X = 0 and Ω D k = 1 -Ω D m -Ω D X = 0.
+ The need for a closure relation is also present in the standard FLRW framework, and it corresponds to the choice of an equation of state for the fluid sources. * Note that the unique solution for n = p (n = -6, p = -2) is physically equivalent to the case n = p = -6: Q D follows the same evolution law in both situations, and so does the total averaged curvature R D which evolves proportionally to a -2 D .

Table 1. Fixed points of the dynamical system [START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF][START_REF] Carloni | A dynamical system approach to higher order gravity[END_REF]. The sign of their eigenvalues determines their stability [START_REF] Arrowsmith | An introduction to dynamical systems[END_REF].

Coordinates Scale factor Stability

(Ω D m , Ω D X ) {eigenvalues} point A (0, 0) a D (t) = H Di (t -t i ) + 1 n < -2, attractor n > -2, saddle {-1, n + 2} point B (1, 0) a D (t) = 3 2 H Di (t -t i ) + 1 2 3 n < -3, saddle n > -3, repeller {1, n + 3} point C (0, 1) a D (t) = -n 2 H Di (t -t i ) + 1 -2 n (n = 0) a D (t) = exp(H Di (t -t i )) (n = 0) n < -3, repeller -3 < n < -2, saddle n > -2, attractor {-n -3, -n -2} line L 1 Ω D m + Ω D X = 1 a D (t) = 3 2 H Di (t -t i ) + 1 2/3 (n = -3) line L 2 Ω D m = 0 a D (t) = H Di (t -t i ) + 1 (n = -2)
There are five situations for which the stability of the fixed points does not change: n < -3, n = -3, -3 < n < -2, n = -2 and n > -2. The fixed point A shares the same properties as a Milne domain (i.e. a void domain with FLRW kinematics and FLRW curvature on average); the fixed point B, corresponds to an Einstein-de Sitter dust domain (i.e. FLRW kinematics and zero FLRW curvature); and the fixed point C to domains filled only with the backreaction fluid. The stability of A and B can change according to the value of the scaling parameter, but they always depict the same cosmological models. In return, C always describes different cosmological models which might also have a different stability. For the bifurcation value n = -3, there is a transfer of stability between B and C which both belong to the line of fixed points L 1 , while at the bifurcation value n = -2, there is a transfer of stability between A and B, both belonging to the line L 2 .

Finally, note that the dynamical system [START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF][START_REF] Carloni | A dynamical system approach to higher order gravity[END_REF] is written according to the evolution parameter N D and not according to a time parameter. Consequently, the phase portraits drawn from this system do not suffice to deduce the time evolution of a cosmological state, since we also need information about the time evolution of N D or, equivalently, a D . We shall therefore also indicate for all phase portraits the behavior of the scale factor.

Phase portraits

The phase portraits of the five cases are shown in figure 1, and the numbers (background colors) of their regions correspond to the evolutions of a D given in figure 2. To deduce the time evolution of a state, one shall refer to these two sets of plots. For the sake of clarity, we provide hereafter a detailed explanation of the correspondence between them in the situation n < -3 (first plot of figure 1).

A state (Ω D m , Ω D X ) belonging to the region I (red) evolves from the fixed point C when its scale factor is zero (plot I (red) of figure 2), through its initial conditions, to infinity when its scale factor attains its maximum. This divergence is due to the singularity of the cosmological parameters for H D = 0 and not due to a physical boundless blow of the domain. Having reached its maximal extension, the domain shrinks and (Ω D m , Ω D X ) comes back from infinity and tends to C, while a D decreases from its maximum to zero. Hence, the fixed point C is both a past repeller and a future attractor.

In the region II.a (green), the states evolve from C, when a D is zero, towards A, when a D tends to infinity (plot II.a (green) of figure 2), without any accelerated period. In this region, C is a past repeller and A is a future attractor.

A state in IV (blue) leaves the fixed point A, while the scale factor decreases from infinity to its minimum in the past, and reaches infinity. It then goes back to A when the scale factor grows to infinity (plot IV.a (blue) of figure 2). The state undergoes an accelerated period when operating its bouncing back from infinity. Here, A is a past repeller and a future attractor.

Finally, in the region V (brown), (Ω D m , Ω D X ) runs from infinity at the minimum of a D to infinity at its maximum, and it oscillates (plot V (brown) of figure 2). The minimum of a D is reached when the orbit runs below the dashed line since it corresponds to a positive acceleration.

All the orbits that go to infinity only describe part of the evolution, i.e. up to a maximum or minimum of the scale factor. The rest of the evolution is described not by reversing the direction of time, but by reversing the monotonicity of a D . In other words, the state travels back the orbit of the phase portraits.

The other generic orbits expounded in figure 1 are the following (see also figure 2): a state in IV.b (blue) follows the same type of evolution as a state in IV.a except that it always undergoes an acceleration; the evolution of a state in II.b (green) differs from that of II.a by the fact that it experiences an acceleration from a finite time; and a state in III (darkest green) evolves from B, when a D is zero, towards C, when a D tends to infinity. In this region, B is a past repeller and C is a future attractor.

Finally, we present the evolution of the scale factor for domains lying on an invariant line. If a state belongs to the segment between two fixed points, it experiences a bigbang and a future expanding phase. If it belongs to a semiline, either the domain undergoes a big-bang and big-crunch evolution when the arrow on the invariant line leaves the fixed point, or the domain undergoes a past and future expanding phase when the arrow is orientated towards the fixed point. m , Ω D X ) of the dynamical system [START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF][START_REF] Carloni | A dynamical system approach to higher order gravity[END_REF]. From left to right and top to bottom: n < -3 (n = -4 for the illustration), n = -3, -3 < n < -2 (n = -2.5), n = -2 and n > -2 (n = -1).

The thick straight lines (dark blue) are the invariant lines Ω D m = 0, Ω D X = 0 and

Ω D k = 1-Ω D m -Ω D X = 0.
Every line parallel to the latter, the inclined line, corresponds to a constant Ω D k , with values increasing downwards. The dashed line shows a vanishing q D ; it is an orbit only for n = -3, and it coincides with Ω D m = 0 for n = -2. A domain is accelerated below this line and decelerated above when n < -2, and this correspondence is inverted for n > -2. The arrows show an increasing N D , the dots are the fixed points of table 1, and the square in the last plot stands for the observational values (see section 4.3). All the states of a region have a scale factor which evolves in the same way. We have chosen a positive H Di to deal with initially expanding domains and a null t i . Regions I (red) show big-bang and big-crunch domains, regions II big-bang and future expanding (with an asymptotically null acceleration) domains, regions III.1 (darkest green) describe big-bang and future expanding (accelerating) domains for -2 < n ≤ 0, regions III.2 (darkest green) big-bang and big-rip domains for n > 0, regions IV (blue) show past and future expanding domains, and regions V (brown) oscillating domains. We distinguish two subcases for the regions II and IV: for the former, regions II.a are never accelerated, while regions II.b are accelerated from a finite time onwards; for the latter, regions IV.a are accelerated during a finite period of time, while regions IV.b are always accelerated.

Summary and discussion

First and foremost, we clarify the definitions of the different types of domains at stake. A domain D is Friedmannian, i.e. locally homogeneous and isotropic, if and only if for all subdomains E ⊆ D one has Ω E Q = Ω E W = 0 or, equivalently, Ω E X = 0. We have focused our analysis on one single domain D without any multiscale considerations, so it is fair to assert that the Friedmannian realization is not achieved in our approach since no additional suppositions are made. A domain with a null Ω D X and with nothing more inferred about the Ω E X of the subdomains, is most probably non-Friedmannian: as noted in section 2, the cancellation of the backreaction fluid can be the result of an exact compensation between the expansion variance and the shear, and inhomogeneities and anisotropies may still be locally present. Nevertheless, such domains are destined to follow on average a Friedmannian evolution since their inhomogeneities do not contribute on average to the kinematics. We shall call these domains Friedmann-like domains. Therefore, we shall deal with Milne-like and Einstein-de Sitter-like domains, instead of locally Milne and Einstein-de Sitter domains. This complies with the actual physical situation: Friedmannian models as dynamical models for the average evolution are only realized for strictly homogeneous distributions of matter and geometry; but there exist situations -called Friedmann-like -in which Friedmannian models do provide the evolution on average, i.e. in which they describe the physical background.

Another terminology we employ makes use of equation ( 9): for n < -2, Ω D X > 0 corresponds to shear-dominated domains, and Ω D X < 0 labels expansion variancedominated domains. This correspondence is inverted for n > -2.

Summary of results

Before delving into details, simply recall that the orbits strongly depend on the leading term in the cosmic trio

Ω D m ∝ a -3 D H -2 D , Ω D k ∝ a -2 D H -2 D and Ω D X ∝ a n D H -2 D .
When n < -2, the constant curvature k D i , useful to compare the effective background with the Friedmann-like background, determines the dynamics of the domain, while in the remaining situations the backreaction fluid is the relevant quantity.

For n < -3 and for a negative constant-curvature k D i , shear-dominated and expansion variance-dominated domains are attracted by a Milne-like state. In the past, the former originates from a state with Ω D X 1, whereas the latter comes from a Milnelike state. Expansion variance-dominated domains are therefore asymptotically, in the past and in the future, dominated by a negative constant curvature. The bounce of these domains happens when the contribution of the backreaction fluid is no longer negligible. For a positive k D i , expansion variance-dominated domains are oscillating (this is the only situation where it occurs): when the volume of the domain is sufficiently small, the backreaction fluid -mimicking a dark energy behavior -becomes preponderant over the other components, thus avoiding the collapse. On the contrary, shear-dominated domains -mimicking a dark matter behavior -evolve towards a domain filled only with Ω D X and collapse in a finite time.

At n = -3, all domains with a negative k D i approach a Milne-like state, and all domains with a positive k D i eventually collapse. The attractor state of these latter depends on the initial conditions: though it always exhibits Ω D k = 0, all the values of Ω D m and Ω D X are attainable provided the Hamilton constraint is satisfied. For the particular state Ω D i k = 1 (dashed line of the second plot of figure 2), the domain experiences a stationary expansion ( ȧD = const.). Ω D k always being equal to 1 in this situation, the dust and backreaction fluid energies exactly compensate each other along the whole evolution. Furthermore, we here confirm that this stationary state is stable: any perturbation always leads back to a stationary state (as was conjectured in [START_REF] Buchert | On globally static and stationary cosmologies with or without a cosmological constant and the dark energy problem[END_REF], see also [START_REF] Barrow | On the stability of the Einstein static universe[END_REF][START_REF] Losic | Initial value problem for second order scalar fluctuations in the Einstein static universe[END_REF] for comparisons).

For -3 < n < -2, shear-dominated and expansion variance-dominated domains have the same qualitative asymptotic properties. For a negative k D i , they originate from an Einstein-de Sitter-like dust state and are attracted by a Milne-like state, whereas for a positive constant curvature they emanate from and converge to an Einstein-de Sitterlike state and eventually collapse. In these last scenarii, even if the backreaction fluid mimics a dark energy behavior over the domain, its intensity is not sufficient enough to counterbalance the collapse contrary to the case n < -3.

In all the previous situations (except for the region V (brown)), a positive constantcurvature domain is bound to collapse, and a negative constant-curvature domain always converges towards a Milne-like state. Therefore, a small perturbation around a null k D i seals a totally different fate according to its sign.

For n = -2, domains with Ω D i m < 1 are attracted by a Milne-like state, and domains with Ω D i m > 1 converge to an Einstein-de Sitter-like state and collapse within a finite time. The constraint (6) here implies Q D = 0 and W D ∝ a -2 D , which means that there is no kinematical backreaction, but also no curvature deviation from a constant curvature stricto sensu. All states follow the kinematics of a Friedmann-like state on average, and the whole drawn phase portrait is physically equivalent to the line Ω D X = 0. Finally, for n > -2, shear-dominated domains emerge from and are attracted by an Einstein-de Sitter-like dust state, and they eventually collapse. Expansion variancedominated domains also originate from an Einstein-de Sitter-like state, but they are attracted by a state filled only with the backreaction fluid, and they experience an accelerating expansion from a finite time onwards.

Discussion: instability sectors

A state leaves the class of FLRW backgrounds and enters into what we define as an instability sector as soon as the backreaction fluid contributes to the kinematics of the domain. The existence of a non-vanishing averaged contribution of inhomogeneities From figure 1 it is obvious that no Friedmann-like state is stable (except the point A), i.e. in those unstable cases no perturbation slightly above the Ω X D = 0 line relaxes to its original point of the phase space diagram. Therefore, we address the more general issue of stability of the class of backgrounds with Friedmann-like behavior.

invokes deviations of a cosmic state from its Friedmann-like characteristics. Given the generic behavior of any domain, we now address the issue of the stability of the FLRW backgrounds: does a Friedmann-like state, lying on the line Ω D X = 0, converge on average to the same or another Friedmann-like state when subjected to perturbations?

A convergence towards any Friedmann-like state or, equivalently, towards the class of FLRW backgrounds, happens when the two following conditions are met: first, X D → 0, which entails that inhomogeneities are negligible on average; second, Ω D X → 0, which states that the energy contribution of the backreaction fluid becomes subdominant on average compared to the other components. Note that for the particular case n = -2, as explained above, the averaged contribution of inhomogeneities vanishes, and all the states are Friedmann-like. This situation does not admit any global inhomogeneity effect, and we shall therefore disregard it in our discussion.

Based on the previous analyses, we conclude that FLRW backgrounds are stable for: (i) n < -2 and k D i < 0, the Milne-like state is an attractor for any inhomogeneous deviations, and (ii) n < -3 and k D i = 0, the Einstein-de Sitter-like dust state is eventually reached. In these situations the FLRW backgrounds are stable, in the situation (i) the background can only be attained asymptotically.

In the other situations, the perturbations of any Friedmann-like state lead either to Ω D X → 1 or X D → ∞ or both, namely to a non Friedmann-like state. The FLRW backgrounds are therefore globally unstable for: (iii) n < -2 and k D i > 0, (iv) -3 ≤ n < -2 and k D i = 0, and (v) n > -2. In these situations they obviously constitute an incorrect approximation to the physical background.

Discussion: interpretation as dark sectors

The different cosmological observations interpreted in the context of FLRW cosmologies constrain the instability sectors to which the dark components could be associated. For instance, we may put a conservative upper bound on the present-day cosmic state, Ω D m,today < 1, i.e. any physical orbit should respect this bound, if we consider smallscale (galaxy clusters and voids) or large-scale (CMB and high-redshift supernovae) cosmological domains. For instance, this restriction forbids some of the orbits for these gravitational systems: they cannot belong to the region V (brown) of figure 1, and so we cannot have oscillating domains.

The dark matter component can be associated with domains that are dominated by shear fluctuations, and which eventually collapse. The instability sectors of the regions I (red) in the first plot of figure 1, in the second and third plots (with Ω D X > 0), and in the fifth plot (with Ω D X < 0) may present an effective dark matter behavior for the backreaction fluid. Concerning the dark energy component, it is effectively mimicked in the instability sectors of the region IV.b (blue) in the second plot, II.b (green) in the third plot, and III (darkest green) in the last plot. In each of these cases, there is a global acceleration of the domain. Note, interestingly, that the sign of the third derivative of a D may also change, allowing for a variety of different dynamical evolution histories of the dark energy component. In the following, we pick two proto-type examples of instability sectors to which the dark components can effectively be associated.

Consider a small domain filled mainly with dust (Ω D i m 1), dominated by its shear fluctuation (Ω D i X 0), with an initially slightly positive constant curvature (Ω D i k 0) and such that n -3. This situation can correspond to negative or positive averaged scalar curvature R D , and it describes a pancake-like configuration of dust, like a proto-sheet (or proto-filament). Such a domain evolves towards a state dominated by the backreaction fluid, passing close to the observational ratio Ω D m /Ω D X = Ω baryon /Ω DM 0.21, and eventually collapses † † (see figures 1 and 2). This example shows that the backreaction terms can turn out to play the role of a dark matter component, thereby accelerating the formation of structures by adding gravity and accreting the dust.

Consider a large domain, say our Hubble sphere, endowed with an initially slightly negative constant curvature (Ω D i k 0) and mainly composed of dust (Ω D i m 1). Suppose also that the expansion variance slowly varies and dominates the backreaction fluid (so -2 < n 0 and Ω D i X 0). Such a domain is always characterized by a negative averaged scalar curvature R D and enters, from a finite time onwards, a phase of accelerated expansion, may pass close to the observational value (Ω D m , Ω D X ) (0.27, 0.73) (the square in figure 1), and eventually ends up being entirely dominated by the backreaction fluid. Recall that the backreaction fluid can be formulated as a scalar field [START_REF] Buchert | Correspondence between kinematical backreaction and scalar field cosmologies -the 'morphon field[END_REF], and here its equation of state would be given by ω = -1 -n/3 -1. These accelerated expansion scenarii include the case found in perturbation theory [START_REF] Li | On the onset of cosmological backreaction[END_REF], and they would be interpreted as the signature of dark energy in the standard model.

These two examples, though not depicting exhaustively all the possible scenarii, show how the instability sectors of FLRW backgrounds can be linked to the dark components.

Concluding remarks

We have studied in the present paper the global gravitational stability of FLRW backgrounds in the class of scaling averaged inhomogeneous models. Our investigation has led to a detailed classification of stability and instability sectors in the phase space of the scaling averaged cosmologies, and it has shown that averaged models are driven away from FLRW backgrounds through structure formation and accelerated expansion. Furthermore, motivated by reasonable physical assumptions, we have been able to relate both dark components of the cosmological concordance model to the instability sectors of the dynamical system.

As for the cases where the FLRW backgrounds are found to enjoy stability, we have to add a disclaimer. Despite their stable character, these FLRW backgrounds might not serve as reliable approximations in the role of physical backgrounds. As an example we look at the region II.b (green) in figure 1: we appreciate that the background experiences a transient acceleration period in between an initial and an asymptotic Friedmann-like state. Such a situation shows that FLRW backgrounds, even if globally stable, do a priori not provide a correct approximation for the physical background during the entire evolution.

A last remark concerns the use of exact scaling solutions for the class of averaged models analyzed. Indeed, there is no proof that scaling solutions depict all the instabilities of FLRW backgrounds, but it is also highly improbable that this is the only class of averaged models where instability occurs. Since our point was to explicitly demonstrate the possible unstable character of Friedmann-like domains, the use of scaling solutions is perfectly legitimate, though not exhaustive. In other words, our study is restrictive in this sense, since scaling solutions, albeit exact, are only indicative for the existing instabilities. One could also speculate that scaling solutions are acceptable building blocks to describe any backreaction behavior, as any analytical function is the sum of a Laurent series; however the correspondence is not that trivial due to the nonlinearity of the problem.

Figure 1 .

 1 Figure 1. Phase space (Ω Dm , Ω D X ) of the dynamical system[START_REF] Carloni | Cosmological dynamics of R n gravity[END_REF][START_REF] Carloni | A dynamical system approach to higher order gravity[END_REF]. From left to right and top to bottom: n < -3 (n = -4 for the illustration), n = -3, -3 < n < -2 (n = -2.5), n = -2 and n > -2 (n = -1).The thick straight lines (dark blue) are the invariant lines Ω D m = 0, Ω D X = 0 andΩ D k = 1-Ω D m -Ω D X = 0.Every line parallel to the latter, the inclined line, corresponds to a constant Ω D k , with values increasing downwards. The dashed line shows a vanishing q D ; it is an orbit only for n = -3, and it coincides with Ω D m = 0 for n = -2. A domain is accelerated below this line and decelerated above when n < -2, and this correspondence is inverted for n > -2. The arrows show an increasing N D , the dots are the fixed points of table 1, and the square in the last plot stands for the observational values (see section 4.3).

Figure 2 .

 2 Figure 2. Evolution of the scale factor of a domain with respect to H Di t. The numbers (colors) refer to the different regions of the phase spaces in figure1. All the states of a region have a scale factor which evolves in the same way. We have chosen a positive H Di to deal with initially expanding domains and a null t i .Regions I (red) show big-bang and big-crunch domains, regions II big-bang and future expanding (with an asymptotically null acceleration) domains, regions III.1 (darkest green) describe big-bang and future expanding (accelerating) domains for -2 < n ≤ 0, regions III.2 (darkest green) big-bang and big-rip domains for n > 0, regions IV (blue) show past and future expanding domains, and regions V (brown) oscillating domains. We distinguish two subcases for the regions II and IV: for the former, regions II.a are never accelerated, while regions II.b are accelerated from a finite time onwards; for the latter, regions IV.a are accelerated during a finite period of time, while regions IV.b are always accelerated.

This is an essential remark when one considers perturbations, the choice of the background being crucial. ¶ For example, this happens for zero-curvature, spherically symmetric Lemaître-Tolman-Bondi solutions, see[START_REF] Buchert | Towards physical cosmology: focus on inhomogeneous geometry and its nonperturbative effects[END_REF] in this volume.

† † Note that our effective model does not take into consideration neither the pressure nor the vorticity of the matter at stake, which obviously cannot be neglected at the end of the collapse process.

Acknowledgements:

The authors are thankful for the valuable comments of the anonymous referees. XR thanks Miv Oto for useful discussions. This work is supported by "Fédération de Physique André-Marie Ampère" of Université Lyon 1 and École Normale Supérieure de Lyon.


Y Ortigoza
email: ortigoza@unizar.es

N Coron

C Cuesta

E García

C Ginestra

J Gironnet

P De Marcillac

M Martínez

C Pobes

J Puimedón

T Redon

M L Sarsa

L Torres

J A Villar

Energy partition in Sapphire and BGO scintillating bolometers

Keywords: Cryogenic detector, Scintillating bolometer, Energy partition, Light yield, Dark matter

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Bolometers are cryogenic detectors widely used in the last decades (see for instance [1,2]) which measure as heat 2 the energy deposited by particle interactions. They consist of an absorber where interacting particles deposit energy 3 producing an increase of temperature ∆T and a thermal sensor (thermally connected to the absorber on one side and 4 to a thermal bath via a heat leak on the other side) where ∆T is converted into an electrical signal. Massive bolometers 5 can be built with dielectric and diamagnetic crystals cooled at temperatures of a few millikelvin because their heat 6 capacity, which is only ascribed to vibrations of the crystal lattice and depends on the cube of the temperature, can be 7 low enough to produce a measurable ∆T .

The simultaneous measurement of heat and light signals was proposed in 1988 as a new tool for particle detection 9 [START_REF] González-Mestres | Proceedings of "the 2nd European Workshop on Low Temperature Devices for the Detection of 162 Low Energy Neutrinos and Dark Matte[END_REF]: the very different scintillation yield of different particles (β/γ particles, α particles and other recoiling nuclei) 10 could be used to identify the type of particle, whereas the heat signal could allow to estimate the deposited energy with 11 high resolution. The viability of this technique was shown in 1992 by the Milano group [START_REF] Alessandrello | [END_REF] using a Si photodiode glued 12 on a CaF 2 (Eu) bolometer to measure scintillation photons. In 1997 a French group [5] developed a new technique for 13 the detection of scintillation photons using a second bolometer. This double bolometer technique is currently being 14 used by the ROSEBUD Collaboration [START_REF] Coron | Proceedings of "Identification of Dark Matter 2008[END_REF], the CRESST Dark Matter Search Experiment [7] and the Milano Group [8] and applied to rare event searches as double beta decay and dark matter. characterized by ROSEBUD (see for instance [START_REF] Coron | Proceedings of "Identification of Dark Matter 2008[END_REF]9]). This work refers to a 50 g Al 2 O 3 and a 46 g BGO scintillating thermally coupled to the mixing chamber of a dilution refrigerator operated at 20 mK. For technical details about the 28 design, experimental set-up and operation of scintillating bolometers see [10,11] and references therein.

29

Events in scintillating bolometers can be represented by points in a light versus heat pulse amplitude scatter 30 plot. In these plots, monoenergetic events like those due to full absorption of γ-rays of a given energy appear as 31 binormal distributions (see figure 2). Both channels, if independent, should not be correlated but we observed, first 32 in a nominally pure Sapphire crystal [10] (with a measured Ti concentration of 6 ppm [12]) and then in a nominally 33 pure BGO crystal [11], a negative correlation between the light and heat signals of these binormal distributions (see 34 table 1) pointing at some energy transfer from one to the other channel. The same effect has also been noticed in Decay experiments [13] and also in a dual phase liquid-gas Xe dark matter detector [14].

37 outputs in the light and heat signals are directly related to the fraction of the deposited energy initially converted into 39 light, α ℓ (we will refer to it as the absolute light yield), and the fraction of the deposited energy initially converted 40 into heat, α h , respectively; the correlation coefficient originated by an energy transfer from one channel to the other is 41 independent of α ℓ and α h ; taking advantage of that, from these values one can estimate the ratio α ℓ /α h . This method, 42 developed for the Al 2 O 3 scintillating bolometer was subsequently applied to the BGO scintillating bolometer [11]. In these previous works, assuming that all the deposited energy is converted into heat and light (i.e., α ℓ + α h = 1), yield of BGO at room temperature to be 6900 ± 140 photons/MeV [16], they have estimated that the BGO light yield From the measured spectrum I m (λ) at 77 K, shown in figure 4, we can estimate the relative intensity of scintillation 78 photons emitted n(λ) at this temperature as

79 n(λ) = I m (λ) η c (λ) • η f (λ) • η g (λ) • S s (λ) ( 1
)
where λ is the wavelength, η c , η f and η g are the transmission efficiencies of the crystal cavity, of the optical fiber and 80 of the diffraction grating, respectively, and S s is the spectral relative sensitivity of the CCD detector (data on S s are 81 only available from 400 nm to 700 nm but it practically covers the full BGO emission spectrum). Figure 5 shows 82 the dependence with λ of these four correction coefficients. Three of them (η f , η g and S s ) have been taken from the 83 manufacturer's datasheets. The cavity transmission efficiency, η c , is defined as the ratio between the light collected by 84 the optical fiber I c (λ) and that emitted by the crystal I e (λ). This coefficient can be estimated in a simple model by where the first term, x, represents the fraction of light directly collected by the optical fiber and the second one, the 87 light collected after successive reflections in the cavity, being R(λ) the reflectivity of the cavity coating [START_REF] Bass | Handbook of Optics[END_REF] (also

85 η c (λ) = I c (λ) I e (λ) = x + R(λ) • (1 -x) • ϕ • ∞ n=0 R n (λ) • (1 -ϕ) n (2) 86 η c (λ) = x + R(λ) • ϕ 1 -R(λ) • (1 -ϕ) • (1 -x) (3)
88
shown in figure 5) and ϕ the probability that a photon reaches the fiber with a correct angle for transmission. With our

89 geometry we have x = S f S c Ω f Ω c and ϕ = S f S c Ω f 2π
, where S f is the cross section of the optical fiber, S c is the cavity total 90 area, Ω f ≈ π • NA 2 is the solid angle for transmission of photons by the fiber and Ω c is the solid angle subtended by 91 the crystal at the optical fiber position.

Figure 5: Transmission efficiencies of the crystal cavity (η c ), of the optical fiber (η f ), and of the diffraction grating (η g ), and spectral relative sensitivity of the CCD detector of the spectrometer (S s ) as a function of the wavelength. Also shown is the Ag reflectivity R(λ) [START_REF] Bass | Handbook of Optics[END_REF] used for the η c estimate.

Values derived for η c (λ) from equation 3 (normalized to one for 700 nm) are shown in figure 5. Figure 6 shows the

E = hc 1 λ = hc          1 N λ f λ=λ i 1 λ • n(λ)          (4)
where N = λ f λ=λ i n(λ) and, in our case, λ i = 400 nm and λ f = 700 nm. We estimate a value of E = 2.436±0.017 stat ± 97 0.068 syst eV, where the systematic error is introduced to take into account the small effects produced by the presence of 98 the teflon pieces and the Al foil on top of the holder cavity, and also by the crystal autoabsorption. It has been estimated 99 as the difference between the mean energy obtained with and without the correction factors inclusion (2.436±0.017 eV 100 and 2.368 ± 0.017 eV). Assuming that, as reported by [15,[START_REF] Itoh | [END_REF], below 100 K the scintillation response remains 101 practically constant and, as suggested by our results shown in figure 4, the BGO scintillation spectrum practically 102

does not change at low temperatures, we can take this value as the average energy of BGO scintillation photons 103 below 100 K and, in particular, at 20 mK (the operation point of our bolometers). In fact, our estimate of E is in 104 good agreement with studies of BGO luminescence at temperatures of a few Kelvin: for instance, in [19] operating 105 at 10 K they report a broad luminescence band at 2.46 eV, whereas in [20] they observe at 5 K an intense emission 106 band at around 510 nm (2.43 eV). This comparison suggests that practically the same scintillation mechanism and, 107 consequently, the same scintillation response are observed in the different BGO crystals analyzed in the literature (see

108
for instance [21][22][23]).

Light yield and energy partition in scintillating bolometers

From the average energy of BGO scintillation photons E estimated in the previous section and the BGO light 111 yield of 23700 ± 2600 photons/MeV at 6 K quoted by [15], we obtain that the absolute light yield of the BGO at 112 20 mK is α ℓ = 0.058 ± 0.006 stat ± 0.002 syst .

113

Carrying out an absolute energy calibration of the light signal with an inner 55 Fe x-ray source placed opposite to 114 the optical Ge bolometer of the BGO scintillating bolometer, we can also estimate its light collection efficiency. We 115 have observed that an x-ray photon of 5.9 keV from the 55 Fe source fully absorbed in the Ge bolometer produces Al 2 O 3 BGO α ℓ 0.112 ± 0.012 ± 0.004 0.058 ± 0.006 ± 0.002 α h 0.778 ± 0.099 ± 0.028 0.464 ± 0.092 ± 0.016 α 0 0.110 ± 0.100 ± 0.028 0.478 ± 0.092 ± 0.016

Conclusions

16 ROSEBUD(

 16 Rare Objects SEarch with Bolometers UndergrounD) is a collaboration between the Universidad de 17 Zaragoza and the Institut d'Astrophysique Spatiale (IAS), which develops and tests different materials as scintillating 18 bolometer prototypes in order to apply them as detectors in nuclear and particle physics, with special emphasis on dark 19 matter searches. Scintillating bolometers of Al 2 O 3 , BGO, CaWO 4 and LiF among others, have been developed and

20

 20

35a

  CdWO 4 scintillating bolometer tested by members of the CUORICINO/CUORE Collaboration for Double Beta 36

74Figure 3 :

 3 Figure 3: a) Experimental set-up of the cryostat and the crystal holder used for measuring the BGO emission spectra at different temperatures. b) Detail of the Cu holder with the BGO crystal. Here ∅ and t stand for, respectively, the diameter and thickness dimensions (given in mm) of the different components.

Figure 4 :

 4 Figure4: BGO differential scintillation spectra I m (λ) measured at different temperatures from room temperature down to 77 K with an x-ray excitation source. All measurements were made with an integration time in the spectrometer of 5 s. Intermediate measurements were performed dynamically while the cryostat was cooling down, measuring the temperature of the surroundings of the crystal. These values are not given because they do not reflect the exact temperature of the crystal (it could be hotter). The last three-four intermediate spectra (with surroundings temperatures around 130-160 K) and the 77 K spectrum show that there is no noticeable change in shape.

93

  measured scintillation spectrum I m (λ) of BGO at 77 K (raw data) and the relative intensity of scintillation photons n(λ) 94 emitted at this temperature (corrected data) as obtained from equation 1 and normalized to I m (λ) area. The average 95 energy of BGO scintillation photons at 77 K can be estimated from n(λ) as 96

Figure 6 :

 6 Figure 6: Raw data stands for the measured spectrum I m (λ) for BGO at 77 K and corrected data for the relative intensity of scintillation photons emitted at this temperature (obtained after corrections explained in the text and normalized to the same area of I m (λ)).

116Table 2 :

 2 the same light pulse amplitude that a photon of 841.3 ± 8.3 keV fully absorbed in the BGO scintillating crystal.117Neglecting the fraction of the x-ray energy lost in traps in the Ge target, we derive an equivalent light energy of 118 7.01±0.07 keV/MeV and a light collection efficiency of 0.121±0.013 stat ±0.004 syst for the double bolometer mounting.119Notice that the Ge bolometer shows an excellent linearity as optical detector of the BGO in the range from 88 to 120 1633.3 keV[24].121In the case of the Sapphire bolometer, since we do not have a measurement of the absolute light yield at low 122 temperatures (of the order of a few Kelvin) for this Ti concentration (measurements at 9 K for others Ti concentration 123 are given in[25]), we do conversely: from the light collection efficiency we estimate the absolute light yield. Since 124 both scintillating bolometers have a very similar geometry, reflecting cavity (Ag-coated Cu) and optical detector, we 125 have assumed that light collection efficiency is the same. On the assumption that Ag reflectivity (ranging from 85 to 126 95%, as shown in figure5) is the dominant effect, influence of the different detector material and surface has been 127 neglected. The absolute energy calibration of the optical Ge bolometer of the Sapphire scintillating bolometer with 128 Energy partition in Sapphire and BGO scintillating bolometers for γ-ray interactions. Here are shown values obtained for the fraction of energy deposited that initially is converted into light (α ℓ ), heat (α h ) or trapped (α 0 ). The first and second errors given correspond to the statistic and systematic ones, respectively.

  Schematic view of a double bolometer configuration. For the measurements presented in this work an inner 55 Fe x-ray source was placed opposite the Ge bolometer to calibrate the energy absorbed on it.

							Op tical detector
							(Ge di s k)
		Scintillating cr ys tal
							55Fe source
		NTDGe
							NTD9Ge
							Therm al link
		Thermal link
		Reflecting cavit y (Agcoated Cu)				Op tical fiber
		0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 1: 0.0 Light signal amplitude (V)	0.5	1.0	1.5	/	2.0 t s n e v e Nuclear recoils 2.5	3.0
				Heat signal amplitude (V)
		Figure 2: Light versus heat discrimination plot obtained with the Al 2 O 3 scintillating bolometer [10]. The lines of 59.5 keV (from a 241 Am source),
		122.1 and 136.5 keV (from a 57 Co source) are clearly seen in the β/γ band at values (in the heat channel) around 1.2, 2.4 and 2.7 V and show a
		negative correlation between the light and heat pulse amplitudes. A similar behaviour is observed in the BGO [11].
	22	bolometers. As shown in figure 1, these scintillating bolometers consist of a scintillating crystal (Al 2 O 3 or BGO) and
	23	a small Ge disk looking at the scintillating crystal, both mounted inside a copper cavity internally coated with Ag
	24	to obtain a good reflectivity. The scintillating crystal and the Ge disk have each one a neutron transmutation doped
	25	(NTD) Ge thermistor glued to it in order to measure thermal increases. The heat produced in the scintillating crystal
	26	is directly measured by its NTD-Ge sensor and the emitted light escaping from it that is eventually absorbed in the
	27	Ge crystal produces a thermal increase that is measured by the other NTD-Ge sensor. The double bolometers are
		Preprint submitted to Astroparticle Physics						December 7, 2010

Table 1

 1 shows the values derived in both bolometers for the ratio α ℓ /α h at different γ-ray energies. One can observe 44 that, in the range of energies analyzed, values obtained for α ℓ /α h in each scintillating bolometer are compatible with

	45
	46	a constant value, independent of the energy, with weighted mean 0.144 ± 0.010 and 0.125 ± 0.021 for Sapphire and
		BGO, respectively.

Table 1 :

 1 Correlation coefficient (ρ) between heat and light pulse amplitudes on full absorption γ-ray lines and the derived ratio between the fraction of the deposited energy initially converted into light α ℓ and into heat α h for scintillating bolometers of Sapphire and BGO operated at 20 mK. a The line of 1633.3 keV corresponds to a measurement made in different experimental conditions (lower gain). The first and second errors given for α ℓ /α h correspond to the statistic and systematic ones, respectively. Data taken from[10] for Al 2 O 3 and[11] for BGO.

		Energy (keV)	ρ	α ℓ /α h
	Al 2 O 3	59.5	-0.61 ± 0.02	0.111 ± 0.006 ± 0.037
		122.1	-0.890 ± 0.005 0.146 ± 0.004 ± 0.012
		136.5	-0.87 ± 0.01	0.148 ± 0.009 ± 0.015
	BGO	88.0	-0.24 ± 0.02	0.145 ± 0.004 ± 0.073
		351.9	-0.39 ± 0.05	0.124 ± 0.010 ± 0.045
		569.7	-0.29 ± 0.03	0.157 ± 0.009 ± 0.070
		609.3	-0.27 ± 0.07	0.103 ± 0.012 ± 0.049
		657.7	-0.54 ± 0.06	0.119 ± 0.013 ± 0.030
		1633.3 a	-0.21 ± 0.08	0.153 ± 0.021 ± 0.082

Notice that this value is consistent with that obtained at 77 K in[16] supporting the observed independence with temperature of the light yield below 100 K.

Acknowledgments 152

This work has been supported by the Spanish Ministerio de Ciencia e Innovación (MICINN) (FPA2008-03228

We have measured the scintillation spectra of BGO at different temperatures from room temperature to 77 K.

141

From these spectra we have observed that the light output of the crystal increases and the spectral shape changes when 142 cooling down. However, at the lowest temperatures both the light output and the spectral shape remain practically


Catherine Feart
email: catherine.feart@isped.u-bordeaux2.fr

Benjamin Allès

Bénédicte Merle

Cécilia Samieri

Benjamin Alles

Pascale Barberger-Gateau

Adherence to a Mediterranean

Keywords: Mediterranean diet, energy intake, nutrient intake

INTRODUCTION

In developed countries, the increasingly higher life expectancy is mostly attributed to a decline of mortality at older ages. Better living conditions and, notably healthy dietary practices, may explain this greater longevity in addition to considerable improvement in health care of older persons [START_REF] Trichopoulou | Mediterranean diet and longevity[END_REF]. Among the general lifestyle recommendations, the traditional Mediterranean diet (MeDi) seems an optimal dietary strategy for health since greater MeDi adherence has been associated with longer survival, reduced risk of cardiovascular or cancer mortality, and reduced risk of neurodegenerative diseases [START_REF] Roman | Effectiveness of the Mediterranean diet in the elderly[END_REF][START_REF] Sofi | Accruing evidence on benefits of adherence to the Mediterranean diet on health: an updated systematic review and metaanalysis[END_REF][START_REF] Sofi | Adherence to Mediterranean diet and health status: meta-analysis[END_REF]. The traditional MeDi is characterized by high consumption of plant foods (vegetables, fruits, legumes and cereals), high intake of olive oil as the principal source of monounsaturated fat but low intake of saturated fat, moderate intake of fish, low-to-moderate intake of dairy products, low consumption of meat and poultry, and wine consumed in low-tomoderate amounts, normally with meals [START_REF] Trichopoulou | Diet and overall survival in elderly people[END_REF][START_REF] Willett | Mediterranean diet pyramid: a cultural model for healthy eating[END_REF]. Therefore, the MeDi combines several foods and nutrients already proposed as potential protective factors against several conditions that may contribute to the development and progression of age-related diseases [6, 12-13, 22-23, 26, 29-30, 34]. However, due to its computation method [START_REF] Trichopoulou | Diet and overall survival in elderly people[END_REF], a similar score on the MeDi does not mean a similar intake of nutrients across populations. Only few studies have yet conducted precise descriptions of its nutritional value [START_REF] Bountziouka | Validity of dietary patterns derived in nutrition surveys using a priori and a posteriori multivariate statistical methods[END_REF]. Thus, a better knowledge of the dietary characteristics of individuals adhering more closely to a MeDi is necessary to support the biological plausibility of its protective effects before promoting this dietary pattern as a universal healthy diet.

The purpose of the current study was to describe the intake of macronutrients, micronutrients, minerals and trace elements according to MeDi adherence among French older communitydwellers, in order to explain the potential contribution of these nutrients to the health benefits of the MeDi.

METHODS

Participants

The data come from the Three-City (3C) study, a prospective cohort study of vascular risk factors of dementia whose methodology has been described elsewhere [START_REF]The 3C Study Group (2003) Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]

Dietary assessment

Participants were visited at home by a specifically trained dietician who administered a food frequency questionnaire (FFQ) and a 24 h dietary recall [START_REF] Feart | Energy, macronutrient and fatty acid intake of French elderly community dwellers and association with socio-demographic characteristics: data from the Bordeaux sample of the Three-City Study[END_REF][START_REF] Samieri | Dietary patterns derived by hybrid clustering method in older people: association with cognition, mood, and self-rated health[END_REF]. During the 24 h recall the dietician registered all the meals and beverages consumed in a period of exactly 24 h before the subject awoke on the day of the interview. No week-end day was recorded. Quantities were assessed according to a book of photographs [START_REF] Su | Portions alimentaires. Manuel photos pour l'estimation des quantités[END_REF] edited for the SUVIMAX study [START_REF] Hercberg | A primary prevention trial using nutritional doses of antioxidant vitamins and minerals in cardiovascular diseases and cancers in a general population: the SU.VI.MAX study--design, methods, and participant characteristics. SUpplementation en VItamines et Mineraux AntioXydants[END_REF].

A table gave the correspondence between the portion size and the weight of the food item.

Photographs of dishes and glasses with the corresponding volume were also available. The same dietician then entered the data of the 24 h recall in the BILNUT® software to obtain an estimation of the daily nutrient intake of each participant. Food composition tables were already included in the BILNUT® software and complemented as earlier described [START_REF] Feart | Energy, macronutrient and fatty acid intake of French elderly community dwellers and association with socio-demographic characteristics: data from the Bordeaux sample of the Three-City Study[END_REF].

Among the whole nutrient database, key macronutrients (proteins, carbohydrates, fats and alcohol) were used to compute total energy intake (EI). The different classes of lipids (saturated (SFA), monounsaturated (MUFA) and polyunsaturated (PUFA) fatty acids) were recorded, as the consumption of carotenes, fibres, retinol, vitamins B1, B2, B5, B6, B12, C, D, E, folate, and of minerals and trace elements (calcium, phosphorus, magnesium, potassium, iron, and zinc).

Computation of the MeDi score

Based on the FFQ, frequency of consumption of 40 categories of foods and beverages for each of the three main meals and three between-meals snacks was recorded in eleven classes.

The number of usual weekly servings, but not portion size, of each of the 148 food items and alcohol, ranging as a result from 0 to 42 was recorded (with a maximum of six meals a day).

The food items were aggregated into 20 food and beverage groups [START_REF] Samieri | Dietary patterns derived by hybrid clustering method in older people: association with cognition, mood, and self-rated health[END_REF] and we identified those considered to be part of the MeDi: vegetables, fruits, legumes, cereals (including bread, pasta and rice, without distinction between whole and refined grains), fish, meat and dairy products. The number of servings/week for each food group was determined and the MeDi score was computed as follows: a value of 0 or 1 was assigned to each food group using sexspecific medians of the population as cut-offs [START_REF] Trichopoulou | Adherence to a Mediterranean diet and survival in a Greek population[END_REF]. Briefly, subjects received one point if their intake was higher than the sex-specific median for a presumed protective component (vegetables, fruits, legumes, cereals, fish and MUFA-to-SFA ratio) and lower than the sexspecific median for a presumed deleterious component (meat and dairy products). For alcohol, one point was given to mild-to-moderate consumers. Cut-offs, chosen to be close to the second quartile of distribution of total alcohol consumption, were defined in men and women separately [START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF]. One point was attributed for men if consumption was within 6-to-14 glasses per week (8.5-to-20g/d) and for women if consumption was within 1-to-4 glasses per week (1.4-to-5.7g/d). Data from the 24 h recall were used to compute the MUFA-to-SFA ratio (ratio of the nutrient intake in g/d). The MeDi score was generated by adding the scores for each food category for each participant. Thus, the MeDi score could range from 0 to 9, with higher scores indicating greater MeDi adherence [START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF].

Covariates

Socio-demographic information included age, gender, education, income and marital status.

Vascular risk factors included measured Body Mass Index (BMI in kg/m 2 ), diabetes, hypercholesterolemia (total cholesterol ≥ 6.2 mmol/L) and hypertension (measured blood pressure ≥ 140/90 mmHg or treated).

Statistical analyses

All statistical analyses were performed with SAS Statistical package (Version 9.1 SAS Institute). Participants were classified according to categories of MeDi adherence (Low MeDi adherence, score 0-3; Middle MeDi adherence, score 4-5; or High MeDi adherence, score [START_REF] Berr | Olive oil and cognition: results from the three-city study[END_REF][START_REF] Bhupathiraju | Coronary heart disease prevention: nutrients, foods, and dietary patterns[END_REF][START_REF] Bountziouka | Validity of dietary patterns derived in nutrition surveys using a priori and a posteriori multivariate statistical methods[END_REF][START_REF] Dai | Association between adherence to the Mediterranean diet and oxidative stress[END_REF] defined to be nutritionally relevant, as previously described and used in the literature [START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF][START_REF] Scarmeas | Physical activity, diet, and risk of Alzheimer disease[END_REF][START_REF] Trichopoulou | Adherence to a Mediterranean diet and survival in a Greek population[END_REF]. Demographic and clinical characteristics were compared between the three categories of the MeDi score. We then explored associations between the MeDi adherence at baseline and EI and macronutrient intakes, as expressed in proportion of total energy intake (% EI) with alcohol. We also compared micronutrient intakes, expressed in g/d, across categories of MeDi adherence. Separate analyses were performed for each gender since the MeDi score was computed according to the sex-specific medians of consumption of food groups.

RESULTS

The study sample consisted of 1595 individuals (607 men and 988 women), aged 76.1y on average (range 67.7-94.9) in 2001-2002.

The MeDi score ranged from 0 to 9 with 1.1% of the total population in the extreme values (0 or 9); 43.5% participants had a MeDi score of 4 or 5. The mean (SD) MeDi score was 4.36

(1.67) and was slightly but significantly higher in men (4.54 (1.72) vs 4.25 (1.63) in women, P=0.0006). High MeDi adherence (score 6-9) was significantly more frequent in men (30.1% vs 23.0% in women, P=0.003). As expected, greater MeDi adherence was characterized by higher intake of vegetables, fruits, legumes, cereals and fish and lower intake of meat and dairy products (Table 1). Greater MeDi adherence was associated with male sex and being married (Table 2). Individuals in the middle or high MeDi categories had a lower mean BMI than those in the lowest category of MeDi score. Greater MeDi adherence was not associated with hypertension, hypercholesterolemia or diabetes (Table 2).

In men, mean EI did not significantly differ across categories of MeDi adherence (Table 3).

However, men in the middle MeDi category (MeDi score 4-5) had a significant higher mean EI computed without alcohol than the others. Regarding total protein intake, there was no significant difference among the three MeDi categories, although the mean protein intake from vegetal origin was higher with greater MeDi adherence. The mean carbohydrate and total fat intakes did not vary with MeDi adherence. MeDi adherence was inversely associated with total SFA intake and consumption of myristic, palmitic and stearic acids in men. Total MUFA intake was higher in men in the middle and highest MeDi categories, and oleic acid intake was only higher in the middle MeDi category compared with the others. Total PUFA, notably total n-6 PUFA and linoleic acid intakes, were higher in the middle MeDi category.

Moreover, there was no association between n-3 PUFA intake and MeDi adherence in men.

The consumption of alcohol was lower in men with the highest MeDi adherence (Table 3).

Regarding the consumption of macronutrients in g/d, similar trends were observed since the MeDi adherence was positively associated with vegetal proteins, total MUFA, oleic acid, total PUFA, n-6 PUFA, and linoleic acid intakes in men. Moreover, an inverse association between total SFA and myristic acid intakes (g/d) and MeDi adherence was observed (data not shown).

In men, the mean linoleic/linolenic acid ratio was positively associated with MeDi adherence, ranging from 8.78 (SD 5.51) among low MeDi adherents to 9.46 (SD 6.09) and 10.42 (SD 6.00) among middle and high MeDi adherents respectively (P=0.034).

In women, the total mean EI computed with or without alcohol was positively associated with MeDi adherence (Table 3). As expressed in percent of EI computed with alcohol (Table 3), MeDi adherence in women was positively associated with intake of vegetal protein, total carbohydrate, polysaccharides, total PUFA, total n-6 PUFA, linoleic acid, total n-3 PUFA, and alpha-linolenic acid, and inversely associated with total SFA and myristic, palmitic and stearic acid intake (Table 3). Regarding the macronutrient intake as expressed in g/d, similar trends between MeDi adherence and higher mean consumptions of total protein, total MUFA and oleic acid were observed (data not shown). In addition, the mean (SD) linoleic/linolenic acid ratios were statistically different among women with low, middle and high MeDi adherence (i.e. 8.46 (5.51), 10.71 (7.93) and 10.61 (7.69) respectively, P<0.0001).

Regarding micronutrient, minerals and trace elements intake (Table 4), there was a positive significant association between MeDi adherence and the mean consumption of fibres, vitamins B6, C, E, folates, magnesium, potassium and iron, both in men and women. In men, there was also a significant association between carotene intake and MeDi adherence, which was not observed in women. Conversely, vitamin B1 intake was positively and significantly associated with MeDi adherence in women only. Finally, there was an inverse association between MeDi adherence and calcium intake which was the lowest in the middle and highest MeDi categories in women, while this inverse association was of borderline significance in men (Table 4).

DISCUSSION

This large population-based cross-sectional study in older persons living in Southwestern France shows a high coherence between dietary data collected by FFQ and used to assess a diet modelled on the traditional MeDi, and the nutrient intake estimated by 24 h recall. As expected, greater MeDi adherence was associated with total mean higher vegetal protein intake (% EI), lower total mean SFA intake (% EI), higher total mean PUFA intake (% EI), but also notably with total n-6 PUFA and linoleic acid intakes, both in men and women. In men, higher total mean MUFA and oleic acid intakes but less alcohol consumption (% EI) were observed with greater MeDi adherence. Total carbohydrate and polysaccharide intakes (% EI) and total mean n-3 PUFA and alpha-linolenic acid intakes (% EI) were positively associated with MeDi adherence in women but not in men. The consumption of several micronutrients, minerals and trace elements (i.e. fibres, vitamins B6, C, E, folates, magnesium, potassium and iron) increased with higher MeDi adherence, in both men and women. Only the consumption of calcium decreased with greater MeDi adherence in women and with borderline significance in men.

The concept of Mediterranean diet has been first introduced in the Seven-Country study by Keys et al. who reported low all-cause and coronary heart disease death rates in cohorts of the Mediterranean basin with olive oil as the main fat [START_REF] Keys | The diet and 15-year death rate in the seven countries study[END_REF]. Not a specific pattern but a collection of eating habits traditionally followed by the populations of the Mediterranean basin first defined the so-called "traditional" Mediterranean diet. However, there is no single Mediterranean diet but several definitions, because dietary habits vary considerably across the Mediterranean countries bordering the sea [START_REF] Bach-Faig | Mediterranean diet pyramid today. Science and cultural updates[END_REF][START_REF] Sofi | The Mediterranean diet revisited: evidence of its effectiveness grows[END_REF]. Compared with usual single-food or nutrient methods, the dietary pattern approach is appealing because analyses based on single nutrients ignore important interactions (additive, synergistic or antagonist effects) between components of diet and because people did not eat isolated nutrients [START_REF] Kant | Dietary patterns and health outcomes[END_REF]. The computation of the MeDi score [START_REF] Trichopoulou | Adherence to a Mediterranean diet and survival in a Greek population[END_REF], as used in the present study, combines several foods providing nutrients which have been identified as potential protective factors against cardiovascular diseases such as fruit and vegetables, fish, as the main provider of long-chain n-3 PUFA, olive oil as main source of MUFA, and a moderate consumption of alcohol. On the other hand, higher total fat and SFA intake have been considered as harmful factors for cardiovascular diseases and related mortality [START_REF] Bhupathiraju | Coronary heart disease prevention: nutrients, foods, and dietary patterns[END_REF][START_REF] Mente | A systematic review of the evidence supporting a causal link between dietary factors and coronary heart disease[END_REF][START_REF] Samieri | Olive oil consumption, plasma oleic acid, and stroke incidence: The Three-City Study[END_REF].

Our findings support the presumed beneficial effects of the MeDi. Regarding macronutrients, higher adherence to a MeDi was characterized by a high intake of vegetal protein but not animal protein, complex carbohydrates hence a lower glycemic index, a low consumption of SFA whose detrimental effects are well documentedalthough debated [START_REF] Astrup | The role of reducing intakes of saturated fat in the prevention of cardiovascular disease: where does the evidence stand in 2010?[END_REF], and a high consumption of MUFA, especially oleic acid, and PUFA. Despite a higher intake of n-3 PUFA with greater MeDi adherence, notably in women, the ratio of n-6/n-3 PUFA precursors increased with MeDi score. This may be in part due to the high proportion of consumers of oils from various vegetal origins, which are providers of linoleic acid, the precursor of n-6 PUFA. Indeed, 60% of the whole sample was regular consumers (i.e. preferred dietary fats used for dressing or cooking) of olive oil, and 44% of sunflower oil, although 32% was simultaneously regular consumers of both oils (data not shown). Hence, the ratio of n-6/n-3 PUFA precursors, high in our sample and far above the recommended value of 5 [START_REF] Martin | The "apports nutritionnels conseilles (ANC)" for the French population[END_REF], might be a restriction to beneficial effects of the MeDi. Higher adherence to the MeDi was also associated with higher intake of various antioxidants (vitamins C and E, carotenes in men, and also probably polyphenols from plants although we could not ascertain their amount) which contribute to protect long-chain n-3 PUFA from lipid peroxidation. The MeDi also provided B vitamins, especially folate. Nevertheless, higher MeDi adherence is characterized by lower intake of dairy products. Therefore, the lower dietary intake of calcium associated with higher MeDi adherence in women, may be a threat to their bone health. This finding suggests that the MeDi as presently defined may not be a universally protective diet.

Refinement in the computation of the score is needed to disentangle the presumed detrimental effects of some SFA from the beneficial effects of calcium found in dairy products.

Overall, the results of the present analysis showed a high coherence between the two methodologies used in the 3C study to describe the anatomy of the Mediterranean-type diet of older participants of the 3C cohort. However, some expected associations between MeDi adherence and various nutrient intakes (for instance, total mean n-3 PUFA in men, carotene in women, vitamins D and B12) have not been highlighted. This may be due to the use of a single 24 h recall which cannot handle high intra-individual variation [START_REF] Feart | Energy, macronutrient and fatty acid intake of French elderly community dwellers and association with socio-demographic characteristics: data from the Bordeaux sample of the Three-City Study[END_REF]. Thus, a reported single day of intake is unlikely to be representative of usual individual intake. However, a sufficiently large sample may allow to determine average intake of a defined subgroups [START_REF] Willett | Nutritional epidemiology[END_REF].

Moreover, a single 24 h recall prevented us to capture the intake of nutrients which are not consumed on a daily basis, but less often. The FFQ used in the present study assessed number of servings but not portion size. The lack of consideration of amount of each food consumed may lead to consider people with same frequency of each food group of the MeDi score, and same MeDi score, as comparable, although they may have different quantitative food consumptions. We cannot exclude that the quantity of presumed protective or deleterious nutrients consumed by men and women with a similar MeDi score may differ, which reduced our chance to evidence associations between food and nutrient intakes. Finally, alcohol was an important component of total EI, notably in men [START_REF] Feart | Energy, macronutrient and fatty acid intake of French elderly community dwellers and association with socio-demographic characteristics: data from the Bordeaux sample of the Three-City Study[END_REF], leading us to consider this nutrient provider of energy in the computation of the total EI in the present analysis. However, a lowto-moderate alcohol consumption is also one of the 9 components of the MeDi score, which may have added bias in the present study.

Analyses of single nutrients using 24 h dietary recalls could help understanding specific mechanisms involved in physio-pathological processes leading to diseases, by establishing biological plausibility. Therefore, the biological basis for the well documented health benefits of the MeDi involves a decrease in oxidative stress and inflammation which also participates in the dysregulation of cellular function, the exacerbation of degenerative diseases, such as cardiovascular diseases or atherosclerosis [START_REF] Bartali | Serum micronutrient concentrations and decline in physical function among older persons[END_REF][START_REF] Bhupathiraju | Coronary heart disease prevention: nutrients, foods, and dietary patterns[END_REF][START_REF] Mente | A systematic review of the evidence supporting a causal link between dietary factors and coronary heart disease[END_REF][START_REF] Solfrizzi | Unsaturated fatty acids intake and all-causes mortality: a 8.5-year follow-up of the Italian Longitudinal Study on Aging[END_REF]. Individuals with higher MeDi adherence have been shown to have higher reduced-to-oxidized glutathione ratio, and lower insulin resistance, oxidized low-density lipoprotein, C-reactive protein and interleukin-18 levels [START_REF] Bach-Faig | Evaluating associations between Mediterranean diet adherence indexes and biomarkers of diet and disease[END_REF][START_REF] Dai | Association between adherence to the Mediterranean diet and oxidative stress[END_REF]15]. It has been hypothesized that the n-3 PUFA and a balanced n-6/n-3 PUFA ratio associated with a high intake of antioxidant in the MeDi could be important determinants in influencing cardiovascular and cerebral diseases [START_REF] Feart | Adherence to a Mediterranean diet and plasma fatty acids: data from the Bordeaux sample of the Three-City study[END_REF][START_REF] Panagiotakos | The current Greek diet and the omega-6/omega-3 balance: the Mediterranean diet score is inversely associated with the omega-6/omega-3 ratio[END_REF][START_REF] Simopoulos | The Mediterranean diets: What is so special about the diet of Greece? The scientific evidence[END_REF]. In the current study, participants with lower MeDi adherence exhibited also greater BMI. Surprisingly, there was no apparent vascular benefits of a greater MeDi adherence in the current sample where the prevalence of hypertension, hypercholesterolemia and diabetes did not differed across MeDi categories. This has already been observed in similar samples of elderly people involved in the 3C study, who exhibited a global healthy lifestyle [START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF][START_REF]The 3C Study Group (2003) Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. Finally, the computation of the MeDi score able to assess associations between a diet modelled on the traditional MeDi, but which is specific to the sample studied. Indeed, the use of sex-specific cut-off points does not measure adherence to a universal traditional MeDi pattern but rather to a specific pattern, which precludes generalising results of the present study to other populations who have on the whole different food intake and MeDi adherence [START_REF] Bach | The use of indexes evaluating the adherence to the Mediterranean diet in epidemiological studies: a review[END_REF]. Moreover, including only 9 food groups, the MeDi score does not consider the overall correlation between all foods [START_REF] Kant | Dietary patterns and health outcomes[END_REF], among which some food groups could reflect specific health concerns and behaviours, such as the consumption of dietary supplements. Despite these limitations, the strengths of the present study are its sample size, the population-based design, and the accuracy of food-intake assessment which validates the methodology employed.

In conclusion, this study conducted in French elderly community dwellers allowed to describe the basis of their Mediterranean-type diet and the actual consumption of nutrients associated with MeDi adherence. As residual confounding by other aspects of lifestyle cannot be ruled out, these nutrients probably do not fully explain the better health of persons who adhere to this MeDi but they likely contribute directly. At least regarding clinical parameters, old consumers conform with MeDi have only a lower BMI and no adverse effects of an elevated n-6/n-3 PUFA ratio. This dietary pattern may indirectly constitute an indicator of a complex set of favorable social and lifestyle factors that contribute to global better health. Indeed, food choices and dietary habits which reflect individual food preferences are highly sensitive to culture, education, socio-economic status, socio-demographic characteristics, environmental and lifestyle determinants, as well as to age-or nutrition-related diseases [START_REF] Drewnowski | The Mediterranean diet: does it have to cost more?[END_REF].

  . The 3C study protocol was approved by the Consultative Committee for the Protection of Persons participating in Biomedical Research at Kremlin-Bicêtre University Hospital (Paris). A sample of 9294 community dwellers aged 65 and over was selected in 1999-2000 from the electoral rolls of three French cities (Bordeaux, Dijon and Montpellier). All participants gave written informed consent. At baseline, data collection included socio-demographic information, lifestyle, symptoms and medical complaints, medical history, blood pressure, smoking status, drug use, anthropometric data, neuropsychological testing, and blood sampling. These data were completed by a comprehensive dietary survey only in the Bordeaux center in 2001-2002. The study sample comprises 1595 participants from Bordeaux, without missing nutritional data.

Table 1 .

 1 Mean number of servings per week for individual food categories, proportion of mild-tomoderate alcohol consumers and mean ratio of daily intake of MUFA-to-SFA by categories of Mediterranean Diet score among older persons living in Bordeaux, The Three-City study (2001-: MeDi Mediterranean diet, BMI, Body Mass Index P value for the Chi-Square test or analysis of variance (ANOVA) among categories of Mediterranean diet score * 2X2 significant comparisons (P<0.05) for means of age and BMI between categories of Mediterranean diet score taking the lowest category as reference group

	2002). (N=1595)
				Low Category	Middle Category	High Category
				(MeDi Score 0-3)	(MeDi Score 4-5)	(MeDi Score 6-9)
				n=492	n=693	n=410
	Food Categories	†
	Mean serving/week (SD)
	Dairy products		20.65 (8.02)	18.08 (7.69) *	16.14 (6.71) * ‡
	Meat			5.36 (2.59)	4.80 (2.41) *	4.18 (2.22) * ‡
	Vegetables			15.56 (6.41)	19.37 (7.11) *	23.44 (6.61) * ‡
	Fruits			10.66 (6.79)	13.84 (6.55) *	16.45 (6.50) * ‡
	Legumes			0.42 (0.68)	0.62 (0.62) *	0.82 (0.65) * ‡
	Cereals			19.78 (6.52)	22.16 (5.76) *	24.54 (5.48) * ‡
	Fish			2.02 (1.49)	2.91 (1.69) *	3.76 (1.73) * ‡
	Mild-to-moderate alcohol (%)		14.63	25.40 *	46.10* ‡
	MUFA-to-SFA ratio, Mean (SD)		0.77 (0.26)	0.88 (0.32) *	1.00 (0.31) * ‡
					test
	(proportion of mild-to-moderate alcohol consumers) among categories of Mediterranean diet
	score.
	* 2X2 significant comparisons (P<0.05) for mean number of servings per week for individual
	food categories, proportion of mild-to-moderate alcohol consumers and mean ratio of daily
	intake of MUFA-to-SFA between categories of Mediterranean diet score taking the lowest
	category as reference group or	‡	between individuals in middle and high categories of
	Mediterranean diet score.

Abbreviations: MeDi Mediterranean diet; MUFA-to-SFA Monounsaturated fatty acid-tosaturated fatty acid ratio (intake g/d) † All P values <.0001 for the analysis of variance (ANOVA) or for the Chi-Square Abbreviations

Table 3 :

 3 Energy, macronutrient and alcohol intakes as expressed in proportion of energy intake by categories of Mediterranean diet score among older persons living in Bordeaux, The Three-City study, 2001-2002 (N=1595) Expressed as % of energy intake (computed with alcohol) P-value for the ANOVA comparing energy intake, macronutrient and alcohol intakes in g/d among the three Mediterranean Diet Score categories * 2X2 significant comparisons (P<0.05) for means of nutrient intake between categories of Mediterranean diet score taking the lowest category as reference group or ‡ between individuals in middle and high categories of Mediterranean diet score.

	Men

a

Table 4 :

 4 Micronutrient intakes, as expressed in mg/d, by categories of Mediterranean diet score among older persons living in Bordeaux, The Three-City study, Expressed as mean intake in mg/d except for fibres, sodium, phosphorus and potassium, both expressed in g/d, and for retinol, vitamin D and folates, both expressed in µg/d P-value for the ANOVA comparing energy, macronutrient, alcohol and micronutrients intakes among the three Mediterranean Diet Score categories * 2X2 significant comparisons (P<0.05) for means of energy and nutrient intake between categories of Mediterranean diet score taking the lowest category as reference group or ‡ between individuals in middle and high categories of Mediterranean diet score.

	2001-2002 (N=1595)
	Men

a

ACKNOWLEDGMENTS

Funding / Support: The Three-City Study is conducted under a partnership agreement between the Institut National de la Santé et de la Recherche Médicale (INSERM), the Institut de Santé Publique et Développement of the Victor Segalen Bordeaux 2 University, and Sanofi-Aventis. The Fondation pour la Recherche Médicale funded the preparation and initiation of the study. The 3C Study is also supported by the Caisse Nationale Maladie des Travailleurs Salariés, Direction Générale de la Santé, Mutuelle Générale de l'Education Nationale, Institut de la Longévité, Regional Governments of Aquitaine and Bourgogne, Fondation de France, and Ministry of Research -INSERM Programme "Cohortes et collections de données biologiques." Role of the sponsors: Study sponsors played no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; and preparation, review, or approval of the manuscript.

Conflict of interest: P. Barberger-Gateau received fees for conferences from Danone, Lesieur, Bauch&Lomb and Aprifel and benefits from research grants from Danone and Lesieur. The other authors declared no support from any institution for the submitted work.