J.-M Virey
email: virey@cpt.univ-mrs.fr

P Taxil

A Tilquin
email: tilquin@cppm.in2p3.fr

A Ealet

D Fouchez

C Tao

Probing Dark Energy with Supernovae : Bias from the time evolution of the equation of state

Keywords: 98, 80, Es, 98, 80, Cq Key-Words : cosmological parameters -supernovae Number of figures : 10

Observation of thousands of type Ia supernovae should offer the most direct approach to probe the dark energy content of the universe. This will be undertaken by future large ground-based surveys followed by a space mission (SNAP/JDEM). We address the problem of extracting the cosmological parameters from the future data in a model independent approach, with minimal assumptions on the prior knowledge of some parameters. We concentrate on the comparison between a fiducial model and the fitting function and adress in particular the effect of neglecting (or not) the time evolution of the equation of state. We present a quantitative analysis of the bias which can be introduced by the fitting procedure. Such bias cannot be ignored as soon as the statistical errors from present data are drastically improved.

Introduction

The general paradigm in cosmology is that we are living in a flat universe, which is dominated by a nearly homogeneous component with negative pressure. This component is often called Dark Energy (DE) and causes the expansion rate of the universe to accelerate.

The recent measurements of type Ia supernovae (thereafter denoted SN) are the most direct evidence of the presence of this component [1,2,3,4]. It is also confirmed by the combination of results from the large-scale distribution of galaxies [6] and the most precise data on the cosmic microwave background (CMB) from the Wilkinson Microwave Anisotropie Probe (WMAP) [7]. A recent combined analysis is presented in [8]. Recently the detection of the late Integrated Sachs-Wolfe (ISW) effect has reinforced the case for DE [9]. All this is most frequently interpreted in the framework of the so-called "concordance Cosmology" [10].

A fundamental problem is the identification of the underlying nature of DE : cosmological constant, quintessence (for a review see [11]) or something else. The most common way is to measure its equation of state (EoS) defined as w = p X /ρ X where p X is the pressure and ρ X the energy density of the DE. The ratio of the DE density to the critical density will be denoted Ω X in a general model and Ω Λ in the simplest case of a Cosmological Constant (w = -1). Ω M is the corresponding parameter for (baryonic+cold dark) matter.

An ambitious SN program is now on the way. Important pieces of information will be provided by large ground-based surveys such as the Supernova Legacy Survey (SNLS) [12] and these investigations will culminate with a space mission as the Supernova Acceleration Probe (SNAP) instrument, part of the Joint Dark Energy Mission (JDEM), which aims at the discovery and follow-up of some 2000 SNIa per year in the redshift range z = 0.2 -1.7 with very precise magnitude measurements [13]. Of course, the validity of the obtained precision will not depend on the size of the SN sample only, but also on the ability to control the systematic uncertainties at the same level [14]. These systematic errors are coming on one side from the instrument itself (calibration, etc, ...) and also from the SN astrophysical environment (evolution, lensing effects, ...). The SNAP collaboration proposes a global strategy to control such effects at the percent level [13].

In this context, it is necessary to analyze very carefully at which precision level it will be possible to draw any conclusions from the expected rich amount of data.

Authors have used the present SN data together with simulated sets to :

• evaluate the accuracy on the Ω Λ parameter to validate the acceleration.

• measure the DE through the equation of state with w constant in redshift.

• evidence a possible redshift dependence w(z) with various parametrizations.

• look at the model dependences and degeneracies and study the possible strategies to break the latter.

There is a large consensus that the future SN data alone will have difficulties to constrain an evolving equation of state and to break model degeneracies. It is mandatory to have a prior knowledge of the values of some parameters. In particular, a precise knowledge of Ω M will be essential if one hopes to pick out the z dependence of w (see e.g. [15,16,17,18,19,20,21]) even in the simplest flat cosmology. However, to use some pieces of information from other sources than SN, it will be essential to combine the data in a coherent way, that is under the same hypothesis, in particular on the DE properties.

On the other hand, some potential difficulties have been pointed out by various authors. Indeed, most papers have been mainly interested in predicting, for each cosmological parameter, the errors around some fiducial value, close to the ones obtained from the present data as Ω M = 0.3, Ω X = 0.7, w = -1, dw/dz = 0. In this procedure, the framework is a particular fiducial model and the chosen fitting function is usually the one used to generate the data. This strategy is valuable for a first estimate but is too restrictive to pin down the underlying physics which is so far unknown.

For example, Maor et al. [16] and Gerke and Efstathiou [20] have stressed the problems which arise if the redshift dependence of the equation of state is neglected whereas this dependence is present. The multiple integral relation between the luminosity distance and w smears out information about w and its time-variation. Assuming in the fitting function that w is a constant, whereas it is not in the fiducial model, can lead to a very wrong estimate of the "constant" or even "effective" w value. At the same time, the central value of Ω M (or Ω X ) is badly reconstructed.

It is unavoidable to get some ambiguities when trying to fit a particular fiducial cosmology with the "wrong" model ! In the following, we will call this problem the "bias problem". For present SN data, as will be shown, this question is not a concern as the statistical errors on model parameters stay large due to the limited statistics. In the perspective of much richer SN samples, this bias problem cannot be ignored. We address the problem in a more quantitative way than the authors of [22] which have only considered some specific classes of models. In this paper, we investigate the full parameter range, in a model independent way, to quantify where the 'bias' creates potential difficulties and where it can be ignored. We also evaluate the impact of priors when only SN measurements are used. We get a handle on the possible bias introduced in the analysis by looking at the different hypothesis which are done at the fitting function level. Results are presented in the parameter space of the fiducial models we have simulated.

This paper is organized as follows. In Section 2 we recall the theoretical description of the magnitude-redshift diagram for supernovae and we present the experimental situation. We identify the relevant parameters which enter as the parameters of the fiducial models and we define the bias which could be introduced by the fitting procedure.

In section 3, we focus on the extraction of the EoS parameter w 0 . We illustrate the confusion created by the bias on examples. Then we analyze the impact of bias on simulated data : we consider future statistical samples corresponding to SNAP, SNLS and also a sample corresponding to present data. We try to find the safest way for extracting an effective w from the data with minimum bias and minimum priors. Our summary and conclusions are given in Section 4.

The experimental and theoretical framework 2.1 The experimental framework

The studies presented in this paper are performed with simulated supernovae samples with statistics equivalent to what we expect to have in the future. We concentrate on three sets of data which simulate the statistical power of the present and future data. We want to emphasize the importance of the future sample of supernovae taken on ground which are limited by the systematics inherent to this approach but are statistically one order of magnitude greater than the present sample.

• We reproduce the published data of Perlmutter et al. from [4]. We take directly the effective magnitude of the sample 3 of this paper, which corresponds to 60 SNIa, corrected from K corrections, extinction and stretch (the values are given in Table 3 of [4]). The errors on magnitude vary in the range [0.16-0.22].

• We simulate some future data coming from ground survey as the large SNLS survey at CFHT [5]. This survey has started in 2003 and the estimation after 5 years of running is to register a sample of 700 identified SNIa in the redshift range 0.3 < z < 1. We simulate a sample as reported in Table 1 in agreement with the expected rate of [5]. We assume a magnitude dispersion of 0.15 for each supernova and constant in redshift after all corrections.

• We simulate data from a future space mission like SNAP, which plans to discover around 2000 identified SNIa, at redshift 0.2< z <1.7 with very precise photometry and spectroscopy. The SN distribution is given in Table 1(from [14]). The magnitude dispersion is assumed to be constant and independent of the redshift at 0.15 for all SN's after correction. On some studies, we include the effect of adding a constant and uncorrelated in redshift systematic error of 0.02 on the magnitude.

A set of 300 very well calibrated SnIa at redshift < 0.1 should be measured by the incoming SN factory project [23]. This sample is needed to normalize the Hubble diagram and will be called in the following the "Nearby" sample. A "SNAP" ("SNLS") sample means in this paper a simulation of the statistics expected from the SNAP like mission ("SNLS" survey) combined with the 300 nearby SN's. To make our point clear, we do not consider in the presented studies the experimental systematic errors, unless otherwise specified.

The theoretical framework

In the standard Friedmann-Roberston-Walker metric, the apparent magnitude of astrophysical objects can be expressed as a function of the luminosity distance :

m(z) = 5 log 10 (D L ) + M B -5 log 10 (H 0 /c) + 25 = M s + 5 log 10 (D L ) (1)
where M B is the absolute magnitude of SNIa, M S may be considered as a normalization parameter and D L (z) ≡ (H 0 /c) d L (z) is the H 0 -independent luminosity distance to an object at redshift z. It is related to the comoving distance r(z) by D L (z) = (1 + z)r(z), where

r(z) =        1 √ -Ω k sin( √ -Ω k J), Ω k < 0 J, Ω k = 0 1 √ Ω k sinh( √ Ω k J), Ω k > 0 (2)
with

Ω k = 1 -Ω m -Ω X ≡ 1 -Ω T , (3)
J = z 0 H 0 H(z ′ ) dz ′ (4)
and

H(z) H 0 2 = (1 + z) 3 Ω m + ρ X (z) ρ X (0) Ω X + (1 + z) 2 Ω k , (5)
with ρ X (z) ρ X (0) = exp 3 z 0 (1 + w(z ′ )) d ln(1 + z ′ ) (6)
Note that we have neglected the radiation component Ω R .

Choice of a fitting function

A fitting function is an expression for H(z) in terms of a number of parameters. H(z) is usually expressed in term of the cosmological parameters contained in Eq.(5) [15,16,17,18,19,20,21] but other possibilities relying on geometrical parametrizations or arbitrary ansatz have been proposed [24]. In this paper, we adopt the first approach where the cosmological parameters have a direct physical interpretation within General Relativity. With such analysis, the constraints are aimed to be model independent and can be used for investigation of a large variety of DE models. However, we have to choose a parametrization for the time (or redshift) variation of the EoS. The current parametrizations of w(z) which can be found in the literature are the linear one [17]: w(z) = w 0 + w 1 z; the one advocated by Linder [25] with a better asymptotic behavior : w(z) = w 0 + w a z/(1 + z); and also w(z) = w 0 -α ln(1 + z) from [20] which is advocated to describe a large sample of quintessence models characterized by a weakly varying EoS. We choose the linear one which allows to compare most easily to previous works. In this case, Eq.( 6) becomes:

ρ X (z) = ρ X (0) e 3w 1 z (1 + z) 3(1+w 0 -w 1 ) (7)
Nevertheless, our results are essentially independent of the choice of this parametrization since the redshift range of the SN data is limited to z < 2 and the sensitivity of the data on w(z) concerns mainly the values taken by w at relatively low redshifts z ≃ 0.2 -0.4 (the "sweet spot" of [18,17,21]) where all parametrizations reduce roughly to the linear one.

Finally, note that models with non-trivial variations of the EoS w(z), like the pseudogoldstone model of ref. [26], cannot be described by such parametrizations. Consequently, our analysis does not apply to this kind of models.

Therefore, five parameters have to be fitted in the most general procedure: M s , Ω M , Ω X , w 0 , w 1 .

Choice of a fiducial model

We define fiducial models which depend on the parameters

M F s , Ω F M , Ω F X , w F 0 , w F 1 .
In order to keep this paper clear, we concentrate on the case where the paradigm Ω T = 1 is verified by the fiducial models. This value is in agreement with the inflationary paradigm and it is supported by the analysis of the rich amount of data from the CMB [7]. Ω F X is then no longer free (Ω F X = 1 -Ω F M ) and, if not stated otherwise, the parameters M F s and Ω F M are fixed at: M F s = -3.6 and Ω F M = 0.3. Our conclusions will not depend on variations of the M F s parameter. The variation due to different Ω F M values has been investigated. If Ω T = 1, it appears that our results can change significantly. Analysis of non-flat models will be presented elsewhere [27].

To focus on the dark energy measurement, we propose to scan a large variety of fiducial models as a function of the couple (w F 0 , w F 1 ). We have scanned the plane (w F 0 , w F 1 ) (hereafter denoted by P) for the values -2 < w F 0 < 0 and -2 < w F 1 < 2. We pay particular attention to the reduced plane P R associated to the ranges -1 < w F 0 < 0 and -1 < w F 1 < 1, which represents a reasonable class of theoretical models. Indeed, in most of the models available in the literature, like the quintessence models [11], the weak energy condition w F 0 > -1 is satisfied (see e.g. [28]). They also possess in general a relatively weak z dependence for the EoS [20,17]. However, we consider the full plane P in order to include in our analysis more exotic models which violate the weak energy condition or which are described by a modified Friedmann equation or by a modified theory of gravity (we refer to [29] for a recent list of models and references).

The fitting method

To analyse the (simulated) data, a minimization procedure has been used [START_REF]Our simulation tool, the "Kosmoshow[END_REF]. A standard Fisher matrix approach allows a fast estimate of the parameter errors. This method is however limited as it does not yield the central values of the fitted parameters. Then we adopt, unless specified, a minimization procedure based on a least square method. The least square estimators are determined by the minimum of the χ 2 = (m-M(z, Ω, w) T V -1 (m-M(z, Ω, w)), where m = (m 1 ...m n ) is the vector of magnitude measurements, M(z, Ω, w) the corresponding vector of predicted values and V the covariance matrix of the measured magnitudes. The error on cosmological parameters is estimated at the minimum by using the first order error propagation technique: U = A.V.A T where U is the error matrix on the cosmological parameters and A the Jacobian of the transformation.

A full 5-parameter fit (5-fit) of the real or simulated data gives the central values and errors for the five parameters. In the following, the normalization parameter M s is left free in the fit, unless specified, as its correlation with the cosmological parameters is strong [19]. To constrain accurately this parameter, which is fixed in most of the papers, we use the very low redshift simulated "Nearby" sample of 300 SN.

Choice of a fitting procedure

There are different ways to choose a fitting procedure. If one only wants to minimize the errors on the fitting parameters, a simple Fisher approach should be sufficient and should give conclusions depending on the initial fit hypothesis. Nevertheless, as we have already emphasized, this approach is fiducial model dependent and can lead to some bias that we want to quantify. Then, after a presentation of different Fisher results, we will expose our strategy to understand and explore the bias introduced by some fitting approach.

The Fisher approach

To perform a Fisher matrix analysis, a fiducial model is chosen and fixed. We take the "concordance model" version of the simplest flat ΛCDM model :

Ω F M = 0.3, Ω F X = 0.7, w F 0 = -1, w F 1 = 0 with M F s = -3.6
We call this model 'Λ' in the following.

Each fitting procedure is defined by a particular choice of the parameters to be fitted: reducing the number of parameters will improve the parameter errors in the fitting function. Priors are introduced by fixing a parameter at a predefined value or inside a given range.

Table 2 gives the errors obtained with the Fisher matrix analysis for various fitting procedures with the SNLS and SNAP simulated data. The numbers are in agreement with the analysis of various authors [15,16,17,18,19,20,21]. Figure 1 gives the contours obtained in the plane (Ω M , w 0 ) at 68.3 % CL with the SNAP statistics, for this fiducial model within the same fitting procedures. As can be seen from the first line of Table 2 and from the black ellipse in Fig. 1, a complete 5 parameter-fit (5-fit)(M s , Ω M , Ω X , w 0 , w 1 ), with no assumptions on the values of the cosmological parameters -including no constraint on the flatness of the universe-yields too large errors on the cosmological parameters to set any definite conclusions even with the high statistics sample expected for the SNAP+Nearby data. Fig. 1 shows clearly that the reduction of the number of fitted parameters or/and the strengthening of the Ω M prior, reduces the contours.

The second line of Table 2 presents errors with the paradigm of a perfectly flat universe (Ω T = 1) and the 5-fit reduces to a 4-fit (M s , Ω M , w 0 , w 1 ). It appears that for the SNAP statistics, one gets better estimates of w 0 and Ω M but the error on w 1 is always large. For SNLS, even the error on w 0 is large.

Adding some prior knowledge of Ω M improves greatly the situation: the same 4-fit (M s , Ω M , w 0 , w 1 ) for SNAP and SNLS with a weak prior on Ω M (0.2 < Ω M < 0.4) yields

F s = -3.6, Ω F M = 0.3, Ω F X = 0.7, w F 0 = -1, w F 1 = 0.
The weak (strong) Ω M prior corresponds to the constraint Ω M = 0.3 ± 0.1 (Ω M = 0.3 ± 0.01). The labels 5-fit, 4-fit and 3-fit corresponds to the fitting procedures 5-fit(M s , Ω M , Ω X , w 0 , w 1 ), 4-fit(M s , Ω M , w 0 , w 1 ) and 3-fit(M s , Ω M , w 0 ), respectively. σ(Ω X ) has been omitted from the table since Ω X = 1 -Ω M is no longer a free parameter except for the 5-fit where σ(Ω X ) is very large. σ(M S ) has been omitted from the table since its value, roughly 1% , changes weakly for the various fitting procedures thanks to the inclusion of the Nearby SN sample.

Fit Ω M prior assumptions 2 ). Only a strong prior on Ω M (Ω M = 0.3 ± 0.01) really improves the situation for w 1 as can be seen from line 4 of Table 2. These conclusions are in agreement with other published results [15,16,17,18,19,20,21].

σ(Ω M ) σ(w 0 ) σ(w 1 ) σ(Ω M ) σ(w 0 ) σ(
One can conclude that the determination of a possible redshift dependence of the EoS remains the most difficult task. All authors agree on the following statement: to get a good precision on the parameter governing the redshift evolution of the EoS, a fit with a strong prior on Ω M ("strong" means at the percent level) is needed within the Ω T = 1 paradigm.

The bias problem

Several comments on the previous approach are in order :

-The setting of strong priors inside the fit should be taken with caution.

First of all, the cosmological parameter Ω M is far from being measured at the percent level [START_REF] Blanchard | [END_REF]. So, this kind of prior cannot be applied blindly, even if there is hope that this parameter might be well measured by the time SNAP provides data.

-Even in a future context, some doubts can be raised about the relevance of very precise priors [15,16] and their use has to be set with some cautions : all the Ω's should be obtained under the same hypothesis (astrophysical, experimental, DE properties) and correlations among fitted parameters have to be taken properly into account for the combined statistical analysis to be relevant. For this fitting procedure, there is almost no difference between the cases "no" and "weak" prior for Ω M .

In this paper, we attempt to extract the informations from SN data, therefore avoiding the (consistency) problems encountered when constraining Ω M with external data.

The results from Table 2 and Fig. 1 lead us to leave aside the determination of the redshift dependence of the EoS and to consider w 0 as an "effective" constant w ef f EoS parameter. Then we concentrate on the best strategy to extract w 0 from present or future SN data with minimal assumptions on priors : we use the paradigm Ω T = 1 either with no priors or with weak priors on Ω M . We focus on a 3-fit (M s , Ω M , w 0 ) or on a 4-fit (M s , Ω M , w 0 , w 1 ) where we know already that the precision on w 1 will be low.

A 3-fit should always provide better constraints than a 4-fit. Table 2 shows, however, that the reduction of the errors is more important for Ω M than for w 0 and since the w 0 error is not strongly improved, the use of the 4-fit seems the best strategy for extracting w 0 [21,14]. We want to point out that this conclusion is only valid for the Λ fiducial model. In some other fiducial models, the 3-fit can increase considerably the constraints on w 0 with respect to the 4-fit: Figure 2 gives the expected contours in the plane (Ω M ,w 0 ) for the two fiducial models (w F 0 = -2,w F 1 = 0) and (w F 0 = -1,w F 1 = 0.2). The best constraints on Ω M and w 0 are given by the 3-fit.

On the other hand, doing a complete minimization procedure shows that the central values given by the 3-fit can be far away from the fiducial values. This fact has been pointed out by several authors [15,16,20,22]. This happens when the fitting hypothesis are not verified by the fiducial model, namely a constant EoS in the 3-fit and a varying one in the fiducial model. In this case bias are introduced on the estimate of the cosmological parameters. This bias problem is due to the degeneracy of the fitting function among the various cosmological parameters [15,16]. A wrong hypothesis on one parameter is compensated in a non trivial way by the value of the others. We know also from the various Fisher analysis presented in the litterature [18,19,21,14] that the errors depend on the central values of the cosmological parameters. Since these last ones are biased, their errors are also biased [21,14]. Then the question of choosing between the 3-fit and the 4-fit is not so obvious and appears to be fiducial model dependent.

In the next section we present in detail a study of this bias by exploring the range of fiducial models which is affected, in order to conclude on the validity of a 3-fit. Let us point out that other bias would be introduced by a departure from the Ω T = 1 paradigm in the fiducial model or if the central value of the Ω M prior used in the fitting procedure is not the same as the fiducial one. These bias will be studied in a future paper [27].

3 How to extract w 0 : a quantitative analysis of the validity of a 3-fit

To be able to choose the best strategy for extracting w 0 , we want to test the relevance of the fitting procedure. We have performed for that purpose a complete study of the bias introduced by the different fitting procedures. After a description of the problem through a short illustration, we present the results of a full scan of the (w F 0 , w F 1 ) plane for SNAP, SNLS and a sample corresponding to present data. The scanning is done by fixing the fitting procedure and varying the various parameters of the fiducial model ; we perform a 3-fit and compare it to the result of the 4-fit where w 1 is left free.

Illustration of the bias effect

Let us start with some illustrations of the bias introduced by the 3-fit procedure. We choose three different models and look at the central values of the parameters obtained from the fit:

• The one of Maor et al. [16] (w F 0 = -0.7, w F 1 = 0.8). The 3-fit applied to this model clearly provides some erroneous results : Ω M = 0.62 ± 0.013 and w 0 = -1.548 ± 0.194. Note that our central values are slightly different from the ones of [16], this is due to small differences between the chosen SN samples.

Figure 3 shows the two elliptical contours from the 3-fit and the 4-fit : the 4-fit ellipse is centered on the fiducial values as expected whereas the 3-fit ellipse is very far away. This is due to the large chosen w F 1 value and this behavior has been already pointed out by Maor et al. (see Fig. 5 of ref. [16]). [16]). The dashed (solid) contour corresponds to the 4-fit (3-fit) contour. A weak Ω M prior has been used. We can observe the large differences in the central values coming from the two 3-fit and 4-fit fitting procedures.

• If we take the model (w F 0 = -1, w F 1 = 0.2), which has a "small" w F 1 value, it leads to a smaller bias : we get Ω M = 0.324 ± 0.016 and w 0 = -1.021 ± 0.067. We see that there is a bias of the order of 1.5σ for Ω M , and a small bias within the statistical error for w 0 . The central values (crosses) displayed in Figure 2 of the ellipses for the 3-fit and the 4-fit are different but the 3-fit ellipse is included in the 4-fit one, indicating a small bias on one parameter (w 0 ) and a not too big bias on the second (Ω M ).

• Finally, with the model (w F 0 = -2, w F 1 = 0) which has no z dependence of the EoS, there is no bias and both ellipses are centered on the same point (see Figure 2). Note that there is no significant bias for the normalization parameter M S in the three cases.

Validity zones for the 3-fit

To quantify the bias, we scan the full plane P (w F 0 , w F 1 ) using the procedure 3-fit, assuming a flat universe. We first define different zones in this plane where the effects of the bias can be quantitatively estimated.

The plane P is divided in three distinct zones :

• The Non Converging Zone (NCZ) where the fit has some detectable problems: the fit is bad either because χ 2 > 3n (n being the number of degrees of freedom), or because the error on one of the fitted parameter is above 1. We also reject fits yielding some Ω M values far away from current expectations, namely we require 0

.1 -σ(Ω M ) < Ω M < 1. + σ(Ω M ).
This zone is shaded in our plots. None of the fiducial models located in this NCZ can be fitted with the procedure 3-fit. Therefore, with real data, the fit procedure will be excluded.

• The Biased Zone (BZ) is the part of P where the fit converges perfectly, but where we know from the simulation that we are far from the fiducial values: We quantify this bias as the difference between the central value of a fitted parameter O and the fiducial value O F . Namely the "bias" is defined as

B(O) = |O F -O| (8)
In general, B = 0 and we define the biased zone as the part of P where B(O) > σ(O) where σ(O) is the statistical error on O.

The important point is that this zone is undetectable with real data. It is the region we want to identify and to minimize through an appropriate choice of the fitting procedure.

• The Validity Zone (VZ) is the remaining part of P where the fit converges perfectly and where B(O) < σ(O). If we require this condition for the full set of fitted parameters together, we can define a "full" validity zone (full VZ).

Consequently, the contours separating the BZ and the VZ in the plane P correspond to the intrinsic limitation of the fitting procedure which is considered.

We now apply this strategy of analysis to the SNAP and SNLS simulated data. The impact of the bias on the present SN statistical sample is also presented.

Fitting procedures for SNAP

Determination of the 3-fit validity zones

We apply this method to determine the validity zones using the 3-fit on the SNAP simulated data. A weak prior on Ω M , Ω M = 0.3 ± 0.1, has been used. Figure 4a,b,c,d give the different validity zones for M s (Figure 4a), Ω M (Figure 4b), w 0 (Figure 4c) and for the three parameters taken together (Figure 4d). In all the figures, the line w F 1 = 0 corresponds to the actual unbiased fiducial models. One can see that the Biased Zone is always limited by the NCZ. The line w F 0 = -1/(3Ω F X ) ≃ -0.48 separates the models which correspond to an accelerating universe today (w F 0 < -0.48) from the decelerating ones (w F 0 > -0.48).

Looking at the validity zone for each parameter individually :

• M s (Figure 4a) is essentially unbiased for models which correspond to acceleration (w F 0 < -0.48) except for large negative values of w F 1 . We notice that M s is always measured with an error better than the percent thanks to the "Nearby" sample. Therefore it is easy to get a bias greater than this error. However this bias has no real consequence on the determination of the other parameters.

• Ω M (Figure 4b), conversely, is unbiased only for a small band around the line w F 1 = 0. It means that for most of DE models with varying EoS, the fitted Ω M is strongly biased in this fitting procedure.

• w 0 (Figure 4c) is valid in a large part of the reduced plane P R , which corresponds to the present acceleration region (w F 0 < -0.48). Otherwise we fall in the Biased Zone which, for instance, contains the previously quoted model of Maor et al. [16]. Outside P R , for large | w F 1 | values, we are in the Biased Zone. Consequently, one has to take some cautions when using the 3-fit for extracting w 0 .

The full Validity Zone is shown in Fig. 4d ans is driven mainly by the bias on the Ω M parameter. One can conclude that the 3-fit is valid on all parameters together for fiducial models which have a small dependence in redshift: when w F 0 > -1, the validity is for

|w F 1 | < 0.2, otherwise it increases to |w F 1 | < 0.4.
One has to emphasize that the experimental error will also contain systematic errors. These latter have been estimated for SNAP [14]. We have simulated this case and the result is given on Fig. 5, where we have included a constant uncorrelated systematic error on each magnitude measurement. The consequence of increasing the error is an increase of the Validity Zones. Compared to Fig. 4c, the VZ for w 0 is substantially enlarged. Note that the NCZ is reduced in the same time. The VZ for M s follows the same behaviour, the one of Ω M being much less affected. We have also looked at the impact of the Ω M prior on our results:

• When there is no prior, the results are very comparable (but slightly worse) to the weak prior case described above.

• When a strong prior is used (e.g., σ(Ω M ) = 0.01), the error on the parameters are reduced but the VZ also. Fig. 6a gives the VZ for w 0 with this prior. Due to the strong prior on Ω M , all the bias effect is reported on the w 0 parameter which is strongly biased as shown on Figure 6a. Looking more closely at the reduced P R , one sees that the VZ is limited to the line w F 1 = 0. Therefore we lose the fact that w 0 is in general well fitted by this fitting procedure.

The conclusion is that the fitting procedure 3-fit used in the case of a high SN statistics experiment like SNAP, can be useful to constrain w 0 for a large part of the plane P, which corresponds to accelerating models, only when no prior or a weak prior on Ω M is used. The other cosmological parameters (Ω M especially) are strongly biased, and strengthening the Ω M prior worsens the situation. Concerning DE models which lead to a present deceleration, it appears clearly from the preceding figures, that the cosmological parameters are biased even if the time dependence of the EoS is weak.

Comparison between the 3-fit and the 4-fit

We have shown previously that the 3-fit procedure may be useful to extract w 0 even in the presence of the bias due to the hypothesis w 1 = 0. The simplest 4-fit procedure (with no constraint on w 1 ) allows the extraction of w 0 without any bias since w 1 is included in the fit, but with an increase of the statistical errors on each parameter. We adress now a comparison of the errors obtained on w 0 with these two different fitting procedures.

To combine the bias error and the statistical error, we choose a fit quality estimator of the procedure as E(w 0 ) = σ 2 (w 0 ) + B 2 (w 0 ) which reduces to σ(w 0 ) for the 4-fit. Figure 7 compares the two procedures by displaying the difference of errors E(w 0 , 3 -fit)σ(w 0 , 4 -fit) with a weak prior on Ω M . The 3-fit is preferred, with this estimator, in a large part of the plane P which corresponds to models with present acceleration. The two black areas correspond to the NCZ of both fitting procedures. Therefore, one cannot conclude for a preference for the 4-fit in this zone. The two hatched zones, corresponding to large values of |w F 1 |, indicate a preference for the 4-fit. This is due to the bias on w 0 inherent to the 3-fit. There is also a narrow hatched band where the 4-fit is still preferred although the bias is small. This appears to be a zone where the correlation between w 0 and w 1 is small and then, the errors from the 4-fit and the 3-fit are similar, the 4-fit being slightly better (less than 2%).

In the white regions, the two parameters w 0 and w 1 are strongly correlated and then the 3-fit has clearly a smaller error.

So, we can conclude that with a weak, thus conservative, prior on Ω M the 3-fit is in general better than the 4-fit to extract w 0 for accelerating models.

Analysis for SNLS

Determination of the 3-fit validity zones

A similar analysis has been performed for a simulated sample corresponding to the SNLSlike statistics.

The various Validity Zones obtained for SNLS with a weak Ω M prior are shown in Figures 8a,b,c,d We deduce from this figure that :

• M S is almost unbiased for accelerating models and also for decelerating ones provided the redshift dependence is small (|w F 1 | < 0.6) (Fig. 8a). • Ω M is not biased in most of the reduded plane P R and it is biased outside this plane if |w F 1 | > 0.6 (Fig. 8b). • w 0 is not biased for an important part of the plane corresponding to acceleration (w 0 < -0.48) (Fig. 8c).

• The 3-fit is a good fitting procedure for reconstructing all cosmological parameters simultaneously for accelerating models (w 0 < -0.48) when the redshift dependence is small |w F 1 | < 0.6 (see Fig. 8d). Otherwise the result is biased.

• The use of a strong prior on Ω M worsens the situation : w 0 is biased for most of the plane P as shown on Fig. 6b.

Comparing these results with the SNAP case shows that the conclusions are quite similar but the validity zones for SNLS are much larger than for SNAP. Due to the lower statistics our criterion B(O) < σ(O) is less constraining. Note that the NCZ zones are strongly reduced as expected with the lower statistics.

Comparison between the 3-fit and the 4-fit

We perform the same comparison of the errors on w 0 obtained from the two procedures 3-fit and 4-fit, still with a weak prior on Ω M . Figure 9 shows the difference of errors between the 3-fit and the 4-fit. The 3-fit is preferred when the bias is small or when the two parameters w 0 and w 1 are strongly correlated. When w F 1 < 0 the hatched region in the accelerating zone has similar origin and properties as the narrow hatched zone of Fig. 7, namely that both fitting procedures provide in fact very comparable errors.

Analysis for a statistical sample corresponding to the SCP data

We can define the validity zones for the statistics of the present SN sample from the SCP collaboration [4] introduced in Section 2.1. Figure 10a gives the full Validity Zone obtained without any prior on Ω M . The large size of this zone shows that the results are not biased (at 1σ) for most of the plane P. In fact only Ω M is affected by a bias.

A weak Ω M prior of σ(Ω M ) = 0.1 has been used to draw Fig. 10b, giving the Validity Zone for w 0 (identical to the full VZ). The VZ is slightly reduced compared to the no prior case but it remains quite important within the plane P. More precisely, it appears that the upper part of the biased zones are comparable for both no and weak prior cases. Only the lower part of the biased zone of Fig. 10b is specific to the weak prior case. Note that changing the error value of the prior has little consequences on the shape of the VZ. This fitting procedure is widely used to extract informations from present SN data [1,2,3,4,32]. The results published in these references are valid for models with constant EoS. From our findings, one can extend the validity of these results and conclude that models with a redshift dependent EoS which verify | w F 1 |< 0.7 may be fitted by this 3-fit procedure with a bias which stays below the statistical error. So, we conclude that we can trust (at 1σ) the results based on present SN data obtained with a 3-fit where we neglect the redshift dependence of the EoS, using a Ω M prior or not. Only (fiducial) models with a strong recent variations of the EoS may lead to biased results.

The comparison of the potentialities of the 3-fit and the 4-fit to extract w 0 does not seem very relevant. However, we have checked this point, and it appears that the 3-fit provides some constraints on w 0 which are roughly 30% better than with the 4-fit for most of P.

Finally, we have remarked that the central values of w 0 and Ω M are shifted as follows: if w F 1 is positive we have w 0 > w F 0 and Ω M > Ω F M , and the converse if w F 1 < 0. We do not recover this simple behaviour with the SNAP and SNLS samples.

Summary and conclusions

In this paper, we have studied the best strategy to extract the cosmological parameters and in particular the dark energy component from the future supernovae data within the Ω T = 1 paradigm. We have compared results from the present statistics with the ones we can expect from future large ground-based surveys and from a future space experiment like SNAP. In order to be as model independent as possible we have looked at a large range of fiducial models by varying the parameters of the equation of state of the dark energy w F 0 and w F 1 where w F (z) = w F 0 + w F 1 z [17]. With present statistics or the one expected from a survey like SNLS, precision on w 1 is very weak and the physical question concerns the extraction of w 0 and Ω M . For a SNAP like experiment a good precision on w 1 is only possible with a strong prior on Ω M .

To avoid the use of strong priors on Ω M , we have focused on the extraction of w 0 only and neglected the w 1 contribution by fixing it in the fitting procedure. We have to face in this case the problem of introducing a bias if the fiducial value of w F 1 is far away from the fixed one. We compare the bias to the statistical error which is expected for each survey.

To study the validity of such a strategy, we have presented an extensive study of fiducial models in a flat universe in which we vary w F 0 and w F 1 in a complete range -2 < w F 0 < 0 and -2 < w F 1 < 2 corresponding to a large variety of DE theoretical models. On the other hand M F s and Ω F M are fixed. In the fit we have used in general a weak prior on Ω M (σ(Ω M ) < 0.1) and we have looked at the bias introduced by fixing w 1 = 0. We call this procedure the 3-fit (M s , Ω M , w 0 ) procedure.

For SNAP (and SNLS) like statistics, we obtain the following conclusions :

• w 0 is not biased for an important part of the scanned plane (w F 0 , w F 1 ) which corresponds to models describing a Universe which is today in acceleration.

• for M S , we get the same conclusion but we emphasize that this parameter is so much constrained in the fitting procedure that a small bias has no visible effect.

• Ω M is strongly biased if the redshift dependence is large (|w F 1 | > 0.2(0.6)).

• If a strong prior on Ω M is used, w 0 is biased for most of the plane. Namely, the 3-fit procedure is relevant to extract w 0 if only a weak Ω M prior used.

We have compared the accuracy of the results on the w 0 parameter to the ones obtained if w 1 is left free as in a 4-fit (M s , Ω M , w 0 , w 1 ) procedure.

We found that, in spite of the bias, the 3-fit procedure, where w 1 is fixed, can be used for SNAP and SNLS to test accelerating models and give better or equivalent results than the simplest 4-fit where w 1 is completely free. This is no longer true if | w F 1 | is large. For decelerating models the 4-fit is mandatory. Note that, concerning the choice of the number of parameters to be fitted, the bayesian method presented in [33] seems promising. These conclusions have been set using only the value Ω F M = 0.3 for the fiducial models. If the true value is different the areas of the different Validity Zones increase if the contribution of DE is smaller, namely if Ω F M is larger. Conversely, if Ω F M < 0.3, the validity of the 3-fit is reduced.

Using the presently available statistics of the SCP collaboration [4], it appears that the procedure where w 1 is fixed can be trusted for almost all the fiducial models considered here. Only fiducial models with strong recent variations of the EoS (| w F 1 |> 0.7) may lead to biased results.

Let us point out that other bias would be introduced, by a departure from the Ω T = 1 paradigm in the fiducial model, or if the central value of the Ω M prior itself (used in the fitting procedure) is not the same as the fiducial value. Indeed a small bias in the prior could induce some strong bias for the other cosmological parameters. These bias will be studied in a future paper [27].

It has been advocated recently [34] that it should be easier to constrain the DE density ρ X (z) and its time derivative ρ ′ X (z), instead of constraining the equation of state w(z). These authors have simulated a large variety of fiducial models in the range -1.2 ≤ w F 0 ≤ -0.5 and -1.5 ≤ w F 1 ≤ 0.5, which is almost completely contained in our Validity Zone for SNAP (see Fig. 5). They interpret physically parts of this plane in term of increasing, decreasing or non-monotonic DE densities ρ X (z).

Their approach is interesting to answer if the DE energy density is a constant or not. Nevertheless, focusing on the equation of state itself remains mandatory if, for instance, one wants to validate precisely the acceleration since, in this case, a precise knowledge of w 0 (and also on Ω M and Ω X ) is necessary. Finally, we would like to point out that trying to answer any particular question on the nature of DE requires an investigation on the choice of the most relevant fitting procedures.

Figure 1 :

 1 Figure1: Fisher contours in the plane (Ω M , w 0 ) at 68,3 % CL for a cosmological constant as the fiducial model. The large dashed contour corresponds to the 5-fit(M s , Ω M , Ω X , w 0 , w 1 ). The three dotted contours correspond to the 4-fit(M s , Ω M , w 0 , w 1 ) with no prior (largest), with a weak Ω M prior (i.e. Ω M = 0.3 ± 0.1) and with a strong Ω M prior (i.e. Ω M = 0.3 ± 0.01, smallest ellipse). The solid contours correspond to the 3-fit(M s , Ω M , w 0 ) with a weak Ω M prior (largest) and with a strong Ω M prior (smallest). For this fitting procedure, there is almost no difference between the cases "no" and "weak" prior for Ω M .

Figure 2 :

 2 Figure 2: Contours in the plane (Ω M , w 0 ) at 68,3 % CL for the fiducial models (wF 0 = -2, w F 1 = 0) and (w F 0 = -1, w F 1 = 0.2). The large dashed (small solid) contours correspond to the 4-fit (3-fit). A weak Ω M prior has been used.

Figure 3 :

 3 Figure 3: Contours in the plane (Ω M , w 0 ) at 68,3 % CL for the fiducial models (w F 0= -0.7, w F 1 = 0.8)(Maor et al. model[16]). The dashed (solid) contour corresponds to the 4-fit (3-fit) contour. A weak Ω M prior has been used. We can observe the large differences in the central values coming from the two 3-fit and 4-fit fitting procedures.

1 FFigure 4 :

 14 Figure 4: Validity Zones(white), Biased Zones (hatched) and Non Converging Zones (black) for a SNAP like data set for M S only (a), Ω M (b), w 0 (c) and all the parameters together (d). The procedure is a 3-fit with a weak Ω M prior. The vertical line at w F 0 = -0.48 separates decelerating models from accelerating ones.

1 FFigure 5 :

 15 Figure 5: Same as Fig.4c for w 0 in the case where we add an uncorrelated and constant in redshift systematic error of 0.02 on the magnitudes.

1 FFigure 6 :

 16 Figure 6: Same as Fig.4c for w 0 in the case of a strong Ω M prior a) : SNAP, b) : SNLS.

1 FFigure 7 :

 17 Figure 7: E(w 0 , 3 -fit)σ(w 0 , 4 -fit) for the SNAP data and a weak Ω M prior. The white part corresponds to a positive difference, the hatched part to a negative difference, the black region corresponds to the Non Converging zones.

1 FFigure 8 :

 18 Figure 8: Validity Zones (white), Biased Zones (hatched) and Non Converging Zones (black) for a SNLS like data set for M S only (a), Ω M (b) and w 0 (c) and all the parameters together (d). The procedure is a 3-fit with a weak Ω M prior.

1 FFigure 9 :

 19 Figure 9: E(w 0 , 3 -fit)σ(w 0 , 4 -fit) on w 0 for a SNLS like data set and a weak Ω M prior. The white part corresponds to a positive difference, the dashed part to a negative difference, the black region corresponds to the Non Converging Zones.

1 FFigure 10 :

 110 Figure 10: Full Validity Zones and Biased Zones for a SCP like data set with a) no prior b) a weak Ω M prior. The NCZ are not plotted due to the large errors on Ω M and w 0 .

Table 1 :

 1 number of simulated supernovae by bin of 0.1 in redshift for SNLS and SNAP respectively.

	z	0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7
	SNLS	-	44 56 80	96 100 104 108	-	-	-	-	-	-	-	-
	SNAP 35 64 95 124 150 171 183 179 170 155 142 130 119 107 94 80

Table 2 :

 2 Statistical errors obtained with a Fisher matrix analysis on the cosmological parameters for various fitting procedures with the SNLS and SNAP data repectively. The fiducial model is a cosmological constant with M

Acknowledgments

We wish to thank A. Mazure and the members of the Laboratoire d'Astrophysique de Marseille, for their support, and also E. Linder, R. Miquel and F. Steiner, for helpful discussions. "Centre de Physique Théorique" is UMR 6207 -"Unité Mixte de Recherche" of CNRS and of the Universities "de Provence", "de la Méditerranée" and "du Sud Toulon-Var"-Laboratory affiliated to FRUMAM (FR 2291). "Centre de Physique des Particules de Marseille" is "Unité Mixte de Recherche" of CNRS/IN2P3 and of the University "de la Méditerranée".


Thérèse Levene
email: therese.levene@univ-lille.frlucie.petit@univ-lille.fr

Lucie Petit

ANALYSE DES BESOINS ET FORMATION EN SITUATION DE TRAVAIL

Keywords: Besoins de formations, Formation en situation de travail, Maraîchage diversifié

Axe 1 : Croiser la perspective des acteurs

Objectifs de la conférence

Apporter de la visibilité aux réalités empiriques de la formation professionnelle

Résumé

 et leur contexte socio-économique.

ANALYSE DES BESOINS ET FORMATION EN SITUATION DE TRAVAIL

Le dispositif de formation au coeur de notre propos aujourd'hui est le Pass installation APTEA (Association pour la Pérennisation et la Transmission des Emplois en Agriculture), filière maraîchage diversifié, mis en oeuvre par l'association Initiatives Paysannes implantée dans les Hauts de France. Ce dispositif ne relève pas d'une AFEST au sens de l'article L.6313-1 1°) -1 1°) mais il propose une formation en situation de travail. Ce qui caractérise ce dispositif1 est qu'il est construit sur une volonté de répondre aux besoins de formation des futurs maraîchers et aux besoins de formation tout au long de la vie des maraichers professionnels. En première partie de cette communication nous reviendrons sur la notion de besoins. En effet, cette question connait une certaine déshérence depuis les travaux de [START_REF] Barbier | Mais qui donc éprouve des besoins de formation[END_REF] ou encore de Malglaive (1981), pour ne citer qu'eux. Dans un second temps nous verrons les difficultés de considérer les besoins de formation. La troisième partie présentera le dispositif APTEA et plus précisément sa genèse et sa conception qui rendent compte de la manière dont les besoins collectifs et objectifs de formation des futurs maraîchers et des maraichers en exercice sont au coeur de la construction du dispositif.

I LA NOTION DE BESOIN

« Il est des confusions et des ambiguïtés dont l'existence ne relève ni de la polysémie et de l'imprécision des termes mais du rôle idéologique qu'elles jouent dans les pratiques sociales. Ainsi en est-il de la notion de besoin de formation » (Malglaive, 1981, p.34).

Pour revenir à cette question essentielle de notre point de vue qu'est la question des besoins des êtres humains, nous reconvoquons des auteurs qui appartiennent maintenant à notre patrimoine scientifique : en sciences de l'éducation, Gérard Malglaive (1981), Bernard Charlot (1976) notamment ; en sociologie Patrice Grevet, Paul-Henri Chombart de Lauwe (1975) et, en philosophie Lucien [START_REF] Sève | Marxisme et théorie de la personnalité[END_REF][START_REF] Sève | Marxisme et théorie de la personnalité[END_REF]. Ces auteurs nous invitent à dépasser deux conceptions qui apparaissent comme des impasses pour penser les besoins : une conception biologique et une conception idéaliste des besoins, conceptions souvent associées. L'approche biologique renvoie à un système de régulation homéostatique : « A la tension du besoin répond l'activité dont le résultat est la réduction de la tension et par conséquent le retour à un certain équilibre, c'est-à-dire au repos » (Sève, 1972, p.395 (Grevet, 1975, p.108). Autrement dit « le moment objectif du besoin est celui de l'apparition des possibilités de sa réalisation dans la formation économique et sociale » (Malglaive, 1981, p.35), c'est-à-dire la société concrète dans l'ensemble de ses rapports (Sève, 1980, p.679).

Il faut donc qu'existent les conditions concrètes et en premier lieu les conditions matérielles de la satisfaction des besoins. Ces conditions nécessaires ne sont pas suffisantes, le moment subjectif est décisif pour qu'en soit assurée la satisfaction (Grevet, 1975, p.110). Ce moment subjectif, Malglaive le définit comme la prise de conscience, l'intériorisation subjective (mais pas nécessairement individuelle) de l'existence objective des besoins de formation.

Le moment social du besoin renvoie à la reconnaissance sociale, politique et financière du besoin objectif [START_REF] Hedoux | L'ingénierie et l'évaluation de la formation[END_REF]. Ce cadre d'analyse des besoins étant posé, on reconnait donc certaines caractéristiques aux besoins :

-« Les besoins peuvent exister objectivement sans être pour autant perçus comme besoins à satisfaire » (Malglaive, 1981, p.36). Les besoins réels ne sont donc pas toujours conscients. -La prise de conscience des besoins se fait sous les effets de l'idéologie dominante (Demunter, 1992). En somme cette prise de conscience peut se trouver empêchée, limitée, dévoyée. C'est probablement la relecture récente de Paolo Freire, son ouvrage tout récemment réédité, La pédagogie des opprimés (2021) qui nous a encouragées à reconsidérer cette notion de besoins.

-Les inégalités structurelles des sociétés humaines sont à l'origine du caractère contradictoire des besoins. Leur satisfaction résulte de compromis entre des systèmes d'intérêts divergents voire contradictoires. -Le décalage entre besoins objectifs et demandes, l'absence de demande, et le caractère contradictoire des besoins nécessitent une analyse scientifique de l'état de développement des rapports sociaux de production et des forces productives.

Renoncer à considérer la question des besoins c'est nous semble-t-il nous priver d'un cadre théorique, scientifique qui permet de penser les problèmes à résoudre et les objectifs à atteindre.

II LES DIFFICULTES D'APPREHENDER LES BESOINS EN FORMATION

Les spécialistes en ingénierie de formation semblent embarrassés avec cette question de besoins de formation. (Hedoux, 1996, pp.84-86).

Barbie et Lesne n'ignorent pas que s'en tenir à l'expression des besoins de formation c'est prendre le risque de laisser les gens là où ils en sont de leurs représentations et de leurs possibilités [START_REF] Hedoux | L'ingénierie et l'évaluation de la formation[END_REF] . Elle dénonce un modèle productiviste qui produit plus que de besoin mais échoue à nourrir tous les habitants de la planète 8 et qui conduit à la disparation des fermes 9 . Elle plaide pour une agriculture de proximité économe, préservatrice de ses propres ressources qui garantit aux paysans un revenu suffisant et des conditions de travail raisonnables sur une surface à taille humaine 10 .

III-3 LES ETAPES DU PARCOURS DE FORMATION

III-3-1 En amont du positionnement, un projet déjà travaillé

Les candidats doivent remplir les prérequis13 à l'entrée du parcours et se présenter avec un projet d'installation en adéquation avec la philosophie de l'association Initiatives Paysannes. Avant le positionnement ils ont effectué un stage de 3 jours : « De l'envie au projet »14 qui permet de « clarifier ses envies et idées en enracinant son projet dans un parcours professionnel et personnel ».

Les candidats présentent la manière dont ils envisagent leur installation, leur production, leur mode de travail… Ce projet sera mis à l'épreuve de leur formation-expérimentation. Ainsi Julie qui projetait d'ouvrir un magasin dans sa ferme optera au final pour une AMAP15 après avoir expérimenté différents circuits de commercialisation. Jacques sensibilisé à la traction animale sur une ferme abandonnera son projet initial de traction motorisée. Oscar, rencontré en fin de parcours, est en mesure de consolider ses choix en termes de protection et de nourriture de son sol.

III-3-2 Le positionnement

Le positionnement soutenu d'entretiens réguliers avec l'animatrice d'Initiatives paysannes dure en moyenne trois mois. Il démarre par une immersion d'une semaine du porteur de projet chez un maraîcher qui aboutira à une co-évaluation des compétences et connaissances du candidat (association, candidat, maraîcher) à l'aide du référentiel d'activités, de compétences et d'évaluation. Puis, le postulant devra trouver trois fermes d'accueil sur chacune desquelles il aura passé une semaine en immersion pour faire connaissance avec ses futurs formateurs et leurs méthodes de culture. Chacun des trois lieux est considéré comme un plateau technique de formation, c'est-à-dire un espace de formation en milieu de travail avec chacun un formateur16 (le maraîcher accueillant) Ensuite, un programme pédagogique personnalisé en cohérence avec son projet d'installation, notamment pour ses besoins de formation technique (comptabilité, système d'irrigation, école des ânes…) est co-construit avec le candidat.

Le passage devant un comité d'admission confirme l'entrée dans le dispositif. Ce long positionnement a pour but de consolider, clarifier, voire réviser le projet, mais aussi de sécuriser le déroulement de la formation pour Initiatives Paysannes, les exploitants-formateurs, le porteur de projet. Au cours des 12 mois de formations trois réunions sont programmées sur chacune des fermes ce qui leur permet de prendre la mesure concrète du modèle des deux autres maraichers du collectif.

III-3-3 Le choix des maraichers

Ce dispositif est donc aussi un dispositif de formation tout au long de la vie visant à répondre aux besoins de formation que non seulement le développement de l'activité fait émerger, mais aussi l'évolution des techniques (ou la redécouverte de techniques), des outils et plus largement la place de l'agriculture paysanne et du développement durable prend dans nos sociétés.

POUR CONCLURE

Le dispositif Pass Installation APTEA consiste en un modèle d'analyse des besoins ancré dans les situations de travail. Il correspond au troisième modèle de détermination des objectifs de formation (Barbier, Lesne, 19177) en ce qu'« il renvoie aux négociations et compromis entre acteurs individuels et collectifs [….] La formation ne résulte ni de la seule exigence de l'organisation, ni de la seule demande individuelle, mais de la confrontation de systèmes de besoins (moment social) pour l'élaboration négociée de projets collectifs de formation » (Hedoux, 1996, p.86). Ce dispositif illustre des manières d'identifier les besoins de formation et d'y répondre dans leurs dimensions objectives, individuelles et collectives.

-Dans leurs dimensions objectives au sens où ce dispositif s'appuie sur les expériences des maraichers professionnels, leurs difficultés et leurs projets de développement dans un contexte de développement durable par définition en évolution : ils ne préparent pas seulement « les futurs professionnels à des pratiques finales,

  ).La conception idéaliste est celle présentée par le célèbre psychologue Maslow. Cette théorie, très connue et qui demeure une référence dans les formations en commerce ou en management définit le besoin comme une énergie vitale qui pousse l'individu à satisfaire une catégorie de besoins dite supérieure. La dernière marche de la pyramide est la manifestation du besoin d'autoréalisation. Ce qui caractériserait l'être humain serait un besoin individuel d'autodépassement. Cette théorie est critiquée à plusieurs titres : elle présente un caractère ordonné trop mécaniste contredit par les faits(Wahba et Bridwell, 1976 et Tat et Diener, 2011 cités par Matzkin, 2018) ; elle relève d'une conception ethnocentrée (impose le modèle américain des classes moyennes aisées des années 1960 comme universel) et enfin fait abstraction des facteurs sociaux en cantonnant le besoin à une pulsion individuelle. Plus récemment[START_REF] Megdiche | Essai sur la notion de besoin[END_REF] propose une articulation entre le besoin biologique et le besoin social : selon lui, de tension instinctuelle, le besoin peut devenir revendication voire révolte politique (p.90).

	On reste là dans un certain biologisme, c'est-à-dire une idéologie où les conditions biologiques
	de l'individualité ou de l'histoire humaine sont prises pour leur base réelle (Sève 1980, p.664)
	De telles approches ne tiennent pas compte de l'histoire et de la dimension contradictoire des
	besoins humains pourtant posée par Sève (1972) et à sa suite Malglaive (1981). C'est-à-dire
	qu'elles ignorent la condition sociale et historique et donc évolutive de l'espèce humaine.
	« Le processus de l'hominisation renverse justement l'essence des besoins humains, l'aspect
	biologique devient un élément subordonné ; le besoin ne résulte plus d'une origine biologique
	interne à l'individu, mais de l'intériorisation d'exigences externes liées aux forces productives
	et aux rapports sociaux. Ces exigences externes forment le moment objectif des besoins, leur
	base en quelque sorte ».

  et donc ne pas considérer qu'une demande peut en cacher une autre, qu'une demande doit être analysée, des représentations déconstruites. Faire ce travail ce n'est pas décider à la place des gens tout au plus c'est désirer pour eux ! Le pédagogue n'est-il pas dans cette posture ? Faire advenir en référence à la maïeutique de Socrate. « La formation à l'analyse de ses besoins en formation doit être un des objectifs essentiels de toute action de formation » avance B.Charlot. Pour lui, seule une analyse des problèmes vécus par les formés (ou les futurs formés) en situation peut déboucher sur l'élaboration de besoins de formation. La formation permet « le repérage des besoins, besoins moins de formation que de Avancer que les seuls besoins réalistes à prendre en compte sont les besoins de l'entreprise ne rend pas l'affaire plus simple. Pourquoi ? Parce que les entreprises ne sont pas non plus conscientes de leurs besoins ! Une enquête auprès de plus de 500 entreprises menée par Inffolor Carif Lorraine et le Centre Inffo notamment en 2006(Drouot, 2007), autour du développement des compétences dans les entreprises et son impact sur la formation a mis en évidence un management à court terme des compétences et "pas toujours en lien avec les évolutions sectorielles" 2 . Les entreprises ont une vision à court terme de leur développement économique, l'explication vient de là. Difficile en effet de prévoir l'imprévisible mais on peut se préparer à des éventualités, être à l'affut, anticiper, se raviser, s'adapter en somme. A cet égard les observations de Patrick Mayen sur « le travail avec le vivant » (2014) sont riches d'enseignements. L'agriculture n'est pas le seul secteur qui connait des phases de transitions ou de mutations. Si l'on admet que toute activité humaine s'inscrit dans le mouvement, alors l'adaptation n'est pas l'adaptation au poste de travail mais au changement. Au-delà de la méconnaissance de l'avenir, rappelons aussi à la suite de Mayen entre autres que le travail réel est méconnu. « Il est méconnu par ceux qui encadrent le travail, ceux qui le définissent, par les formateurs, et même dans une certaine mesure par ceux qui l'exercent » Le maraîchage, souvent en agriculture biologique ou paysanne, est une des filières du secteur agricole 3 qui attire de nombreux candidats au changement de métier et de vie. En effet, les nombreux aspirants maraîchers non issus du monde agricole ne bénéficient pas d'une socialisation primaire ni de la passation d'un patrimoine familial (expérience, savoirs nécessaires pour exercer, et outil de production) ni d'une mémoire du métier (Olry etChrétien, 2012 ;[START_REF] Chrétien | Transmission professionnelle et recours à la mémoire du métier. Analyse croisée de deux domaines professionnels[END_REF] 4 . « La transmission du métier n'est donc pas chose simple ni pour l'agriculteur, ni pour l'apprenant, car la transmission de connaissances en situation de travail en agriculture […] implique toute la personne, son intégration au « métier-exercice » comme au « métier-vie ».[START_REF] Chrétien | Apprendre et se tester avant de s'installer en maraichage biologique[END_REF]. Les futurs maraîchers ont donc besoin d'un accompagnement plus personnalisé que celui offert par les dispositifs de formation existants 5 . C'est ce que propose le dispositif de formation continue, le Pass installation APTEA (Association pour la Pérennisation et la Transmission des Emplois en Agriculture), filière maraîchage diversifié, mis en oeuvre par l'association Initiatives Paysannes implanté dans les Hauts de France. Initiatives paysannes 6 est une association qui rassemble des paysans, des porteurs de projet, des artisans et des citoyens. Elle oeuvre au développement et à la promotion d'une agriculture paysanne. L'agriculture paysanne est née de la réflexion des paysans de la Confédération Paysanne 7

	III LE DISPOSITIF PASS INSTALLATION APTEA
	III-1 HISTOIRE ET CONTEXTE

réflexion et d'action » poursuit Dominique Beau dans le même numéro thématique de la revue Education permanente

de 1976 (p.66)

. Plutôt que des enquêtes de satisfaction en fin de formation ne devrait-on pas faire une enquête sur les besoins qu'elle a fait émerger ?

(Mayen, 2014, p.20)

. D'où l'importance d'une analyse des situations au sens d'« environnement expériencié » donné par Dewey

(1993( , cité par Mayen, 2014, p.29, p.29

) « L'ingénierie de formation se concrétise principalement dans deux pratiques : l'analyse de besoins et la construction de dispositifs de formation [...] On peut donc dire que c'est dans le prolongement des démarches d'analyse des besoins que vient s'établir ce qui va constituer un des piliers fondateurs de la didactique professionnelle, l'analyse du travail »

[START_REF] Pastré | La didactique professionnelle[END_REF]

.

La conception du dispositif que nous avons observé s'est appuyée sur l'analyse du travail par les maraîchers eux-mêmes, il est sans doute aujourd'hui un peu atypique dans un contexte de concurrence accrue. Il a été pensé et conçu (et continue d'être pensé) en lien avec son contexte et l'évolution de celui-ci : le développement durable, l'agriculture paysanne, et plus précisément le maraichage diversifié sur petites exploitations, la formation des futures générations de maraichers, l'apprentissage continu entre pairs et l'entraide. Ici le futur public est acteur du processus de définition des objectifs issus de l'analyse de ses besoins collectifs et individuels.

APPRENDRE DES PAIRS : LE PASS APTEA, UN DISPOSITIF DE FORMATION TOUT AU LONG DE LA VIE ET UN COLLECTIF D'ENTRAIDE Le

  Le choix des maraichers est une étape importante dans le processus de positionnement car elle conditionne la satisfaction des besoins individuels de formation Les candidats sont incités à choisir des plateaux techniques diversifiés en termes de surface, outillage, production, etc. Retenons l'exemple de Robin qui a choisi trois fermes représentant trois modèles très différents. Il a choisi un maraicher en couveuse 17 « pour voir un modèle en train de se construire », une ferme expérimentale gérée par une multinationale « pour voir le côté salarié, ça fait partie de la réflexion » et un maraicher installé depuis 5 années sur une parcelle qu'il exploite seul et qui « a du recul sur son expérience ». Marie se projette en permaculture mais elle a choisi trois plateaux techniques qui se distinguent par leurs outils de production pour le travail du sol, l'entretien des cultures et leur transport : le recours à des engins motorisés, à des animaux domestiques ou à l'outillage manuel comme la « chtit bine » ou lit de désherbage. Trois sites, trois manières de travailler et de vivre le métier : « On voit, on compare et donc on analyse les différents systèmes. Je pense que pour apprendre c'est super intéressant, parce que on voit pas en disant « ça c'est la solution ultime, c'est ça qu'il faut faire », mais non, ça permet de réfléchir en disant « ah non, y a différents systèmes, ça aussi ça fonctionne très bien. Donc ça c'est intéressant dans tel contexte […] d'un point de vue technique et professionnel, c'est pas mal, ça permet aussi de voir différentes ambiances de travail, ça c'est aussi un point intéressant je trouve. » (Robin) La planification est la dernière phase avant la mise à l'épreuve de la réalité de l'installation. Elle vient témoigner des réponses obtenues aux besoins en formation identifiés à la phase de positionnement et aux besoins émergés au cours de la formation et/ou des lacunes qu'ils restent à combler. Elle correspond à la formalisation du projet d'installation et à sa mise en chiffres (coût, surface, quantité). Cette formalisation est inscrite au référentiel des compétences (bloc de compétences n°8) 18 , il s'agit de « présenter un dossier d'installation solide aux différents interlocuteurs ». Ce dossier devra être validé par l'association et les maraichers accueillants.La planification se prépare bien en amont de l'installation surtout quand elle peut être testée concrètement au cours de la formation ;Enfin dernière caractéristique de ce dispositif, il permet la formation continue des maraichers et favorise l'entraide. stagiaire est vecteur d'informations d'un paysan à l'autre, il provoque des échanges et des réflexions avec les maraîchers accueillants en rapportant ce qu'il découvre et expérimente sur les autres fermes : les outils parfois fabriqués par les maraîchers eux-mêmes, les techniques déployées, les aléas rencontrés, les astuces trouvées, les solutions mises en place. Par son intermédiaire les paysans se prêtent des outils, se dépannent (suite à un gel inattendu sur une de ses parcelles Anne, paysanne qui accueille Oscar, s'est vue proposé des plants de remplacement par un autre maraîcher accueillant) ou encore développent leurs connaissances. Le stagiaire donne vie et sens au collectif. Cette organisation centrée sur le porteur de projet a été pensée ainsi dès le départ avant même que le collectif ne finalise le projet de formation en situation de travail. Et ce afin de mettre en réseau de manière active des maraîchers volontaires.

	III-4
	III-3-4 La planification et les premiers pas dans le maraichage professionnel

Bénéficiant d'un financement régional, l'association a sollicité l'équipe Trigone du laboratoire CIREL de l'Université de Lille pour modéliser les méthodes pédagogiques innovantes dans le cadre de ce dispositif.

Plus de 60% des entreprises déclarent avoir une vision de leur développement des ressources humaines à moins d'un an dont 23% à moins de 6 mois et ce quelle que soit la taille de l'entreprise.

Le recensement agricole de 2020 indique que dans la région des Hauts-de-France, on compte par rapport à 2010 62 % de structures supplémentaires « cultures de légumes et champignons » (maraîchage quel que soit le type d'agriculture).

Selon Purseigle (Le un, Hebdo. Ces jeunes qui choisissent la terre, février 2020,n°284) : « Seulement 13 % des jeunes qui sont dans les écoles agricoles viennent de ce milieu […] 30 % de ceux qui s'installent ne sont pas issus du monde agricole ».

Nous pensons au Brevet professionnel responsable d'entreprise agricole (BP REA) et au certificat de spécialisation (CS).

https://initiatives-paysannes.fr, repérée le 9 juin 2022

La Confédération paysanne est un syndicat agricole à vocation générale, représentatif au niveau national. Elle a obtenu 19,63 % des voix aux élections professionnelles de 2007, derrière l'union FNSEA* et JA* et devant la Coordination rurale et le Modef

-Les prérequis sont : un niveau 4, une expérience en lien avec le milieu agricole et/ou une formation telle que le BP REA (Brevet professionnel responsable d'entreprise agricole) ou le certificat de spécialisation de conduite de production maraîchère, avec si possible une mention agriculture biologique.

https://envieauprojet.fr, repéré le 10 juin 2022 : « faire le clair sur sa situation, sur le projet. Travailler sur la viabilité, réfléchir sur l'impact du projet sur sa vie et ses proches, réussir l'insertion du projet dans le territoire et se donner de la méthode ».

AMAP : associations pour le maintien d'une agriculture paysanne. Un groupe de consommateurs s'engage à prendre chaque semaine pendant 6 mois ou un an un panier de récoltes locales auprès d'un paysan. Ce panier est prépayé et assure ainsi aux producteurs locaux des revenus stables.

Nous reprenons les termes utiliser par Initiatives Paysannes (plateau technique de formation pour la ferme maraîchère, formateur pour le maraîcher accueillant), car ils indiquent clairement l'objet et le projet (la formation en situation de travail de futurs maraîchers) du Pass installation APTEA

Dans le cadre d'un contrat d'appui au projet d'entreprise, le démarrage en « couveuse » permet au maraicher débutant (comme à tout entrepreneur débutant) de tester son projet en grandeur nature en bénéficiant d'une couverture juridique : numéro de SIRET avec lequel il peut exercer, protection sociale et d'un appui comptable et administratif : https://chrysalide.apetitspas.net/entree-en-couveuse/ récupéré le 3 juin 2022

https://formations-en-hauts-de-france.c2rp.fr/formation/2019647126, récupéré le 6 juin 2022

L'opérationnalisation de ces principes s'est traduite notamment en région Hauts de France par le regroupement de paysans autour de la question des besoins en main d'oeuvre, de la formation de celle-ci mais aussi du devenir des terres, c'est-à-dire des exploitations après le départ en retraite des paysans. L'observation du dispositif Pass installation APTEA en vue de la modélisation des méthodes pédagogiques qu'il déploie s'est appuyée sur des entretiens auprès de porteurs de projets et de maraîchers accueillants et des observations filmées de ces binômes en situation de travail, suivies d'entretiens en autoconfrontation.

III-2 LA CONCEPTION DU PASS INSTALLATION APTEA

La conception du Pass installation APTEA a bénéficié d'une ingénierie de formation de type bottom up, c'est-à-dire une démarche d'ingénierie ascendante. En effet, le parcours ainsi que le référentiel d'activités, de compétences et d'évaluation 11 auquel il est adossé ont été conçus par un groupe de travail composé de paysans en exercice et de porteurs de projet (futurs maraîchers). Pour un certain nombre d'entre eux, les paysans ont connu avant leur installation la même situation que le public visé. À l'époque de leur reconversion, ils ont été insatisfaits de la formation suivie trop éloignée des réalités du terrain et insuffisamment ancrée dans les problématiques spécifiques à l'installation. Cette ingénierie s'est appuyée sur une étude qui a fait ressortir la nécessité d'aborder en formation : les savoir-faire techniques, organisationnels, de gestion ; les savoirs sociorelationnels propres au milieu ; l'anticipation de nombreux paramètres avant l'installation (taille de la ferme, méthodes de culture, de mécanisation et de commercialisation, constitution d'un réseau social de pairs et de fournisseurs, etc.). Ces apprentissages devaient se faire en situation dans différentes exploitations 12 . Cette ingénierie de formation vise un double objectif : l'installation en tant que chef.fe d'une exploitation maraîchère, et l'apprentissage du maraîchage. Pour atteindre ce but, elle propose une formation sur douze mois couvrant ainsi les quatre saisons d'un cycle entier de production sur trois lieux de stage différents. Avant l'entrée en formation, le stagiaire dépose un projet, consacre environ 3 mois à coconstruire sa formation, c'est ce qu'Initiatives paysannes appelle le positionnement qui aboutit à l'établissement d'un collectif dit « collectif APTEA » comprenant les trois formateurs (maraîchers accueillants), le porteur de projet, et une animatrice pédagogue d'Initiatives Paysannes. Voyons ces étapes et en quoi elles consistent à identifier les besoins en formation maraichère 11 L'objectif central de la formation : Être capable de définir et d'orienter son futur système de production agricole en toute cohérence avec ses contraintes environnementales et personnelles Le référentiel décrit les huit blocs de compétences à acquérir à l'issue de la formation en filière maraîchage : https://formations-en-hauts-de-france.c2rp.fr/formation/2019647126, récupéré le 6 juin 2022 12 D'après le projet FOIN : Favoriser le développement des compétences des publics éloignés de l'emploi. L'APTEA : une innovation organisationnelle en matière de pédagogie qui s'appuie sur des situations de travail, et les entretiens réguliers avec l'animatrice pédagogue.


Juliette Stehlé

Alain Barrat

Ginestra Bianconi

Dynamical and bursty interactions in social networks

Keywords: numbers: 89.75.-k, 64.60.aq, 89.65.-s

We present a modeling framework for dynamical and bursty contact networks made of agents in social interaction. We consider agents' behavior at short time scales, in which the contact network is formed by disconnected cliques of different sizes. At each time a random agent can make a transition from being isolated to being part of a group, or viceversa. Different distributions of contact times and inter-contact times between individuals are obtained by considering transition probabilities with memory effects, i.e. the transition probabilities for each agent depend both on its state (isolated or interacting) and on the time elapsed since the last change of state. The model lends itself to analytical and numerical investigations. The modeling framework can be easily extended, and paves the way for systematic investigations of dynamical processes occurring on rapidly evolving dynamical networks, such as the propagation of an information, or spreading of diseases.

Recently, technological advances have made possible the measure of social interactions in groups of individuals, at several temporal and spatial scales and resolutions showing that human activity obeys scaling law and statistical features which reveal long time correlations and memory effects. Evidence comes from data on email exchanges [START_REF] Eckmann | Proc. Natl. Acad. Sci[END_REF][START_REF] Barabási | [END_REF][START_REF] Rybski | Proc. Natl. Acad. Sci[END_REF], mobile phone communications [START_REF] Onnela | Proc. Natl. Acad. Sci[END_REF][START_REF] González | [END_REF], spatial proximity [START_REF] Hui | Proceedings of the 2005 ACM SIGCOMM workshop on Delay-tolerant networking[END_REF][START_REF] Eagle | [END_REF][8][START_REF] Pentland | Honest Signals: how they shape our world[END_REF], web browsing [START_REF] Vázquez | [END_REF] and even face to face interaction [11][12][START_REF] Alani | 8th International Semantic Web Conference[END_REF]. In this respect the traditional framework of models used for risk assessment and communication, which describe human actvity as a series of Poisson distributed processes, need to be changed in favor of new models which take into account the occurrence of burstiness in many aspects of human activity.

Social interactions give rise to social [START_REF] Granovetter | [END_REF][START_REF] Wasserman | Social Network Analysis: Methods and applications[END_REF] and collaborative [START_REF] Newman | [END_REF] networks characterized by a complex evolution. In these networks, links are constantly created or terminated, and the social network of an individual evolves at different levels of organization. After the pioneering papers on complex networks showing that many social networks are small world and display heterogeneous degree distributions [START_REF] Dorogovtsev | Evolution of networks: From biological nets to the Internet and WWW[END_REF], and that these network topologies strongly influence the dynamics taking place on the networks [START_REF] Barrat | Dynamical processes on complex networks[END_REF], a number of papers have been devoted to modeling the dynamics of social interactions. Issues investigated in this context are in particular community formation [START_REF] Kumpula | [END_REF]20] and the evolution of adaptive dynamics of opinions and social ties [21][START_REF] Marsili | Proc. Natl. Acad. Scie[END_REF][START_REF] Holme | [END_REF][24].

The evidence coming from the analysis of social contact data calls for new frameworks that integrate these models with the bursty character of social interactions. The duration of contacts between individuals or groups of individuals display indeed broad distributions, as well as the time intervals between successive contacts [START_REF] Hui | Proceedings of the 2005 ACM SIGCOMM workshop on Delay-tolerant networking[END_REF]11,12,25]. Such heterogeneous behaviors have strong consequences on dynamical processes [START_REF] Onnela | Proc. Natl. Acad. Sci[END_REF]26], and should therefore be correctly taken into account. It is therefore necessary to introduce this fundamental aspect on human activity in models of social interactions, possibly reconstructing then social networks by aggregating the network of contacts over a certain period [26][27][28]. The modeling literature in this area being still in its infancy [25,[START_REF]Adaptive Networks: Theory, Models and Applications[END_REF][START_REF] Tang | [END_REF][START_REF] Gautreau | Proc. Natl. Acad. Sci[END_REF][START_REF] Hill | [END_REF], it is important to develop simple, generic, easily implementable models of dynamical networks which reproduce the empirical facts observed in contact duration and inter-contact intervals.

In this Letter, we take a step in this direction, focusing on short timescales such as the ones involved when people interact in social gatherings (e.g., scientific conferences). We define a simple agent-based model for rapidly evolving sparse dynamical networks, aimed at describing the dynamics of human social interactions in the context of small discussion groups. In particular we are interested in investigating basic mechanisms which could be responsible for various contact duration distributions. The model is kept simple, so that it can be easily simulated. It is accessible to analytical investigations in a certain number of cases. It can also be easily extended or modified. For instance, the population of agents is considered homogeneous (i.e. every agent is assumed to have the same dynamical parameters) and an extension to heterogeneous populations can easily be envisioned.

The dynamical network under study is formed by disconnected groups of agents which evolve by successive mergings and splittings. In particular at each time step an agent can either leave or remain in its group, or introduce an isolated agent to its group. The general formulation of the model allows to describe a variety of behaviors of the dynamical networks. In particular, the duration of contacts between individuals can display either narrow or broad distributions. A narrow distribution is for instance obtained by simply assuming that each agent leaves a group or invites a new agent in its group with a time-independent probability. On the contrary, broad distributions of contact durations, similar to those observed in empirical studies [6-8, 11, 25], are obtained through a reinforcement dynamics of the interaction, that can be summarized as "the longer an agent interacts with a group, the less it is likely to leave the group; the more the agent is isolated the less likely it is to interact with a group". This dynamics, reminiscent of the preferential attachment in the context of complex networks [START_REF] Barabási | [END_REF], could be argued to stem from Hebbian-like mechanisms at the underlying cognitive level. In general, for both narrow and broad distributions of interaction times, larger groups are found to be less stable than smaller ones. This is also observed in the data [11] and can be simply explained: the lifetime of a group depends on the decisions of all its members. In a first approximation these decisions correspond to independent events, therefore groups with more agents become less stable. Interestingly, our model also exhibits a dynamical transition towards the formation of large size group. This transition, supported by some measurements in animal behavior [34,35], is not observed in human behavior, and corresponds thus to parameter values where the model loses its applicability to the description of human social interactions. Note, nevertheless, that the formation of large social organizations and cities, demonstrate that in humans, large group formation occurs at a different level of organization.

The model we propose considers a fixed population of N agents, interacting in a limited space, as for example in a conference venue [8,[START_REF] Pentland | Honest Signals: how they shape our world[END_REF][11][12][START_REF] Alani | 8th International Semantic Web Conference[END_REF]. Therefore, in a first approximation we neglect the spatial dispersion of the agents and assume a well mixing dynamics. Each agent can either be isolated or belong to a group with other agents, and the groups define an instantaneous contact network. During the dynamics, agents can join other agents or on the contrary leave the group they belong to. More precisely, each agent i is characterized by two variables: the number p i of other agents with which it is in contact (i.e. its degree in the network) and the time t i at which p i last evolved. At each time step t, an agent i is chosen at random. If i is isolated (p i = 0), i changes its state with probability b 0 f (t, t i ). In this case, another isolated agent j is chosen with a certain probability Π(t, t j ), and i and j form a pair (p i → 1, p j → 1 and t i → t, t j → t). If on the other hand i is part of a group G of size greater than one (i.e. i has degree p > 0), a change of state occurs with probability b 1 f (t, t i ). When this occurs, agent i can either leave the group (probability λ) or introduce an isolated agent in the group (probability 1 -λ). If i leaves the group G and becomes isolated, p i → 0, and p j → p-1 ∀j ∈ G \ {i}, and as a consequence of this event the time of the modified nodes is reset to t, i.e. t ℓ → t ∀ℓ ∈ G. If i introduces to the group an isolated agent j, chosen again with probability Π(t, t j ), then p ℓ → p + 1 ∀ℓ ∈ G ∪ {j} and each agent ℓ in G ∪ {j}, changing state at time t, sets t ℓ → t. The parameters b 0 and b 1 determine the tendency of the agents, respectively isolated or in a group, to change their state, while λ controls the tendency either to leave groups or on the contrary to make them grow. The model's dynamical behavior depends also on the functions f and Π.

In order to make contact with empirical data, the main quantities of interest concern the time spent by agents in each state, the duration of contacts between two agents, and the time intervals between successive contacts of an agent. We can gain insight into these properties by writing rate equations for the evolution of the number N p (t, t 0 ) of agents which are at time t in state p since t 0 :

∂ t N 0 (t, t 0 ) = - N 0 (t, t 0 ) N b 0 [f (t, t 0 ) + Π(t, t 0 )(r(t) + (1 -λ)α(t))] + p≥1 π p,0 (t)δ t,t0 ∂ t N 1 (t, t 0 ) = -2 N 1 (t, t 0 ) N b 1 f (t, t 0 ) + (π 0,1 (t) + π 2,1 (t))δ t,t0 ∂ t N p (t, t 0 ) = -(p + 1) N p (t, t 0 ) N b 1 f (t, t 0 ) + (π p-1,p (t) + π p+1,p (t) + π 0,p (t)) δ t,t0 , p > 1
where π p,q (t) is the average number of agents going from state p to state q at time t, and

r(t) = t ′ N 0 (t, t ′ )f (t, t ′ ) t ′ N 0 (t, t ′ )Π(t, t ′ ) (1)
α(t) = p≥1,t ′ N p (t, t ′ )b 1 f (t, t ′ ) t ′ N 0 (t, t ′ )b 0 Π(t, t ′ ) (2)
where in the sums t ′ < t. These equations can be simplified and solved in certain cases, and the distribution P p (τ ) of (normalized) times τ = (t -t 0 )/N in which an agent remains in a given state p can then be deduced. Let us for definiteness assume that f and Π are stationary functions so they depend only on t -t 0 ; it is then natural to look for a stationary state, reached at large enough times, such that α, r, (π p,q ) are constants, and N p (t, t 0 ) = N p (t -t 0 ). If for instance f is a constant, it is easy to see that the {N p } p∈N decay exponentially with time, so that the {P p (τ )} p>0 are as well exponentially decaying functions. We consider the more interesting case of f and Π decaying with t -t 0 : the more an agent is in a state, the less probable it becomes to change state, as previously described in the self-reinforcement mechanism. For sake of simplicity, we focus on the case f = Π, so that r = 1, which allows to simplify the computations [36]. Computations can be carried out completely for instance in the case f (t -t 0 ) = Π(t -t 0 ) = (1 + τ ) -1 , where τ = (t -t 0 )/N . The choice of this scaling is consistent with the scaling of email communications and other The lines show Eq. ( 5).

human activity [START_REF] Barabási | [END_REF][START_REF] Vázquez | [END_REF]. In particular, N p (t) for p ≥ 1 is readily seen to decay as a power-law with exponent (p + 1)b 1 . More involved computations are needed to obtain the decay exponent of N 0 . Writing {π p,q } as functions of {N p } p∈N , f and Π, we can obtain recurrence relations for (π p,0 ) p>0 and deduce α = (2λ -1) -1 , so that

P 0 (τ ) ∝ (1 + τ ) -1+b0 3λ-1 2λ-1 (3) P p (τ ) ∝ (1 + τ ) -1+(p+1)b1 , p ≥ 1 . (4)
The previous analytical results are obtained under the conditions b 1 > 1/2, λ > 1/2, b 0 > (2λ -1)/(3λ -1), which determine the phase diagram of the model: outside these boundaries, the hypothesis of stationarity is violated.

It is also possible to compute the average state of the agents in the stationary state, as

p = π 1,0 2λ i≥1 i(i + 1) (i + 1)b 1 -1 1 -λ λ i-1 (5)
where

π 1,0 =   1 2 b 0 -2λ-1 3λ-1 + 1 2λ i≥2 i ib 1 -1 1 -λ λ i-2   -1
.

For λ → 0.5 + the average group size p diverges indicating that, in this limit, the non-stationary state is dominated by the formation of a large group of size O(N ).

We have performed numerical simulations of dynamical networks generated by the present model, with different f , Π, values of the parameters b 0 , b 1 , λ, and sizes N . We will here show the simulations corresponding to f = Π = (1 + τ ) -1 , in order to compare with the analytical predictions presented above. We first show in Fig. 1 the average agent state as a function of the different parameters, recovering the behavior predicted in Eq.( 5). The average state increases with b 0 , decreases with λ, and presents a non-monotonous behavior with b 1 . Figure 2 displays the distributions {P p } 0≤p≤4 of time spent in the various states. These distributions are power-laws, in perfect agreement with the analytics. We also note that, for f = Π = (1 + τ ) -ν with ν = 1, {P p (τ )} p∈N can be shown analytically to become either stretched exponentials (ν < 1) or power-laws (ν > 1), and we have also checked this behavior numerically. The broadness of the distributions is therefore not limited to the particular case described above, but is quite robust with respect to changes in the microscopic rules.

In Fig. 3, we also show the distribution of contact durations between two agents (which is different from P 1 : two agents remain in contact when they are joined by a third, but leave the state p = 1), of triangle durations, and of the time intervals between the starting times of two successive contacts [11]. This last quantity is highly relevant in the context of causal processes, as it gives the time scale on which an agent can propagate an information or a disease after receiving it. All these distributions are broad, similarly to empirical observations [11,25].

Let us finally mention that, when considering parameter values outside the validity of the stationary state analytical computations, different scenarios are observed, depending on λ: if λ > 0.5, the average state slowly decreases (towards 0 if b 0 < 0.5, and 1 if b 1 < 0.5) while, for λ < 0.5, a large cluster appears, with size is proportional to N , and lasting on a diverging timescale. Interestingly, even in this non-stationary case, the shape of the distributions P p (τ ) may remain stationary (not shown). This is particularly relevant as most empirical data are neces- sarily obtained in non-stationary environments.

In this Letter, we have proposed a modeling framework for dynamical networks in the context of interacting social agents. Both broad and narrow distributions can be obtained, corresponding to different social situations. The present framework can be developed in several research directions. First, many variations of the microscopic rules may be thought of and implemented, in order to model more precisely mechanisms of social contacts in various contexts or even of animal behavior. For instance, merging and splitting of groups could be introduced, as well as heterogeneity between agents to take into account different propensities to interact or to create groups. Moreover, it will be interesting to investigate how the properties of the interaction durations shape the resulting aggregated networks on various timescales. Finally, model dynamical contact networks can be used as a support for the simulation of dynamical processes taking place on dynamical networks, such as information spreading in a conference: the spreading process, although taking place on an extremely sparse network which is at any time formed of disconnected groups, may overall concern the whole population of agents, thanks to the dynamics of the agents who move from one group to another. The fact that the various network characteristics (such as the broadness of the distribution of contact durations and inter-contact times) can be controlled by changing the model's parameters will then make it possible to understand better the effect of these characteristics on the dynamical processes under scrutiny.

FIG. 1 :

 1 FIG. 1: Average state p in the stationary state for different sets of parameters and a population of N = 2000 agents. (a) b1 = λ = 0.7; (b) b0 = λ = 0.7; (c) b0 = b1 = 0.7. In (c) we observe the divergence of the average group size as λ → 0.5 + . The lines show Eq. (5).

FIG. 2 :

 2 FIG. 2: Distribution Pp(τ ) of times during which an agent does not change connectivity p. N = 10000, b0 = b1 = 0.7, λ = 0.8, and the simulation is run for T = 10 5 N time steps. The lines are the analytical predictions.

FIG. 3 :

 3 FIG. 3: Distributions of (a) duration of a contact between two agents; (b) time intervals between the beginnings of successive contacts of an agent A with two different agents B and C; (c) duration of a triangle. b0 = 0.7, λ = 0.8, b1 = 0.6 and 0.9. N = 1000, T = 10 5 N .

Acknowledgments This work has been partly supported by the FET Open project DYNANETS (number 233847) funded by the European Commission.