Francis Bernardeau

Constraints on higher-dimensional gravity from the cosmic shear three-point correlation function

Keywords: numbers: 98.80.Cq, 98.80.Es, 04.80.Cc, 04.50.+h

With the developments of large galaxy surveys or cosmic shear surveys it is now possible to map the dark matter distribution at truly cosmological scales. Detailed examinations of the statistical properties of the dark matter distribution reveal the detail of the large-scale structure growth of the Universe. In particular it is shown here that the behavior of the density field bi-spectrum is sensitive to departure from normal gravity in a way which depends only weakly on the background evolution. The cosmic shear bispectrum appears to be particularly sensitive to changes in the Poisson equation: we show that the current cosmic shear data can already be used to infer constraints on the scale of a possible higher-dimensional gravity, above 2 h -1 Mpc.

Almost all cosmological observations point toward a coherent model in which there is a non-vanishing dark energy [1]. The existence of such a component is very puzzling from a high energy physics point of view and signals physics beyond the standard model in way we do not yet apprehend. The resolution of such an issue may lie either in the introduction of a new form of cosmic fluid -a dark energy -with a negative pressure that would correspond to two third of the actual energy density of the universe or maybe in a modification of the Einstein gravity equations at cosmological scales. The two scenarios have obviously to be distinguished [2]! The first route is usually taken in models evoking quintessence component. However it cannot excluded that a non-zero cosmological constant observation actually signals a modification of the Einstein law at cosmological scales. This idea is actually fuelled by recent developments in string theory phenomenologies. Indeed the introduction of branes in the context of superstring theories [3,4], led to concepts of higher dimensional spacetimes. Usually it is assumed that the interaction gauge fields are localized on a 3-brane (i.e. a 3+1 dimension spacetime) whereas gravity propagates in all dimensions. In any of such string inspired models, one expects both the existence of Kaluza-Klein gravitons implying a non standard gravity on very small scales and light bosons, which can manifest as a new fundamental small scale force. It cannot be however excluded that gravity might be modified at large cosmological scale as well [5,6,12,13]. This possibility has been explicitly linked to the evidences of a cosmological constant [14,15]. It asks for precise test of gravity law at cosmological scales. This is the aim of this letter in which we pursue investigations on how cosmological observations, namely cosmic shear observations, could be used to test gravity on large distances.

Deviations form Newtonian or Einstein gravity have been looked for in the past mainly at rather small scales (from a cosmological point of view) whether it is in laboratory experiments [START_REF] Fischbach | Tamadge The search for non Newtonian Gravity[END_REF][START_REF] Long | [END_REF] or in stellar systems. That put severe constraints on the post-Newtonian parameters [START_REF] Ciufolini | Gravitation and inertia[END_REF][START_REF]For a recent review of experimental tests of GR see T. Damour[END_REF] but tells little on the validity of the Einstein equations at cosmological scales (say at scales above 1 h -1 Mpc). At best one can invoke studies of galaxy clusters. Comparisons between X-ray emissivity and gravitational lensing, which appears to be an indirect test of the Newton law through the equation of hydrostatic equilibrium, show no dramatic discrepancy below 2 Mpc [11] if normal gravity is assumed. On larger scales, there is no possible test on the Poisson equation (although other aspects of GR can be tested, see [16]) but by the mechanism of structure formation through the development of gravitational instabilities. This is the object of this letter.

Following [17] we simply assume that the gravitational force is changed at large-scale. Other possible modifications of the large-scale gravity have been considered in [18]. We assume in particular that the background metric is that of a Friedmann-Lemaître metric with an expansion factor a(z) having a redshift dependence assumed to be that of a Univers with a cosmological constant. Note that it does not mean the Friedmann equations is assumed to be correct, e.g. that there is a dark energy component, but that we take the evidences for a non-zero cosmological constant at face value. The latter being mainly of cosmographical nature, e.g. they come from the redshift dependence of the expansion parameter and quantities that are directly related to it such as angular or luminosity distances, they do not provide us with an actual weighing of the matter/energy content of the Universe.

As a further consequence, in what follows, as long as we are dealing with subhorizon scales, we can take the metric to be of the form,

ds 2 = -(1 -2Φ)dt 2 + a 2 (1 + 2Φ) dχ 2 + q 2 (χ)dΩ 2 , ( 1
)
where t is the cosmic time, a(t) the scale factor, χ the comoving radial coordinate, dΩ 2 the unit solid angle and q(χ) = (sin χ, χ, sinhχ) according to the curvature of the spatial sections. In a Newtonian theory of gravity, Φ is the Newtonian potential Φ N determined by the Poisson equation

∆Φ N = 4πGρa 2 δ ( 2
)
where G is the Newton constant and ∆ the three dimensional Laplacian in comoving coordinates, ρ the background energy density and δ ≡ δρ/ρ is the density contrast. If the Newton law is violated above a given scale r s then we have to change Eq. ( 2) and the force between two masses distant of r derives from Φ = Φ N f (r/r s ) where f (x) → 1 when x ≪ 1. This encompasses for instance the potential considered in [5,20] for which f (x) = 1/(1 + x) (in that case f ∝ 1/x and 5D gravity is recovered at large distance). Using (2) it leads, with r = ax, to

Φ(x) = -Gρa 2 d 3 x ′ δ(x ′ ) |x -x ′ | f |x -x ′ | x s , (3)
which, making use of ∆

[f (x)/x] = -4πδ (3) (x) + f s (x/x s ) with f s (x/x s ) ≡ (∂ 2 x f )/x, gives ∆Φ = ∆Φ N -Gρa 2 d 3 x ′ δ(x ′ + x)f s (x ′ /x s ). ( 4
)
The motion equations of the field are better handled in Fourier space. In Fourier space, Eq. ( 4) reads

-k 2 Φ(k) = 4πGρa 2 δ(k)f c (kr s ) (5)
where f c (k r s ) ≡ 1 -2π 2 f s (kr s ), f s (kr s ) being the Fourier transform of f s (r/r s ). It is unity for large values of k r s , it behaves like k r s for small. To explore the physics of large-scale structure growth the now standard method [START_REF] Peebles | The Large Scale Structure of the Universe[END_REF] is to expand the local density field with respect to the initial density field. Such an expansion can be made either in real space or in Fourier space,

δ(k, t) = δ (1) (k, t) + δ (2) (k, t) + . . . (6)
where δ (1) (k) is the linear density field, δ (2) (k) is quadratic in the linear density field, etc. The motion equation for the density field can be solved order by order and the resulting expansion can be used to infer the evolution of the stochastic properties of the local scalar field, whether it is the local density field or the gravitational potential. More quantitatively for any stochastic field X (statistically homogeneous and isotropic) we define its power spectrum P X by

X(k) X(k ′ ) ≡ P X (k)δ (3) (k + k ′ ) (7)
where δ (3) is the Dirac distribution, X the Fourier transform of X and the brackets refer to ensemble averages [START_REF] Peebles | The Large Scale Structure of the Universe[END_REF] over the stochastic process that gave birth to X. Note that the poisson equation then implies,

P ∆ΦN (k) = 4πGρa 2 2 P δ (k). ( 8
)
It is usually assumed that the initial metric fluctuation are Gaussian distributed. In the early stage of the dynamics it is therefore enough to consider the power spectrum for the scalar fields to fully characterize them. The subsequent evolution of the field however is bound to induce non-Gaussian effects. The second order δ (2) (k, t) in particular contains mode coupling terms that induce non-zero third moments (whether it is in real or in Fourier space). One way to quantify those effects is to introduce the field bispectrum, B X (k 1 , k 2 , k 3 ) defined as,

X(k 1 ) X(k 2 ) X(k 3 ) ≡ B X (k 1 , k 2 , k 3 )δ (3) (k 1 + k 2 + k 3 ). ( 9
)
In cosmology it is common to define the reduced bispectrum as,

Q X (k 1 , k 2 , k 3 ) = B X (k 1 , k 2 , k 3 ) P X (k 1 )P X (k 2 ) + sym. ( 10
)
the reason being that Q δ turns out to be roughly scale independent.

With modified gravity laws we can already note that,

P ∆Φ (k) = 4πGρa 2 2 P δ (k)f c (k r s ) 2 (11)
and

Q ∆Φ (k) = Q δ (k) f c (k r s ) (12)
for equilateral wave vector configurations. As a consequence, a simple way to test the validity of the Newton law is to compare power spectra of the density field and that of ∆Φ to see if they follow the same scale dependence. That was the point made in [17]. This is possible however only if one can measure δ and Φ independently. That supposes in particular that galaxy counts can give a reliable account of the mass distribution at large-scale, which as some level is always questionable due to the unavoidable presence of biasing effects[31] Note that constraints on modified gravity law can also be obtained from the shape of the power spectrum alone [26] but such constraints rely on assumptions on the shape of the spectrum of the primordial metric fluctuations.

The aim of this letter is to show that it is possible to test departure from standard gravity from cosmic shear survey alone taking advantage of the behavior of the reduced bispectrum.

As long as we have not entered too much into the nonlinear regime, the amplitude and shape of the power spectrum is determined by the growth rate of the linear term, δ (1) (k, t). In such model, and contrary to cosmological models with standard gravity, the growth rate is scale dependence. More precisely we have,

δ (1) (k, t) = D k (t)δ lin. (k) ( 13
)
where D k (t) is the linear growing rate of mode k that satisfies the equation [START_REF] Peebles | The Large Scale Structure of the Universe[END_REF],

Dk (t) + 2H Ḋk (t) = 3 2 H 2 Ω(t) f c (kr s /a(t)) D k (t). (14)
The solution of this equation have already been explored in [17]. It leads to a slowing of the growth rate for the scales beyond r c . For the 2nd order terms the structure of the conservation and motion equations leads to the following functional form (see also the approach adopted in [27]),

δ (2) (k) = d 3 k 1 d 3 k 2 δ Dirac (k 1 + k 2 ) (2π) 3/2 [F 0 (k 1 , k 2 )δ lin. (k 1 )δ lin. (k 2 )+ F 1 (k 1 , k 2 )δ lin. (k 1 )δ lin. (k 2 ) k 1 .k 2 k 1 k 2 + F 2 (k 1 , k 2 )δ lin. (k 1 )δ lin. (k 2 ) (k 1 .k 2 ) 2 k 2 1 k 2 2 - 1 3 ( 15
)
where the coefficients F 0 , F 1 and F 2 represent the amplitude of respectively the monopole, dipole and quadrupole terms that appear of the expression of the 2nd order local density field. For cosmological models with a standard gravity, F 0 , F 1 and F 2 are scale independent. Moreover they appear to behave like D 2 (t) with a fixed coefficient that depends extremely weakly on the cosmological parameters.

Lest us now explore the case of a modified gravity dynamics. The continuity equation imposes that δ (2) = 0 which implies that F 0 -F 1 + 2/3F 2 ≡ 0. The behavior of the 2nd order local density field will then be obtained from the evolution equation for these 2 functions,

F0 -2H Ḟ0 - 3 2 H 2 Ω(t) f c |k 1 + k 2 | r s a(t) F 0 = 4 3 Ḋk1 Ḋk2 + 3 4 H 2 f c k 1 r s a(t) + f c k 2 r s a(t) D k1 D k2 (16) (17)
and 4/3 Ḋk1 Ḋk2 in the r.h.s. of the previous equation is changed into 2 Ḋk1 Ḋk2 for the evolution equation of F 1 . These two equations have a very similar structure and can be solved numerically for whichever background we consider. To get the behavior of F 0 and F 1 in the following we assume we live in a universe in which, which corresponds to the expansion factor behavior for a universe for a given value of Ω 0 and no curvature (in the following we do the calculations with Ω 0 = 0.3). Note that we found that the results described in the following are very weakly dependent on the background behavior.

a(t) = a 0 sinh 3H 0 t 2 (1 -Ω 0 ) 1/2 (18)
Compared to the standard gravity case, F 0 , F 1 and F 2 now depend on the wave vectors through both their length and their relative angle. It introduces further dependence of the bispectrum on the triangle configuration. As an illustration we present on Fig. 1 the behavior of F 0 (k 1 , k 2 )/D k1 /D k2 as a function of time and for different angle between the two wavevectors (of the same length). The angular modulation is exhibited at late time (at early time this ratio is 17/21 irrespectively of the angle). It is found to be actually modest given the precision with which such a quantity can be measured. Similar results can be found for F 1 .

The resulting behavior of the bispectrum are show on Fig. 2. It shows the time evolution of the reduced bispectrum Q as defined in Eq. ( 10) for either the density field or the potential Laplacian compared to the standard gravity case. The density bispectrum can be measured for instance in galaxy catalogues. It has been done in particular in the PSCz galaxy survey [28]. It does not show however strong dependence on the modified gravity effects. This is not the case for the Laplacian field whose reduced skewness is amplified by modified gravity effects. That was indeed to be expected from the scaling relation shown in Eq. ( 12) which suggests that such a quantity is expected to grow at large scale.

It happens that quantities directly related to the reduced third moment of the potential Laplacian have been measured in cosmic shear catalogues [29]. The three-point correlation of the shear field (or rather some geometrical average of it) has been detected with a significant confidence level (through a rather elaborate procedure) at different angular scales [30]. If one applies the scaling law one expects for modified gravity to the predictions of standard gravity (in a Λ dominated universe) it is possible to put constraints on r s . From Fig. 3 it appears that current data already disfavor values of r s below 2h -1 Mpc although it is probably impossible to put a definitive statement from this data set. The constraints on the scale at which modified gravity intervenes are however strongly dependent on the modelling of the observations, and on the detailed shape of the 4D to 5D transition.

Cosmic shear surveys appear nonetheless very attractive to test the validity of the Einstein gravity law at cosmological scales. The fact such surveys provide us with genuine potential distribution at large scale makes them precious probes of the gravitational instability mechanisms in general. Undoubtedly, much stringent constraints should be obtainable from the coming cosmic shear surveys such as the CFHTLS or on the long term satellite missions like JDEM/SNAP.

Note that should such a modified gravity be the reason of the observed evidences of an accelerated universe r s would be expected to be the order of the Hubble size. Obtaining constraints on r s of this order form large-scale surveys may not be as hopeless at it seems. For the potential shape derived in [14] parameterized by a scale r c whose estimated value is r c = 1.2 c/H 0 [15], the amplification of the three-point correlation is not as small for the available scales as the simplified model used here suggests. At 10 h -1 Mpc scale the effect would be at percent level; at 100 h -1 Mpc scale, it would reach 7% (a valid theory of gravity has however to be done in those models). If this approach reveals the only way to distinguish between quintessence type and modified gravity models to account for the dark energy evidences, it calls for the realization of very large -possibly whole sky -cosmic shear surveys.

FIG. 1 :

 1 FIG.1: Time dependence of F0(k1, k2)/D k 1 /D k 2 for k1 = k2 and for different value of the relative angle between k1 and k2. This angle varies from 0 to π from top to bottom lines.

FIG. 2 :

 2 FIG.2: Shape of the bispectrum for the density field (long dashed lines) and the potential Lagrangian (solid lines). The computation has been made for k1 = rs/10 and k2 = 2rs/10 and assuming P (k) ∼ k -1 for the density field.

FIG. 3 :

 3 FIG. 3: The measured reduced shear skewness (square with error bars) compared to cosmological models with standard gravity (dotted line for Λ-CDM, upper dashed line for open CDM and lower dashed line for standard CDM) or with modified gravity for Λ CDM, rs = 10h -1 Mpc (lower solid line) and rs = 2h -1 Mpc (upper solid line).


Mario Alves

Santos Junior

Damien Voyer

Carlos Antonio

França Sartori

Djonny Weinzierl

Ronan Perrussel

Christian Vollaire

Laurent Krähenbühl

José Roberto Cardoso

Investigation of Electronic Stirring Chamber Phase-shifting Excitation and Load Effects

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers,

I. INTRODUCTION

Canonical chambers -that is Reverberation Chambers (RC) and TEM Chambers -are generally used for electromagnetic immunity testing despite their particular operational restrictions. RCs using mechanical paddles or frequency stirring provide a statistical E-field uniformity in all the directions inside the work volume [START_REF]Electromagnetic Compatibility (EMC), Part 4, Testing and Measurement Techniques -Section 21: Reverberation Chamber Test Methods[END_REF]- [START_REF] Klingler | On the use of 3D TEM Cells for Total Radiated Power Measurements[END_REF]. Nevertheless, the frequency operation of RCs is inversely proportional to the chamber dimensions and it is a constraint for low frequencies tests. The International standards recommend the RC configuration for immunity tests over 80 MHz frequencies [START_REF] Klingler | On the use of 3D TEM Cells for Total Radiated Power Measurements[END_REF]. TEM Chambers present for low frequencies a deterministic E-field uniform over a work area parallel to the septum, but not in all directions in the chamber volume [START_REF] Perini | Extending the frequency of mode stir chambers to low frequencies[END_REF]. Recently, a concept called Transmission Line Excitation Chamber (TLEC) has been proposed, based on a phase-shifting excitation of several transmission lines (TL). For a sake of illustration, a configuration constituted of three-conductor phaseshifting excitation has been investigated in [START_REF] Hill | Electronic mode stirring for reverberation chambers[END_REF].

In this work, we present semi-analytical and numerical approaches for evaluating the performance of a TLEC. Basically, a chamber excited by several TL presents several TEM modes inside the closed metallic cavity. The resulting standing waves depend on the position of the TL as well as the phase-shifting excitation and the loading. Those parameters are important in the search for a suitable chamber working volume; they can be modified electronically, resulting in a random standing wave profile. Based on this, a set of parameters combination can be chosen to improve and satisfy the pre-defined uniformity criteria within a wide frequency range, even at frequencies lower than 80 MHz.

II. SEMI-ANALYTIC AND NUMERICAL APPROACHES

A. Analytic expression of a single TL Considering the TL geometry given by Fig. 1, the E-field of a TEM mode inside the chamber can be evaluated by the following analytical expressions:

( ) ( ) ( ) ∑ ∞ + = ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ < ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + > ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 m 2 m 2 1 m 1 0 X TEM 0 y l y a m sh 0 y l y a m sh x a m cos a 2 y , x E π α π α π η (1) ( ) ( ) ( ) ∑ ∞ + = ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ < ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + > ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 m 2 m 2 1 m 1 0 Y TEM 0 y l y a m ch 0 y l y a m ch x a m sin a 2 y , x E π α π α π η (2) with 2 1 1 1 2 m m th l a J m m ch l a m m th l th l a a π π α π π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = ⎛ ⎞ ⎛ ⎞ + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ; 1 2 2 1 2 m m th l a J m m ch l a m m th l th l a a π π α π π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = - ⎛ ⎞ ⎛ ⎞ + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (3)
where η 0 is the vacuum wave impedance and J m the harmonic coefficients related to the current density on the central conductor. Assuming the conductor as an infinitely thin wire along z axis, one finds:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 0 0 m x a m sin a 2 I J π (4)
The dependence with z direction is decoupled and equal to e -j k 0 z , when an incident wave is considered.

Investigation of Electronic Stirring Chamber Phase-shifting Excitation and Load Effects M.A. Santos Jr. 1,4 , D. Voyer 2 , C. A. F. Sartori 1 , D. Weinzierl 3 , R. Perrussel 2 , C. Vollaire 2 , L Krähenbühl 2 , J.R. Cardoso 1

1 Laboratório de Eletromagnetismo Aplicado LMAG/PEA/EPUSP. 05508-900. São Paulo-SP, Brazil 2 Ampère (CNRS UMR5005), Université de Lyon, Ecole Centrale de Lyon, 69134, Ecully Cedex -France 3 Centro Universitário de Jaraguá do Sul -Rua dos Imigrantes 500, 89254-430, -Jaraguá do Sul/SC -Brazil 4 Centro Tecnológico da Marinha em São Paulo -05508-000 -São Paulo/SP -Brazil mario_alves@pea.usp.br, Damien.Voyer@ec-lyon.fr, sartori@pea.usp.br, Laurent.Krahenbuhl@ec-lyon.fr

B. Phase Shifting and Load Effects Modeling

Several TL can be treated by superposition. When the TLs are ended with the same load, the E-field can be written using the separation of variables in the transverse plane and in the longitudinal direction. The phase shifting between the TLs has then an effect on the repartition of the fields in the transverse plane:

( ) ( ) i j i TEM i TOT e y , x E y , x E ϕ ∑ = (5)
where E i TEM is the E-field due to the i th TL and ϕ i represents the applied excitation phase.

The load introduces a reflection coefficient Γ that affects the longitudinal repartition of the field:

( ) ( ) ( ) z jk z jk TOT TOT 0 0 e e y , x E z , y , x E + - + × = Γ (6)
If we consider when |Γ | = 1, a stationary wave is expected but the maximum of E-field can be moved changing the phase of Γ. Thus, it is possible to homogenize the field in z direction even if the TL is unmatched.

C. Numerical Approach

Numerical evaluation was performed using FIT [5]. The loads are imposed by boundary conditions while the phase-shifting excitation is implemented by a postprocessing approach.

D. Indexes of Merit calculation

Standard deviation for the distribution of E-field in the chamber can be calculated from the definitions in [START_REF]Electromagnetic Compatibility (EMC), Part 4, Testing and Measurement Techniques -Section 21: Reverberation Chamber Test Methods[END_REF] [START_REF] Hill | Electronic mode stirring for reverberation chambers[END_REF] and [START_REF] Weinzierl | Numerical evaluation of non-canonical reverberation chamber configurations[END_REF]. Due to the phase and load shifting, the E-field considered at any point of the chamber is the average field.

III. APPLICATION AND RESULTS

A TLEC with dimensions of 0.6m×0.6m×1.2m has been considered. The semi-analytical approach has been applied in the transverse plane {x, y} since the variation of E-field in z direction can be canceled using a suitable load-shifting. The area under evaluation is a rectangular 0.3m×0.3m centered at the middle of the chamber. The study concerns the influence of the phase-shifting when several TL are considered. Results are reported in Table I. It appears that the number of TL is important: the standard deviation decreases of 3 dB when 4 TL are introduced instead of 2 TL. Moreover, the phase-shifting improves of 0.5 dB the performance. A TEM stripline like the one used in TEM Chambers has also been simulated and the E-field uniformity is presented. 3D simulations using commercial software have also been carried out. The interest is that this approach is more realistic since it takes into account the connection of the TLs outside the chamber, which introduces a discontinuity. Table II presents the statistical E-field indexes calculated when the TLs are terminated by a 50 Ω load. The volume is defined with the same cross section as previously, and a 0.6m length in the z direction. The influence of the phase-shifting on E-Field profile is also given at Fig. 2 in a 2 TL configuration. The effect of the load is presented in Fig. 3: the maximum of E-field moves between a -j500Ω and j500 Ω load.

Fig. 1 :

 1 Fig. 1: geometry of a TL in the transverse plane

  a) Position of two TL in the chamber (b) E-field distribution in the horizontal plane at the middle of the chamber (load 50 Ω).

Fig. 3 E

 3 -field distribution with different loads (a) -j500Ω (b) j500 Ω.

ACKNOWLEDGMENT

This work was partially supported by Capes-Cofecub (07/0568), and FAPESP (2007/51192-6).


Mario Alves

Santos Junior

Sergio Luciano Avila

Carlos Antonio

França Sartori

Djonny Weinzierl

Laurent Krähenbühl

Luiz Lebensztajn

José Roberto Cardoso

Carlos Antonio França

Conductor Positions Optimization of a Transmission Line Excitation Chamber

Keywords:

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

In previous works the Transmission Line Excitation Chamber (TLEC) performance evaluations have been carried out using different approaches. These were presented in [START_REF] Weinzierl | Numerical evaluation of non-canonical reverberation chamber configurations[END_REF] and [START_REF] Santos | Maximum working volume evaluation in a non-canonical reverberation chamber[END_REF] in which FIT and TLM, and also an optimization tool were used to evaluate the field uniformity within the working volume on a predefined TLEC configuration. The TLEC configuration here presented is based on three-conductor phase-shifting excitation configuration and proposed as an alternative when low frequencies are taken into consideration [START_REF] Perini | Extending the frequency of mode stir chambers to low frequencies[END_REF]. Fig. 1 shows a sketch of a three-wire reverberation chamber configuration. Thus, the optimization problem aims to maximize the working volume and the E-field, which is restricted to uniformity constraints.

In this work, the main focus is to find the better transmission lines configurations, i.e., wire positions, which present the best TLEC performance related to the aforementioned indexes. A good TLEC performance is associated to a high working volume and E-fields average values. To obtain the wire configuration and the corresponding TLEC performance, a numerical approach has been implemented by applying multi-objective optimization. Deterministic and statistical approaches are applied in the analysis.

II. OPTIMIZATION PROCEDURE

A working volume is assumed to be a region inside the reverberation chamber (Fig. 1) were the homogeneity of the electric field attains a prescribed level. EMC tests could be performed only with this minimum homogeneity level, and the mean value within the working volume should be as high as possible considering low input power.

In this work, the main interest is to understand how the reverberation chamber works when associated to a set of transmission lines positions and its related working volumes. It should be mentioned that when the working volume increases the mean electric field decreases. This means that a multi-objective optimization algorithm should be employed (two or more antagonistic goals).

Here, the optimization problem has twelve optimization variables: three pairs coordinates (TL positions), two equatorial radii, polar radius and three parameters associated to the ellipsoid translation (shift) from the chamber's centre (these six last variables define the working volume). In this set, inside some working volume the electric field is high, whereas the volume is low.

There are other working volumes, which belong to this set, but have low electric field and high volume. Inside the working volumes the field homogeneity is constrained by pre-defined values. When the chamber characterization is performed, the engineer could make a choice to obtain the most suitable set of TL to design the TLEC.

In order to solve it, the optimization problem is assumed as an unconstrained optimization problem, where inequality constraint was treated as a penalty function.

A. Modeling the chamber

The mean E-field ( x E ), the Standard Deviation (σ x ) and Standard Deviation in dB ( ˆx σ ) are calculated by ( 1), ( 2) and ( 3)

1 1 N x xk k E E N = = ∑ (1) ( ) 2 1 1 1 N x x xk k E E N σ = = - -∑ (2) 10 ˆ20log x x x x E E σ σ   + =     (3)
Conductor Positions Optimization of a Transmission Line Excitation Chamber M. Santos Jr. 1,4 , S. Avila1 , C. A. F. Sartori 1 , D. Weinzierl3,1 , L. Krähenbühl2 , L. Lebensztajn 1 , J. R. Cardoso 1

Similar equations could be obtained to the y-axis, z-axis and to the combined E-field [START_REF] Weinzierl | Numerical evaluation of non-canonical reverberation chamber configurations[END_REF], [START_REF] Santos | Maximum working volume evaluation in a non-canonical reverberation chamber[END_REF] and it is assumed a value equal to 4 dB as the standard deviation for defining the working volume at frequencies lower than 80 MHz.

B. The Optimization Problem

The conductor position problem could be written as an optimization problem as follows:

   z y x E , , Max Volume Working Max S. t: Max ( ) dB 4 z y x z y x ≤ , , , , , ˆσ σ σ σ (4)
In order to bounder the optimization parameters, Fig 2 illustrates and shows a cross-section of the chamber to be analyzed. First of all, it is assumed that each wire is parallel to one of the three axis of the chamber as Fig 1 shows. So, each wire could only be placed in the grey region, because it could not be close to the wall, due to the standards. On the other hand, if we put it on the center of the chamber (the forbidden area) the associated working volume will be small. Thus the TLs can only be moved inside a space "close" to the walls in order to avoid solutions that are impossible for practical utilization.

Forbidden Area

Wire Fig. 2 Chamber Cross Section: the position wire Multi-objective optimization (4) seeks to optimize the components of a vector-valued cost function. Unlike single objective optimization, the solution to this problem is not a single point, but a family of efficient points called Paretofront. Each point on this surface is optimal in the sense that no improvement can be achieved in a cost vector component that does not lead to degradation in at least one of the remaining components. Each element in the efficient set constitutes a non-dominated (non-inferior or non-superior) solution to the multi-objective problem. With this set of solutions it is possible to understand the dependence between each objective.

The Pareto-set of this multi-objective problem was obtained by using the Multi-objective Genetic Algorithm (MGA) [START_REF] Avila | Sensitivity Analysis in the Parameters Space Applied to Decision Making in Multi-objective Evolutionary Optimization[END_REF]. The MGA is derived from the Genetic Algorithm, which is a stochastic procedure based on the concepts of natural selection in genetics.

C. Coupling the Field Computation Software and the Optimization Environment

The adopted procedure has six fundamental steps: i) Random positions for TL are generated, as well the working volumes and ellipsoid radius;

ii) TL positions are charged to the field computation software, that uses FIT [START_REF] Cst | Advanced Topics[END_REF]; iii) The field computation inside the reverberation chamber is performed with FIT [START_REF] Cst | Advanced Topics[END_REF]; iv) The values of the E-field are exported considering a regular grid; v) The working volume is calculated. For the determination of a particular working volume, Eq. ( 1) should be solved, and vi) The indexes of merit are applied to decide which one are the better TL configurations. The third and fourth steps are computed by applying the FIT [START_REF] Cst | Advanced Topics[END_REF] At the end of step iv a file is exported to our optimization environment.

The EMC standards adopt a parallelepiped as a working volume [START_REF]Electromagnetic Compatibility (EMC), Part 4, Testing and Measurement Techniques -Section 21: Reverberation Chamber Test Methods[END_REF]. In this work we have adopted another solid to characterize the volume: an ellipsoid [START_REF] Santos | Maximum working volume evaluation in a non-canonical reverberation chamber[END_REF].

III. RESULTS

Fig. 2 shows some results which allow us to know the compromise among the searched objectives (max E-field and max working volume), the constraints and the conductor positions. By inspection of Fig. 2 the maximal working volume, taking into account a defined E-field level, can be chosen. For instance, it yields approximately 15e10 -3 m 3 for an Efield level of 15V/m considering the total chamber volume equal to 0,432 m 3 .

Fig. 1 .

 1 Fig. 1. Sketch of the three-wire reverberation chamber configuration

Fig. 2 .

 2 Fig. 2. Compromise between working volume and E-field.

Laboratório de Eletromagnetismo Aplicado LMAG/PEA/EPUSP. 05508-900 -São Paulo/SP -Brazil

Ampère (CNRS UMR5005), Université de Lyon, Ecole Centrale de Lyon, 69134, Ecully Cedex -France

Centro Universitário Jaraguá do Sul -Rua dos Imigrantes 500, 89254-430 -Jaraguá do Sul/SC -Brazil

Centro Tecnológico da Marinha em São Paulo -05508-000 -São Paulo/SP -Brazil mario_alves@pea.usp.br, leb@pea.usp.br


G Di Molfetta

A Pérez

Quantum Walks as simulators of neutrino oscillations in vacuum and matter

We analyze the simulation of Dirac neutrino oscillations using quantum walks, both in vacuum and in matter. We show that this simulation, in the continuum limit, reproduces a set of coupled Dirac equations that describe neutrino flavor oscillations, and we make use of this to establish a connection with neutrino phenomenology, thus allowing one to fix the parameters of the simulation for a given neutrino experiment. We also analyze how matter effects for neutrino propagation can be simulated in the quantum walk. In this way, important features, such as the MSW effect, can be incorporated. Thus, the simulation of neutrino oscillations with the help of quantum walks might be useful to illustrate these effects in extreme conditions, such as the solar interior or supernovae.

I. INTRODUCTION

Quantum simulation is important for the ability to explore the behavior of a quantum system under conditions that are experimentally difficult to access. A clear example is the simulation of the Dirac equation, which allows one for the visualization of effects like the zitterbewegung or the Klein paradox [1][2][3], which can in fact be easily simulated on a classical computer, but are hard to verify on the laboratory. Related to this application of quantum simulations, we will consider in this paper another problem, that concerns the simulation of neutrino oscillations using the quantum walk (QW). Neutrino oscillations were proposed by B. Pontecorvo in 1957 [4], under the form of neutrino-antineutrino oscillations, in analogy to Kaon oscillations, and afterwards as "flavor transitions" by Maki, Nakagawa and Sakata in 1962 [5]. This kind of oscillations appear because the three types of flavor states for neutrinos (ν e , ν µ and ν τ ) that have a definite interaction in the Standard Model of particle interactions (SM) do not coincide with the mass eigenstates of the Hamiltonian described by the same model. Instead, there is a unitary transformation that relates both sets of states. As a consequence, a neutrino which is produced e.g. as an electron neutrino ν e can be detected, with a given probability, as any of the three flavors at a later time (see, for example [6,7]). Neutrino oscillations have given rise to very rich phenomena, ranging from the solution to the Solar neutrino problem, supernovae [START_REF] Raffelt | Stars as laboratories for fundamental physics: The astrophysics of neutrinos, axions, and other weakly interacting particles[END_REF], reactor neutrinos [START_REF] Kim | [END_REF], the Early Universe [START_REF] Lesgourgues | Neutrino Cosmology[END_REF], or atmospheric neutrinos [START_REF] Kajita | [END_REF].

QWs are quantum cellular automata in the one particle sector that can be viewed as formal generalizations of classical random walks. They have been first considered by Feynman [START_REF] Feynman | International Series in Pure and Applied Physics[END_REF] as a possible discretization of the free Dirac dynamics in flat space-time, and later introduced in the physics literature in a systematic way by Meyer [START_REF] Meyer | [END_REF], following the seminal work of Aharonov [14]. A continuous-time version first appeared in [15]. QWs have been realized experimentally with a wide range of * giuseppe.dimolfetta@lif.univ-mrs.fr physical objects and setups [16][17][18][19][20][21][22], and are studied in a large variety of contexts, ranging from fundamental quantum physics [22,[START_REF] Giulini | Decoherence and the appearance of a Classical World in Quantum Theory[END_REF] to quantum algorithmics [START_REF] Ambainis | [END_REF][START_REF] Magniez | SIAM Journal on Computing -Proceedings of the thirty-ninth annual ACM symposium on Theory of computing[END_REF], solid-state physics [START_REF] Aslangul | [END_REF][27][28][29] and biophysics [30,31]. Following the Feynman's idea, several authors have studied the connection between quantum automata and quantum field theory [32,33]. In particular, it is well known that the continuous limit of various QWs formally coincides with the Dirac equation [34][35][36]. It has been shown recently that several QWs can model the dynamics of free Dirac fermions coupled to electromagnetic [37,38] and relativistic gravitational fields [3,[39][40][41].

Here, we will make use of the capabilities of the QW as discretizations of quantum field theories, to establish a connection between QWs and the phenomenology of neutrino oscillations in different scenarios. Given the known bounds to the neutrino masses, we will concentrate in relativistic neutrinos, an approximation that holds in the vast majority of experimental situations. Neutrino oscillations have been demonstrated by an optical analogical experiment based on the two-state system of polarized photons traveling through a birefringent crystal [START_REF] Weinheimer | Astro, Particle and Nuclear PhysicsInternational Workshop on Nuclear Physics[END_REF]. In [START_REF] Noh | [END_REF] the authors discuss the idea of simulating the coupled Dirac equations describing the neutrino propagation with trapped ions. A different proposal [44] consists in using waveguide arrays. The QW, however, offers the possibility to perform the simulation by using a suitable modification of some of the experimental setups already available, or proposed, for the QW dynamics. Experimental procedures allowing the implementation of DTQWs with wave functions having more than two components have been proposed in [45,46]. In these procedures, the DTQWs are implemented with single photons or classical light, for example in optical cavities or with atoms in optical lattices.

A previous work has already motivated such study [47]. Here, we extend this work in two crucial directions. First, we analyze the continuous version (in space and time) of the proposed model. In this way, we can obtain the resulting equations that describe the well-known Hamiltonian dynamics of (relativistic) neutrinos. This result, not only is a verification of the model as describing neutrino oscillations, but also allows one to establish a clear correspondence between the parameters of the simulation and the neutrino data for a given experiment. Secondly, we show how matter effects can be incorporated into the simulation. As it is well known, matter effects are crucial for the explanation of the Solar neutrino problem, specially for the higher energy 8 B neutrinos [7], and for long baseline experiments [48,49]. A quantum simulator can then be used as an extra tool to visualize these effects from the point of view of a discretized field theory.

This paper is organized as follows. In Sec. II we give a short introduction to the phenomenon of neutrino oscillations, and illustrate it for the case of two flavors. In Sec. III we define the dynamics of the QW model, and we show that, in the continuous space-time limit, we recover the dynamics of a set of coupled Dirac equations that can be put in correspondence with the Hamiltonian formulation of neutrino oscillations, thus allowing one to identify the value of the parameters needed to simulate a given neutrino experiment. We make use of this correspondence, in Sec. IV, to give an example of the simulation of three flavors in vacuum. In Sec. V, we show how matter effects in the neutrino propagation can be incorporated in the QW model. Our main conclusions are summarized in Sec. VI. Along this paper we use natural units, defined by = c = 1.

II. NEUTRINO OSCILLATIONS

Neutrinos are produced as "flavor states" via charged or neutral currents in nuclear reactors, stars, cosmic rays and many other scenarios, in the way described by the SM. These flavor states Ψα (α = e, µ, τ ) are related, at a given space-point (t, x), to the mass eigenstates by an unitary transformation R, such that

Ψα (t, x) = i R αi Ψ i (t, x), (1)
where Ψ i (t, x) is the neutrino field with mass m i . The field can, in principle, take either a Dirac or a Majorana form although we will consider, for definiteness, a Dirac field: The precise nature of the neutrino field is still undetermined and, although important for the fundamental knowledge of neutrinos, is irrelevant in the case of relativistic neutrinos considered here. Eq. ( 1) actually refers to the left chiral components of the neutrinos, since only these components interact within the SM. We will restrict ourselves to propagation along one spatial dimension, then x refers to the coordinate along that dimension.

We will concentrate on neutrino eigenstates, with momentum k and energy E i = k 2 + m 2 i of the Hamiltonian H of the theory:

H |ν i = E i |ν i . (2)
Suppose that, at time t = 0, a flavor neutrino |ν α is produced, then at time t the neutrino state evolves according to

|ν α t = e -iHt 3 i=1 R * αi |ν i = i |ν i e -iEit R * αi . (3)
Therefore, at time t, the initial neutrino can be detected as any flavor ν β . The probability of the ν α → ν β transition after a time t is then given by the following expression

P (ν α → ν β ; t) = | 3 i=1 R βi e -iEi t R * αi | 2 . ( 4
)
To illustrate the latter formula, let us analyze the case with only two neutrino states (some interesting cases, such as solar and atmospheric neutrinos, can be effectively described in this way). The matrix R can be taken as

R = cos φ 12 sin φ 12 -sin φ 12 cos φ 12 , (5)
φ 12 being the mixing angle. By approximating

E i ≃ | k| + m 2 i 2| k|
, Eq. ( 4) gives, for β = α:

P (ν α → ν β ; t) = sin 2 2φ 12 sin 2 ϕ(t), (6)
where

ϕ(t) ≡ ∆m 2 12 t 2| k| , (7)
with the notation ∆m 2 12 = m 2 2m 2 1 . Notice that, in the latter equation, one can replace | k| by E within the same order of approximation as above, where E = E 1 ≃ E 2 . Also, it is customary to use the distance L ≃ t traveled by the neutrinos from the production to the detection point. Then, the argument of the oscillatory function in (6) becomes

ϕ(L) ≃ 5.08 ∆m 2 12 L 2E , (8)
when ∆m 2 12 is expressed in eV 2 , L in Km and E in GeV units.

III. THE MODEL

Consider a QW defined over discrete time and discrete one dimensional space, labeled respectively by j ∈ N and p ∈ Z. This QW is driven by an homogeneous coin acting on the 2n-dimensional tensor-product Hilbert space H = H spin ⊗ H n , where H spin is 2-dimensional and H n describes the n-flavor Hilbert space of the walker. The evolution equations read

  ψ 1 j+1,p ... ψ n j+1,p   =   h=1,n SQ h ǫ     ψ 1 j,p ... ψ n j,p   , (9)
where Q h ǫ ∈ SU (2), h = 1...n, and

Q h ǫ = cos(ǫθ h ) i sin(ǫθ h ) i sin(ǫθ h ) cos(ǫθ h ), (10)
is the quantum coin acting on each flavor state of the walker depending on dimensionless real parameters (ǫ, θ h ). The operator S is the usual spin-dependent translation acting on each flavor component, ψ h j,p = {ψ ↑h j,p , ψ ↓h j,p } ∈ H spin and defined as follows:

Sψ h j,p = ψ ↑h j,p+1 , ψ ↓h j,p-1 ⊤ . ( 11
)
We note Ψ = ( h=1,n ψ h ) ⊤ ∈ H n . Equations ( 9) describe the evolution of n independent two-level systems, and it has been shown that each of them recover, in the continuous limit, the Dirac equation [3,37,39], where the parameter θ h corresponds to the mass of the fermion. Let us now consider an additional unitary operator R and its inverse R -1 = R † acting on

H n ⊗ H 2 .
The specific role of this operator is to mix the flavor degrees of freedom, as in Eq. ( 1). The evolution equation becomes:

Ψj+1,p = R   h=1,n SQ h ǫ   R † Ψj,p (12)
where we call Ψj,p = RΨ j,p the flavor eigenstates. In the next section we will prove that this simple model recovers, in the continuous limit, the oscillatory dynamical behavior of Dirac neutrinos in vacuum.

A. Generalized Dirac equation for neutrinos

In order to compute the continuous limit of equations ( 9), we first consider that Ψj,p are the 'values' Ψ(t j , x p ) taken by a two-component wave-function Ψ at space-time point (t j = jǫ, x p = pǫ). We assume that Ψ is at least twice differentiable with respect to both space and time variables for all sufficiently small values of ǫ. Assuming the existence of the continuous limit imposes the following constraint on the coin:

lim ǫ→0   R   h=1,n SQ h ǫ   R †   = I 2n . ( 13
)
The above equation ( 13) is directly verified because Q i = I 2 as ǫ → 0 and RR † = I 2n by definition. When we Taylor-expand (12) at first order in ǫ, the zeroorder terms cancel each other, since ( 13) is satisfied, and the first-order terms provide the following system of partial differential equations (PDEs):

  ∂ t -   h=1,n σ z   ∂ x -iM   Ψ(t, x) = 0, (14)
where M r,s is the mass tensor, with indices r, s = {1, ..., n}:

M r,s = c=1,...,n R r,c Qc R -1 r,c Qc = θc σ x . (15)
Equation ( 14) are standard Dirac equations for n relativistic flavor neutrinos in (1+1) continuous space-time. Notice that, because Ψ(t, x) is a solution of the Dirac equation, it is also a solution of the Klein-Gordon (KG) equation. As usual, we can expand the neutrino field in plane waves of the form Ψ(t, x) = Ψk (t)e ikx , so that equation ( 14) transcribes in (∂

2 t + k 2 + M † M) Ψk (t) = 0, where M † M = diag{ θ2 1 , ..., θ2 n }.
To describe relativistic neutrinos, however, the condition k >> θi has to be fulfilled. In this limit, one can linearize the previous KG equation as follows:

i∂ t Ψk (t) -Ω k Ψk (t) = 0, Ω k = (k + R † |M| 2 2k R), ( 16
)
where we have considered only the positive-energy probability amplitudes (neutrinos), and defined |M| = √ M † M. For negative-energy states (antineutrinos) a global minus sign appears in equation (16). Following a similar procedure as in Sec. II one can obtain, starting from Eq. ( 16), the transition probability, which can be written as

P (ν α → ν β ; t) = | k ψα * k (0) ψβ k (t)| 2 . ( 17
)
For example, in the case of two neutrino flavors, one would arrive to Eq. ( 6), but with the phase ϕ(t) given by

ϕ(t) = θ2 2 -θ2 1 2k t. ( 18
)

IV. SIMULATION OF NEUTRINO OSCILLATIONS IN VACUUM

The above equations were derived for any number of neutrino generations. Writing them explicitly in terms of mixing angles is extremely cumbersome in general. Let us consider the three neutrino generations {e, µ, τ } in vacuum, respectively the electron, the muon and the tau neutrino. The transformation R recovers the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix and depends on three real parameters [50]: R = e iφµτ λ7 e iφeτ λ5 e iφeµλ2 (19) where the λ are the Gell-Mann matrices that correspond to the spin-one matrices of SO (3). Each angle φ ij corresponds to the mixing between two neutrino species. Let us remark that when R is the identity, the solutions of Eq. ( 16) are propagating plane waves ∝ e -i(Et-kx) , where

k 2 = E 2 -θ2 h , h ∈ {e, µ, τ }.
The comparison between Eqs. ( 7) and ( 18) allows us to establish a criterion to simulate a given neutrino oscillation experiment, by requiring that the accumulated phase ϕ(t) takes the same value in both cases, where t represents the number of timesteps in the QW simulation. For example, one can choose the values θ2 h as given the neutrino square masses m 2 i , expressed in eV 2 , and k given by the energy in GeV. This will, in turn, fix the number of required timesteps, for a given distance L. A second condition, of course, is that the mixing angles φ ij correspond to the observed values (see [51] for recent results).

In Fig. 1 we can observe the oscillatory behavior of three flavor neutrinos starting from a pure electronneutrino initial state:

ψi * k (0) = 1 √ n n-1 p=0 e -i(k-k0)xp ⊗ (1, 0, 0, 0, 0, 0) ⊤ . ( 20
)
One can observe that these plots reproduce the ones corresponding to actual calculations of neutrino oscillations (see for example [START_REF]Invisibles: Neutrino, dark matter and dark energy physics[END_REF]).

V. NEUTRINO OSCILLATIONS IN MATTER [START_REF] Wolfenstein | [END_REF] was first to recognize that the medium induces a modification of the neutrino dispersion relation [START_REF] Wolfenstein | [END_REF], and then Mikheyev and Smirnov showed that oscillations can be resonant when neutrinos pass through a density gradient, so that the flavor branches of the dispersion relations have an avoided crossing [54]. This Mikheyev-Smirnov-Wolfenstein (MSW) effect is very important in astrophysics because neutrinos are naturally produced in the interior of stars and stream through a density gradient into empty space. In order to model neutrino oscillations in a medium, let us consider the case with only two flavors, which we freely denote as ν e and ν µ . Notice that, as already mentioned, many realistic experiments can be effectively reduced to a two-neutrino oscillation. In that case we consider one mixing angle and a 2-dimensional matrix R:

R = cos φ sin φ -sin φ cos φ ⊗ I 2 (21)
Matter interaction is modeled by introducing a position-dependent phase:

Ψ j+1,p = V p R   h=1,2 SQ h ǫ   R † Ψ j,p , (22)
where

V p = diag(e iǫρp , 1) ⊗ I 2 (23)
By the same procedure presented in section A we can derive the following Dirac equations: (20).

i∂ t Ψ(t, x) -H m Ψ(t, x) = 0, (24)
H m = i (σ z ⊗ I) ∂ x -M + V(x) (25)
where the potential V reads as follows:

V(x) = γ 5 I 4 ρ(x), (26)
with I 4 the 4-dimensional identity matrix, and γ 5 = 1 2 (1+ σ z ). The transition probability from an electron neutrino to the muon neutrino is given by:

P (ν e → ν µ ; t) = | h U eh e -iω h t U * µh | 2 (27)
where U = U (x) is position-dependent and reads:

U (x) = cos Φ(x) sin Φ(x) -sin Φ(x) cos Φ(x) , (28)
and the mixing angle Φ(x) is related to the mixing angle in vacuum φ via the well-known relation:

sin 2 2Φ(x) = sin 2 2φ A(x) , (29)
with

A(x) = cos(2φ) -2Eρ(x) ∆ 2 m
2 + sin 2 2φ the resonance factor. Notice that, if matter density is not constant, it is necessary to take into account the effect of ∂ x U (x) in the evolution equation in mass eigenstates. In fact:

U H m U -1 = iU (σ z ⊗ I) ∂ x U -1 + diag. terms (30)
As mentioned above, the eigenvalues E 1m and E 2m of the Hamiltonian (25) present an avoided crossing (see Fig. 2) close to tthe resonance. The effect of the offdiagonal terms proportional to ∂ x Φ(x) in the evolution is to generate transitions between the energy states close to the resonance. However, if the derivative is much smaller with respect to the diagonal terms such transitions are negligible. To quantify the strength of the off-diagonal terms it is useful to introduce the adiabaticity parameter:

γ = ∆m 2 eµ 4k|∂ x Φ(x)| . (31)
If γ ≫ 1 for all x the evolution is adiabatic, otherwise the probability of flavor transition reads:

P (ν e → ν µ ; t f ) = - 1 2 + ( 1 2 -P c ) cos 2Φ(t f ) cos 2Φ(t i ) (32
) where t f and t i are the final and initial times (which can be approximated by the final and initial position x f and x i , for relativistic neutrinos), P c is the crossing probability for x f → ∞:

P c = exp(-π 2 γ r ) -exp(-π 2 γr sin φ 2 ) 1 -exp(-π 2 γr sin φ 2 ) , (33)
and γ r is the adiabaticity parameter γ at the resonance [55]. In Fig (3) we have shown that a QW can mimic the time evolution of two neutrino flavors in matter with a linear density and for γ r ≪ 1. In the long time behavior the transition probability simulated by the QW converges to the asymptotic probability P (ν e → ν µ ; t f ), which confirms the agreement between the QW's and the neutrino's dynamics in matter.

VI. CONCLUSIONS AND OUTLOOK

In this paper we have analyzed the simulation of Dirac neutrino oscillations both in vacuum and in presence of matter effects using quantum walks, which can therefore be regarded as a discretization of the underlying field theory. We showed that, in fact, in the continuous limit one recovers a set of coupled Dirac equations that describe flavor oscillations. This fact allows us to establish a clear connection with the neutrino phenomenology for a particular scenario, such as detection of solar, atmospheric, reactor neutrinos, etc..., so as to fix the relevant parameters of the simulation (e.g. the initial state, or the necessary time steps). We have also introduced a way to simulate neutrino propagation in matter, an element which is crucial for some experiments, such as detection of solar neutrinos [7], or long baseline experiments [48,49]. As discussed in Sec. V, the simulation correctly reproduces the flavor conversion probability for a wide range of values of the adiabaticity parameter. The above results show that quantum walks can be used to simulate these phenomena, thus allowing one for a visualization of the neutrino phenomenology in scenarios like the solar interior or supernovae.

Figure 1 .

 1 Figure 1. (Color online) Time evolution of the probability P (να → ν β ; t), β ∈ {e, τ, µ} of a three flavor neutrino oscillation in vacuum, simulated by a QW. (Top) Short (inset) and medium time range (200 time steps). (Bottom) Long time range (1000 time steps). The mass differences are ∆m 2 eµ = 0.003 rad, ∆m 2 µτ = 0.32 rad, ∆m 2 eτ = 0.31 rad, and the mixing angles φ12 = 0.34 rad, φ13 = 0.54, φ23 = 0.45 rad, whereas k0 = 100. The initial condition is defined in Eq. (20).

Figure 2 .

 2 Figure 2. Level crossing scheme. Dependence of the eigenvalues of the Hamiltonian in matter with linear density ρ(x) = x : E1m (lower curve) and E2m (upper curve) on the position x for two different values of the mixing angle φ = 0.12 (solid blue) and φ = 0.84 (dashed red line). The mass difference is ∆m 2 eµ = 0.72 rad (θe = 0.1 rad and θµ = 0.2 rad), and k0 = 100.

Figure 3 .

 3 Figure 3. (Color online) Time evolution of the probability P (να → ν β ) in matter with linear density ρ(x) = x simulated by a QW in 125 time steps. The mass difference and the initial energy is the same of Fig.2. The dashed line (black) represents the asymptotic crossing probability at the resonance given by formula(32), for different adiabaticity parameters γr. The initial and final time steps are ti = 0, t f = 125. The initial condition is as defined in Eq.(20).

VII. ACKNOWLEDGEMENTS

This work has been supported by the Spanish Ministerio de Educación e Innovación, MICIN-FEDER project FPA2014-54459-P, SEV-2014-0398 and Generalitat Valenciana grant GVPROMETEOII2014-087. We acknowledge valuable discussions with E. Roldán and A. Verga.


Julien Guillaumond

Denis Dumortier

THE REPUTATION OF IRELAND IN FRANCE: ARE OLD CLICHES WITHERING AWAY AMONG YOUNG FRENCH PEOPLE? 1

This present work is part of a progress report for the GIS EIRE diaspora group commissioned by the Irish Embassy in Paris.

Introduction

Among other issues, Ireland's international reputation seems to have become a key component of the Irish government's global vision and strategy as one of its latest publications can attest 2 . "Ireland's international reputation", it says, "takes time to build, but can be easily damaged. Irish representatives work hard to project and maintain a positive image of Ireland as a good place in which to live, to do business, to invest, to study or to visit" 3 . Then follows a list of actors and organisations which are central to Ireland's positive global footprint.

History together with commercial and cultural exchanges 4 and the arts 5 (travel writings from the early 17 th century 6 , literary works, films and festivals) have contributed to shape the image French people have of Ireland, and vice versa. More recently, in the 1960s particularly, with the development of international tourism, the promotional campaigns of Bord Fáilte have very much shaped French people's views of Ireland, giving it a positive image as a country immersed in traditions and rurality 7 . Such campaigns, argued Dumortier in 1992, reinforced a positive vision of Ireland among French people 8 as well as a romantic view of Ireland, seen more as

The Reputation of Ireland in France an island of the past 9 using three main characteristics of the country: the magnificent scenery of the west of Ireland, the pace of life and its people 10 . The rural aspect has also been promoted together with various pictures of an Irish landscape, while a second image presented Irish people as welcoming and communicative hosts 11 . In the Irish case, according to B. Dumortier writing in the early 1990s, the countryside has been one key promotional tool of a tourism strategy to counter the negative effects of the weather or the cost of living 12 . Considering the Imaginaire Irlandais festival and recent French tourists' images of Ireland, Sheridan and O'Leary (2007) argued that French people still have a positive image of Ireland, but suggested that the tourism strategy should be updated following the transformation of Irish society and should take into account the longing for less traditional types of images among 21 st century tourists 13 . Seen from a French perspective, a case study on some pubs and businesses selling Irish products in the Auvergne region in 2014-15 has shown that their marketing and commercial strategies still hung on to old clichés about Ireland, and strengthened those same old images. They are thus contributing to maintain some distant 'invented tradition' providing an experience to their customers 14

Methodology

This report is based on an online survey conducted between 13 to 20 January 2020 among 1 st year Applied Foreign Languages undergraduates at Clermont Auvergne University15 , in the Auvergne region (central France). The survey was conducted in French to avoid any possible confusions on 9 Ibid. 10 Geraldine Sheridan and Sinéad O'Leary, "French Tourist Images of Ireland and L'Imaginaire Irlandais", in Irish Studies Review,vol. 13,no. 2,p. 151. 11 D. Dumortier, "L'Image de l'Irlande en France à travers la publicité de Bord Fáilte », op. cit., pp. 87-88. 12 Brigitte Dumortier, "Les Clés de l'essor touristique irlandais », in Hommes et Terres du Nord, 1992, vol. 2, pp. 79-85, p. 79. 13 G. and S. O'Leary, "French Tourist Images of Ireland and L'Imaginaire Irlandais", op. cit., p. 159.

14 Julien Guillaumond, "Marketing Irishness Today: A Study on Authenticity in French Businesses", in Frank Healy and Brigitte Bastiat, (eds), Voyages between France and Ireland. Culture, Tourism and Sport, Bern, Peter Lang, 2017. terms and to allow students to express themselves as much as possible in their native language. It was anonymous and respondents could fill in the survey at University or at home16 .

Students graduating with a BA in Applied Foreign Languages from Clermont Auvergne University will have studied equally foreign languages to the same level as well as economics, marketing and human resources. They will be proficient in both languages and will be able to work for businesses abroad. From the very beginning, undergraduate students are told that such a 3-year degree also comes with internships in France or abroad, and a whole semester of studies in a partner university abroad in their last few months. At the time of undergoing that survey, first-year undergraduates had not yet had any classes on Ireland as part of their present curriculum. That does not mean that they had not been to Ireland or that they had not studied some aspects of Irish history or Irish culture previously (at secondary schools or high schools for instance)17 .

The survey was divided into 5 sections offering the possibility to narrow down to a certain number of elements the profile of our respondents and identify more precisely their views about Ireland 18 . The first section identified the profile of respondents while the second was devoted to finding out what respondents knew about Ireland. For that, students had to associate words/ideas with "Ireland". Note here that no definition of what Ireland was, politically or geographically, was given to students. That section allowed a ranking of the most often-quoted words related to Ireland (word-occurrence system) to be established. Section 3 was a complement to section 1 (profile) while it also offered the possibility of nuancing what was answered by respondents. Section 4 aimed at identifying some common knowledge about Ireland in association with students' previous knowledge, singling out an author, a music group and/or a film. The final section acted as a concluding part in which respondents could offer some extra comments, and say whether they were likely to visit Ireland in the near future.

When the survey was closed, 154 students had responded which represents an 84% response rate based on all students enrolled in the first-year of the academic year 2019-2020. A first filter was applied to find out if all surveys were complete, if information was correct or if they were all genuine first-year undergraduates 19 . In the end, the cohort was made up of 122 students (98 women, 24 men) aged from 18 to 21. 48% came from the Puy-de-Dôme department, while others came from neighbouring areas, the second largest being Allier, followed by Haute-Loire.

Findings and analysis

Most common ideas about Ireland

A ranking based on word occurrences was established. To that end, a broad categorization of key words was defined as expressed by respondents when asked about three ideas coming to their minds when thinking about Ireland (Quelles sont les trois premières choses que vous associez à l'Irlande ?). The first big category was "folklore and festive days", containing 34 hits relating to Saint Patrick, or Saint Patrick's day, and leprechauns (see appendix 3). The second biggest category was geography/capital city with 17 hits (among them, 14 hits for Dublin, 1 for Belfast), then drinks with 15 answers associating Ireland with beer (11 hits), Guinness, pubs and bars 20 . The "historical events" and "symbols" categories followed. Then came the sport category (10 hits) with 8 hits for rugby, 1 for Gaelic football and one mention of MMA fighter Conor McGregor. On second and third counts (see appendix 3), elements corresponding to the same top categories previously identified were also given by respondents which suggest that there remains a rather strong correlation among respondents between the "Ireland" they imagine and ideas referring to its patron saint, its feast day and festive events together with the drinks, the symbols and the landscape. Is this representation of Ireland different if students had visited Ireland? There does not seem to be conclusive evidence on that particular point, and our sample of students who have already travelled to the island of Ireland (school trips or family holidays) is too small (less than 1%).

Literature, music and films associated with Ireland

What about cultural markers? Are young undergraduate students able to identify some cultural elements about Ireland in the three following domains: literature, music and the cinema? Is James Joyce one of the most often quoted traditional Irish writers among the list? 45 respondents (out of 19 See appendix 1 for cohort size and successive filters. 20 Interestingly, there appears to be a slight gender bias here as our survey reveals that among all answers associating Ireland with drinks, two-thirds of those answers came from female students surprisingly. The week before the survey took place, I think the Ulster team played ASM Clermont in Clermont.

The Reputation of Ireland in France 122, around 37%) gave an Irish author's name but two-thirds mentioned Oscar Wilde as an Irish author, and The Picture of Dorian Gray or the Canterville Ghost among the main literary works. Then came Samuel Beckett, Joseph O'Connor, Bram Stocker and Eoin Colfer. Such findings have to be considered in the light of respondents' previous syllabi in high school 21 . In terms of films, only 15 replies were received with a selection of old and more recent films such as a biopic, Michael Collins, or a drama, The Wind that Shakes the Barley, fantasy cartoons such as Brendan and the Secret of Kells, or the Renegade. To the question Pourriez-vous me donner le nom d'un groupe de musique irlandais ?, more than half of respondents offered at least one music group22 : U2 (44 times), followed by the Cranberries (32) and others such as Niall Horan, Sinead O'Connor, rather more modestly Realtá, the Dubliners, the Pogues, the Corrs, Hozier or Ed Sheeran. Their main source of information on Ireland appears to be strongly correlated with their education since a great majority (80% of respondents) were taught about Ireland at secondary and/or at high schools (see appendix 4).

Likelihood to visit Ireland

The survey ended on a final question: "souhaiteriez-vous visiter l'Irlande dans les prochaines années ?"or would you like to visit Ireland in the coming years, and if so, could you briefly explain why? 111 out of 122 respondents replied positively (among them, 8 people who had already been to Irelandsee previous question). The often quoted reason was to discover the country, though it is not officially named or defined as such), its culture and the language. What is interesting though is that respondents tend to move away from traditional stereotypes regarding the Irish as being welcoming and nice people, focusing more on the landscapes and the culture as key determining factors for of a visit to Ireland in the future. In more than half of the answers, one can find the words "culture" and "landscape" as reasons to visit Ireland. One might say that it could be associated with the specific profile of students, Applied Foreign Languages students, in other words students who might be more inclined towards a more international perspective of their future careers and of course who are possibly attracted by the idea of working abroad. Quite often, respondents gave as the main reasons the idea of discovering the world, visiting an English-speaking country and improving linguistic skills, but the core of the answers remained culture and landscape. At the same time, such a question was purely indicative as they may change their minds.

However, there are some specific adjectives which still refer to the Irish as welcoming people, friendly, impressive landscape, an intriguing culture, landscape looking fantastic. Even in the open final comments section, the words landscape and a welcoming people, a nice country and one added the filming of Games of Thrones, traditions, local products and Guinness come up again. It would be interesting to measure more precisely how they have learnt about Ireland. Interestingly, there does not seem to be huge differences on ideas about Ireland between students who have been to Ireland (school trips or family holidays) and those who have not, once they are back in France. When asked about the most visited places23 , those students did not follow a particular pattern, some visiting both parts of the island, others staying in the South: Dublin, the city and some of its museums, the Wicklow mountains for one, the National Leprechaun Museum for another, Glendalough, the Guinness museum, the West with Galway, the Cliffs of Moher and Connemara, Belfast, the Titanic Museum and "la place [sic] où s'est produit le Bloody Sunday", said one respondent.

Limitations and concluding remarks

While categories used to identify trends among young respondents' ideas about Ireland could be questioned because there could be a personal bias (for instance the word "castle" was put in the "history" category because of the context of the answer and not into "landscape"), most of these categories have also been used by other studies looking at tourist images of Ireland among the French24 . However, it should be noted that the purpose of that study was also to establish more clearly what young undergraduates enrolled in their BA in Applied Foreign Languages knew about Ireland 25 .

Ireland and L'Imaginaire Irlandais", in Irish Studies Review,vol. 13,no. 2, To our knowledge, this is the first time such a study was conducted on a large scale. Previously, a short questionnaire was done during the first lecture and students could interact with the lecturer during the first session, and such a survey gave way to a short session in which the lecturer tried to offer some answers to most common stereotypes on Ireland among French students. In that sense, the objective of previous short surveys was a teaching objective: identify some common held stereotypes which could be used for the structure of the different lectures during the 3 years the students would be taught about Irish society. In that sense, the idea was to find out about their knowledge of Ireland in order to tailor the different classes they are going to have in the course of their BA in Applied Foreign Languages to their knowledge of Irish society, north or south.

Our analysis suggests that young French students see Ireland very positively. Its reputation among young French undergraduates is safe and strong: it is an attractive destination for many of them. However, old clichés remain strong among those young French adults. Even though our sample is not representative of all French students, whether in terms of geographic profiles, academic curricula or professional aspirations, Ireland remains a place they associate with its Patron Saint, its festive character, its pubs and drink culture, and its landscape. However, for such a generation born between 2000 and 2002, the first generation born after the development of the Internet and also future tourists, they seem to be stuck into a rather old perception of what Ireland is. The way those 18-year olds see Ireland could of course be explained by their age, but it is all the more interesting to see that hardly any respondent talked about Northern Ireland or scarcely mentioned elements which could be associated with it (apart from some mentions of Belfast and others to be found in the "history" category) at a time when Brexit has taken so much space in the media in relation to the border between the North and South. In other words, they have no interest in the geographical or political definition of Ireland. What they have in mind remains an imagined Ireland with some elements of historical truth mingled with it.

Further studies could be conducted to find out whether such elements form part of a wider background or knowledge background of each and every French person. Such a conclusion is all the more valid today as the survey results suggest that there exists a view of Ireland which, though positive, remains static among the French public, and that old visions of Ireland seems to have passed through to the young. Another question could be whether or not a geographic dimension shaping commonly held stereotypes about Ireland exists since the Auvergne region does not have any real Celtic connection or association with Ireland compared to Brittany. Are there specificities of people from the Auvergne region regarding knowledge about Ireland? At the same time, could a gender dimension be isolated to look at Ireland's reputation abroad and as an addition to the age segmentation used by the tourism agency?

In their research paper, Sheridan and O'Leary (2007) wondered whether time had come to review the marketing policy and publicity of Ireland following its latest economic development and update its presentation when environmental issues as well as transport issues for instance have become more pressing issues 26 ? What our study suggests a few years later is that if such a change has happened, there has not been any obvious trickle-down effects yet. Some respondents explained that they learnt about Ireland through the web and videos. Will the movie industry (films and series on video on demand platforms) act as new agents spreading updated images of Ireland or will it reinforce previous stereotypes? At a time where film induced tourism has become a valuable product of the tourism industry 27 , should Irish tourism authorities watch out over ideas that could be used in their future marketing campaign to avoid that other stereotypes held in films do not become ideas associated with Ireland for the new generation? APPENDICES Appendix 1 * First year students who studied outside metropolitan France (or who did not properly answered the question regarding their previous school year location).

** Respondents were excluded either because they were repeating students or coming for a previous year in another college or diploma.

irlandais ? Si oui, Vous pouvez inscrire votre/vos groupe(s) ou chanteur(s) ci-dessous : Section 5 : Conclusion Enfin, souhaiteriez-vous visiter l'Irlande dans les prochaines années ? Si oui, pourriez-vous m'expliquer brièvement les raisons pour lesquelles vous souhaiteriez visiter l'Irlande ? Si vous souhaitez ajouter quelque chose d'autre, vous pouvez le faire ci-dessous :

  based on what they expect, reinforcing existing stereotypes. In such a context where reputation does matter, what are the views young French people have of Ireland? What do they know about Ireland, and what shapes young French people's views of Ireland in 2020?

Table :

 : Cohort size and gender together with successive filters

					Respondents details
		Filter		Total	Men	Women
				154	32	122
	1)	Incomplete survey		150	31	119
	2)	Geographic	origin	142	28	113
	missing or inappropriate*
	3)	First-year only**		122	24	98

A BA in Applied Foreign Languages is a three-year degree where students specialize in 2 languages (English and a second foreign language) and also study economics, law and marketing among the main subjects as well as international business.

In addition, students were briefly introduced to the survey when it was presented to them at the end of a class led by another lecturer hours before it was launched, and they received emails reminding them of this on-going survey twice during this period in January.

Their first lecture on Ireland was planned following the end of the survey.

That might have to do with the baccalaureate diploma as well.

The question there was slightly off the mark as I asked for groups while I should have asked about groups/individual musicians.

The question was not put to respondents, unless they said they had been to Ireland before. The idea was to avoid the type of answers that would be influenced too much by tourism brochures, thus the focus was much more on basic knowledge.

See in particular Geraldine Sheridan and Sinéad O'Leary, "French Tourist Images of

G. Sheridan and S. O'Leary, « French Tourist Images of Ireland and L'Imaginaire Irlandais", op. cit., p. 152.

See Visit Scotland, Insight Department: The Outlander Effect and Tourism, Edinburgh, 2020, retrieved https://www.visitscotland.org/binaries/content/assets/dotorg/pdf/research-papers-2/outlander-effect-2019-updated-february-2020.pdf on 2 March 2020.

Next year, a question will try to find out if respondents know where Ireland is. In the late 1960s, a study done in France showed that French people did not know wher eexactly Ireland stood geographically, and that people had difficulties between North and South (at a time of the Troubles though). See D. Dumortier, "L'Image de l'Irlande", op. cit., p. 86.

The Green colour Green (4/5) but no specificity

The Green colour Green 6/6 History 3/5 potato blight


P Dular

R V Sabariego

C Geuzaine

M V Ferreira Da Luz

P Kuo-Peng

L Krähenbühl

9. Numerical Techniques -(g) hybrid methods Finite Element Magnetic Models via a Coupling of Subproblems of Lower Dimensions

Abstract⎯ Model refinements of magnetic circuits are performed via a subdomain finite element method based on a perturbation technique. A complete problem is split into subproblems, some of lower dimensions, to allow a progression from 1-D to 3-D models. Its solution is then expressed as the sum of the subproblem solutions supported by different meshes. The procedure simplifies both meshing and solving processes, and quantifies the gain given by each model refinement on both local fields and global quantities.

I. INTRODUCTION

The perturbation of finite element (FE) solutions provides clear advantages in repetitive analyses and helps improving the solution accuracy [START_REF] Dular | Sub-domain finite element method for efficiently considering strong skin and proximity effects[END_REF]- [START_REF] Dular | Magnetic Model Refinement via a Perturbation Finite Element Method -From 1-D to 3-D[END_REF]. It allows to benefit from previous computations instead of starting a new complete FE solution for any variation of geometrical or physical data. It also allows different problem-adapted meshes and computational efficiency due to the reduced size of each subproblem.

A subproblem FE method is herein developed for coupling solutions of various dimensions, starting from simplified models, based on ideal flux tubes defining 1-D models, that evolve towards 2-D and 3-D accurate models. It is an extension of the method proposed in [START_REF] Dular | Perturbation Finite Element Method for Magnetic Model Refinement of Air Gaps and Leakage Fluxes[END_REF]- [START_REF] Dular | Magnetic Model Refinement via a Perturbation Finite Element Method -From 1-D to 3-D[END_REF], applied to refinements up to 3-D models. From the so calculated field corrections, the associate corrections of global quantities inherent to magnetic models, i.e. fluxes, magnetomotive forces (MMFs), currents and voltages, are also evaluated. The developments are performed for the magnetic vector potential FE magnetostatic and magnetodynamic formulations, paying special attention to the proper discretization of the constraints involved in each subproblem. The method is illustrated and validated on test problems.

II. COUPLING OF MAGNETIC MODELS OF VARIOUS DIMENSIONS

A. Series of coupled subproblems

Instead of solving a complete problem, generally with a 3-D model, it is proposed to split it into a sequence of subproblems, some of lower dimensions, i.e. 1-D and 2-D models. Its solution is then to be expressed as the sum of the subproblem solutions.

Each subproblem is defined in its own domain, generally distinct from the complete one. At the discrete level, this aims to decrease the problem complexity and allow distinct meshes with suitable refinements. Each subproblem approximates at best its contribution to the complete solution. The domains of the subproblems can overlap [START_REF] Dular | Perturbation Finite Element Method for Magnetic Model Refinement of Air Gaps and Leakage Fluxes[END_REF], [START_REF] Dular | Perturbation finite-element method for magnetic circuits[END_REF] or not [START_REF] Dular | Sub-domain finite element method for efficiently considering strong skin and proximity effects[END_REF], [START_REF] Dular | Magnetic Model Refinement via a Perturbation Finite Element Method -From 1-D to 3-D[END_REF]. Herein, non-overlapping subdomains are considered. They are separated by interfaces This work was supported by the F.R.S.-FNRS (Belgium), the CNPq (Brazil), the Belgian Science Policy (IAP P6/21) and the Walloon Region.

Γ f,p , through which a sequence of boundary conditions (BCs) or interface conditions (ICs) is to be defined.

B. Canonical magnetostatic or magnetodynamic problems

Each subproblem p is defined in a domain Ω p , with boundary ∂Ω p = Γ p = Γ h,p ∪ Γ b,p . It is governed by magnetostatic or magnetodynamic equations with volume and surface sources or constraints. Classical volume sources fix remnant inductions in magnetic materials and current densities in stranded inductors. Similar volume sources can also express changes of permeability and conductivity from one problem to another [START_REF] Dular | Perturbation finite-element method for magnetic circuits[END_REF]. Also, the usually homogeneous surface sources, i.e. BCs or ICs on the traces of the magnetic field h p and flux density b p , respectively n × h p | Γ f,p and of n ⋅ b p | Γ f,p , with n the unit exterior normal, can be extended to non-zero constraints calculated from previous problems. ICs have the general forms

[ n × h p ] γ p = j f,p , [n ⋅ b p ] γ p = b f,p , (1a-b

C. Sources at subproblem interfaces

Portions of a 3-D structure satisfying a translational or rotational symmetry can be first studied via 2-D models. This consists in neglecting some end effects, zeroing either n

× h p | Γ f,p or n ⋅ b p | Γ f,p .
Besides, if the field is chosen to be zero out of Ω p , a discontinuity of one of its traces is then voluntarily defined through Γ f,p .

With such assumptions, two subproblems 1 and 2 with adjacent non-overlapping sudomains Ω 1 and Ω 2 share a common interface Γ f,1 = Γ f,2 through which a field discontinuity occurs. A third subproblem, 3-D, serves then to correct the field distribution in a certain neighborhood Ω 3 on both sides of the interface, then denoted Γ f,3 . This is done via ICs

[ n×h 3 ] Γ f,3 = j f,3 , [n ⋅ b 3 ] Γ f,3 = b f,3 . (2a-b)
with the surface sources

j f,3 = -(n×h 1 | Γ f,1 -n×h 2 | Γ f,2 ) , b f,3 = -(n⋅b 1 | Γ f,1 -n⋅b 2 | Γ f,2 ) . (3a-b)
These sources compensate the traces 1 and 2 to recover the continuity of the total solution. Note that Γ f,1 , Γ f,2 and Γ f,3 are similar and only differ at the discrete level due to their different supporting meshes. The ICs (2a) and (2b) in a magnetic vector potential (a) FE formulation are considered via natural and essential constraints respectively [START_REF] Dular | Perturbation finite-element method for magnetic circuits[END_REF]. The essential constraint strongly fixes the discontinuity of the trace of a through Γ f,3 (continuity if b f,3 = 0), whereas the natural constraint weakly acts via a surface integral term in the FE formulation. This surface term, with (2a) and (3a), involves the traces of previous solutions, each one being actually involved in similar surface terms in the associated previous FE formulations, thus linked with their other volume integrals. At the discrete level, these surface integrals must be substituted with those volume integrals, limited to one single layer of FEs touching the interface [START_REF] Dular | Sub-domain finite element method for efficiently considering strong skin and proximity effects[END_REF]- [START_REF] Dular | Magnetic Model Refinement via a Perturbation Finite Element Method -From 1-D to 3-D[END_REF]. Because each solution is calculated in a different mesh, mesh-to-mesh projections of solutions are necessary. They can be profitably limited to the single layers of FEs. This procedure is of key importance for ensuring consistency between all the formulations and their coupling. It will be detailed in the extended paper and it will be shown to allow the accurate calculation of the global quantities (flux, MMF, current, voltage) at each step of the series, in particular the correction due to the end effects.

III. APPLICATION EXAMPLES

As a primary illustration, two flux tubes are first separately considered before being connected in series (Fig. 1). The solutions in each separate tube are simply calculated via 1-D models. When the tubes are connected, their junction surface acts as an interface Γ f,3 , with continuity of the normal magnetic flux density ([n ⋅ b 3 ] Γ f,3 = 0) and discontinuity of the tangential magnetic field ([n×h 3 ] Γ f,3 ≠ 0). This gives the requested source for a 2-D model, calculating the field correction limited to a certain neighborhood Ω 3 on both sides of the interface, with a locally refined mesh. A stranded inductor is then studied via the coupling of a 2-D plane model for its portion with a translational symmetry, a 2-D axisymmetrical model for its end winding and a 3-D model for the 3-D correction on both sides of the interface separating the portions (Figs. 2 and3). Because the correction is local to the interface, the associated 3-D mesh only needs to be refined in its vicinity.

Y X Z + Y X Z = Y X Z Y X Z + Y X Z = Y X Z
Various results and discussions will be given in the extended paper, in particular regarding the correction of both local and global quantities, the way to consider additional regions (e.g. the magnetic or conducting plate below the inductor in Fig. 2; based on [START_REF] Dular | Sub-domain finite element method for efficiently considering strong skin and proximity effects[END_REF] and [START_REF] Dular | Perturbation finite-element method for magnetic circuits[END_REF] for magnetostatic and magnetodynamic models), the way the fields decrease at infinity with the different models and the adaptation of the domain of each subproblem with its effect on the convergence of the complete solution. Parameterized analysis modifying some subproblems (e.g. end windings) while keeping the others constant will be shown to benefit from the developed method.

IV. CONCLUSIONS

The developed subdomain FE method allows to split magnetic models into subproblems of lower complexity with regard to meshing operations and computational aspects. A natural progression from simple to more elaborate models, from 1-D to 3-D geometries, is thus possible, while quantifying the gain given by each model refinement on both local and global quantities.

  )where the notation [ ⋅ ] γ = ⋅ | γ + -⋅ | γ -expresses the discontinuity of a quantity through an interface γ (with sides γ + and γ -) in Ω p . The associated surface fields j f,p and b f,p are generally zero, defining classical ICs for the physical fields, i.e. the continuities of the tangential component of h p and of the normal component of b p . If nonzero, they define possible surface sources that account for particular phenomena occurring in the idealized thin region between γ + and γ -.

Fig. 1 .

 1 Fig. 1. Field lines (top) and magnetic flux density (bottom) of the initial problem with two ideal flux tubes in series (b 1 and b 2 in both tubes left), its local correction at the junction (b 3 , middle) and the complete solution (b, right).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Field lines generated by a stranded inductor (half geometry): solution of a 2-D plane model in the XY plane (z = 0) (b 1 , portion on the left) and of a 2-D axisymmetrical model in the YZ plane (x = 0) (b 2 , portion on the right); the interface between the two portions is shown.


Jesse Donham

Alexander Wezel

Paola Migliorini

IMPROVING ECO-SCHEMES IN THE LIGHT OF AGROECOLOGY Key recommendations for the 2023-2027 Common Agricultural Policy

Forward

On the 25th of June 2021, the EU finalised its negotiations for the new Common Agricultural Policy (CAP) that is set to come into power on the 1 January 2023 and run until 2027. While this CAP is being championed by its creators as a radical new framework for tackling agricultural and environmental issues, it has left many civil society organisations to wonder if it will deliver on the commitments found in other legislations such as the Farm to Fork Strategy of the EU Green Deal.

The Horizon 2020 project, Agroecology for Europe (AE4EU), aims to take part in this discussion and provide insights for policy makers and Member States in order to ensure that this new CAP is as robust as possible and delivers on its promises for change.

As Agroecology Europe (2021) has proposed earlier this year, it is important to separate practices (i.e. buffer strips, winter cover crops) and production systems (i.e. agroecology, agroforestry, organic farming) for a more cohesive integration of Eco-schemes. Production systems such as conservation agriculture, agroforestry, and extensive silvo-pastoral systems should be subsidized by basic premiums, as organic farming is. Practices that can be implemented on their own, and are not production systems themselves, should be reclassified into three separate measures, those that: increase input efficiency, substitute inputs and redesign the production system. Such categories can be further classified according to the function they fulfil within agroecosystems: soil fertility, weed management, pest and disease control, pasture management, animal welfare, biodiversity and pollinator conservation, and climate change mitigation and adaptation. A further description of what such a system would look like can be seen in Table 1, where each measure is represented by a different colour: efficiency (E) in orange, substitution (S) in blue, and redesign (R) in green.

| 1

Integrating Eco-schemes according to agroecology

One of principal changes within the new CAP has been the inclusion of Eco-schemes -voluntary programmes linked to the first pillar which will be available to farmers with the hope to incentivize more ecological and environmentally-friendly farming practices. While agroecology holds an eminent space within this list by being listed as one of the primary recommendations, it does so within a role of just another practice to achieve a more sustainable farming system.

As stated by many before, such as [START_REF] Hill | Redesigning the food system for sustainability[END_REF], [START_REF] Gliessman | Transforming food systems with agroecology[END_REF] and Agroecology Europe (2021), agroecology is not just the substitution of one practice for another, it is a restructuring of the entire agricultural and food system. It is not just a tool to increase efficiency, it is a paradigm shift that uses food, health and the environment as a starting point to create a system that is inherently resilient. Further, it is important to remember that agroecology consists of three major elements: a set of practices, a science and a social movement [START_REF] Wezel | Agroecology as a science, a movement and a practice. A review[END_REF], Agroecology Europe, 2020). 3.

4.

5.

6.

| 4

Recommendations for Eco-schemes

An analysis by the AE4EU project on the inclusion of agroecology related Eco-schemes in the draft national strategic plans, shows a low number of agroecology-related policies (Table 2). While on average, countries have around three Eco-schemes with agroecological elements, with five being the highest (Poland) and zero the lowest (Belgium -Wallonia, Cyprus), there is a lot to be said for the strength of the existing Ecoschemes, which as mentioned above, are found by many civil society organisations as poor and unlikely to deliver on environmental benefits. The most popular Eco-schemes in the strategic plans are those relating to extensive grasslands management, use of cover/catch crops and organic farming. The least popular, with no schemes found in any countries, are "mixed cropping -multi cropping" and "improved rice cultivation to decrease methane emissions" (although Spain does mention rice production in one of their schemes, the scope remains unclear). Multidimensional Eco-schemes are the most likely to deliver, nevertheless they are found in only five countries -Czech Republic, Estonia, Latvia, Slovakia and the Netherlands. Interestingly, while the Netherlands has only chosen to include a single Eco-scheme within its strategic plan, there are nevertheless four different agroecological elements found in the scheme.

| 5

Assessment of draft Eco-schemes

An assessment of the Eco-schemes shows that only 19 % of schemes are likely to deliver their environmental objectives, with 40 % needing significant improvements and 41% either concerning or completely greenwashing. Welldesigned schemes are underfunded, while their less demanding counterparts remain more financially attractive (BirdLife Europe et al., 2021). This is dangerous for true environmental benefits to biodiversity, soil health, and climate mitigation and adaptation.

Further, the forementioned assessment has found many policy gaps within the proposed strategic plans including only two countries creating schemes to reduce antimicrobial use (although both schemes were deemed poor and as potential hidden subsides for intensive animal farming); only one scheme reducing herd size; only a few schemes focusing on growing feed to reduce feed import (a major solution for climate mitigation); only one scheme ceasing farming on drained peatlands (another major source of climate mitigation) and none to incentivize paludiculture; minimal support for agroforestry; and finally, the inclusion of precision farming without any rules for the reduction of fertiliser and pesticide use (BirdLife Europe et al., 2021).

Agroecology related Eco-schemes in draft national CAP strategic plans

The way forward

It is clear that many Eco-schemes have not been created with enough coherence, some barely going beyond basic practices and conditionality, and unlikely to sufficiently deliver on needed ecosystem services. What are needed are multi-dimensional Eco-schemes with robust funding, clear targets and proven benefits in order to improve the sustainability of farming in Europe.

The formal review process by the European Commission of the national CAP strategic plans is taking place in early 2022 and marks a key milestone to pave the way towards a consistent agricultural policy that is beneficial to the climate, biodiversity and health. The Commission should encourage and support Member States to restructure their draft national strategic plans in order to set clear objectives and roadmaps that are in line with other major EU legislations and agroecology.

Table 1 :Table 1 :

 11 Classification of the Eco-schemes proposed by the Commission according to the logic of "efficiency -substitution -redesign" (letter and colour code) and the logic of classification of measures in relation to agroecosystem service management (columns). Each measure is represented by a different colour: efficiency (E) in orange, substitution (S) in blue, and redesign (R) in green. (source: Agroecology Europe 2021).

	| 2

Authors