Application of a Vortex Penalization Method in Solid-Porous-Fluid Media to Passive Flow Control

Chloé Mimeau 1 * , Iraj Mortazavi 2 and Georges-Henri Cottet 1 1 Univ. Grenoble-Alpes, LJK, 51 rue des Mathématiques, F-38041 Grenoble FRANCE 2 IMB Univ. Bordeaux, M C 2 INRIA Bordeaux Sud-Ouest, F-33405 Talence FRANCE Abstract: A coupling of vortex methods with penalization method is proposed in this work in order to accurately and easily handle solid-fluid-porous media. This immersed boundary approach indeed maintains the efficiency and the robustness of vortex methods and allows to model the three different media without prescribing any boundary condition. In this paper, we propose an application of this immersed boundary method to passive flow control past a semi-circular cylinder, realized adding a porous sheath around the obstacle in order to smooth the flow dynamics.

The numerical model

In this work, flow simulations are based on particle methods. The fluid particles which are displaced by convection, stretching and diffusion are characterized by their position and their vorticity. The vorticity transport is expressed by the Vorticity Transport Equation, obtained taking the curl of the incompressible Navier-Stokes equations and expressed as follows, where one can distinguish the advection, the stretching and the diffusion terms:

∂ω ∂t + (u • ∇)ω -(ω • ∇)u = 1 Re ∆ω in D, (1)
where ω, u, Re and D respectively denote the vorticity, the velocity, the Reynolds number and the computational domain. This equation is approximated using a remeshed Vortex method [START_REF] Cottet | Vortex Methods[END_REF]. The latter is very robust and low-cost to simulate high Reynolds number recirculating flows. It relies on approximating separately the different terms at each time step. The convective part is solved using a "Vortex-In-Cell" (VIC) method with a semi-Lagrangian resolution: the fluid particles are displaced with the convective velocity and then remeshed on the original grid in order to avoid Lagrangian distortion. The stretching term is solved on the grid through a 4th order FD scheme. Finally diffusion and Poisson equation, given by ∆u = -∇ × ω and allowing to recover velocity from vorticity field, are also solved on the grid, using Fast Fourier Transforms. This work is devoted to the study of flows in solid-porous-fluid media and the method used to handle such problems is the vortex penalization technique. This method appears as a very good approach since it involves a unique equation for the whole domain. The latter is the non-dimensional Brinkman-Vorticity Transport Equation obtained from Eq. ( 1) by adding a penalization term and expressed as:

∂ω ∂t + (u • ∇)ω -(ω • ∇)u = 1 Re ∆ω + ∇ × [λ χ S (u s -u)] in D, (2)
where λ essentially depends, in the inverse proportion, on the intrinsic permeability k of the medium. Varying the λ value thus directly defines the different media. Indeed, in the fluid, the intrinsic permeability coefficient k goes to infinity, thus the fluid can be considered numerically as a porous media with a very high permeability. We set λ = 0 in this region. On the contrary, the solid has a permeability coefficient k which goes to zero, it can be consequently modeled setting the penalization parameter λ to a very high value. In this study λ equals 10 8 in the solid. It was proved in [START_REF] Angot | A Penalization Method to Take Into Account Obstacles in Incompressible Viscous Flows[END_REF] that solving Eq. ( 2) with such a value of λ was equivalent to solve Darcy's law in the solid. Furthermore, setting the λ parameter to an intermediate value, reasonably chosen between these two extreme values (λ = 0 and λ = 10 8 ), would model a porous medium in which the flow has a Darcy velocity. The accuracy and efficiency of the penalization method come from its capability to take into account these variations of λ and to capture the induced steep velocity variations at the different interfaces with a minimum number of discretization points.

Application to passive flow control in 2D

Here, the solid-porous-fluid configuration is applied to cover a semi-circular cylinder geometry with a porous coating in order to decrease the shear forces applied on this body and to smooth the flow dynamics [START_REF] Bruneau | Numerical Modelling and Passive Flow Control Using Porous Media[END_REF]. The semi-circular cylinder can be considered as a simplified section of an outside rear-view mirror of a car or a motor cycle. The subsequent flow control simulations are performed at transitional (Re = 550) and highly transitional regime (Re = 3000) in a 2D computational domain. In order to analyze the effects of our control approach, we compare global flow quantities like the drag force (F D ) and the enstrophy (Z), expressed as Z = D |ω| 2 dx, allowing to measure the dissipation effects in the flow. A parametric study considering different values of λ in a surrounding porous and homogeneous layer was achieved in order to determine the influence of the permeability on the control efficiency. This study shown that setting λ = 1 (high permeability) inside the layer, clearly appears as the best solution compared to uncontrolled case for both regime in terms of flow regularization (F D : -29% at Re=550 and -6% at Re=3000, Z: -38% at Re=550 and -29% at Re=3000). Nevertheless, the position of the permeable zone also has a great importance in passive control effects. To illustrate this statement, we perform a comparison between the porous layer configuration and the "porous poles" configuration where the permeable zone is only located on top and bottom of the semi-circular cylinder, i.e in the vicinity of the detachment points. For manufacturing easiness we choose to set these poles to an intermediate permeability, λ = 10. And surprisingly, the results obtained with the "porous poles" device are very satisfactory. Figure 1 indeed shows that it allows an eddy detachment from the wall and a decrease of the transverse dimension of the wake implying a drag force reduction, which is about 23% at Re=500 and 7% at Re=3000 (more than the λ = 1 porous layer case).

Conclusion and Future Work

In this work, a hybrid vortex-penalization method is proposed in order to easily carry out complex flows involving solid-fluid-porous media. An application to passive flow control past a semi-circular cylinder is performed at transitional and highly transitional regimes, showing the emergence of a very suitable "porouspoles" device. This preliminary 2D study can be useful to supply information on general trends for a control. However, such a flow is three-dimensional and this 2D study should be complemented by 3D computations to get qualitative physical results. These on-the-way 3D results will be presented at the conference.

Figure 1 :

 1 Figure 1: Zoom of the mean vorticity fields and isolines at Re = 550 and Re = 3000.

This work is supported by NSFC No. 91016001.


Luis Gonzalez-Mestres
email: luis.gonzalez-mestres@cosmology.megatrend.edu.rs

Ultra-high energy physics and standard basic principles Do Planck units really make sense ?

has not yet been elucidated whether the observed flux suppression for ultrahigh energy cosmic rays (UHECR) at energies above ≃ 4 x 10 19 eV is a signature of the Greisen-Zatsepin-Kuzmin (GZK) cutoff or a consequence of other phenomena. In both cases, violations of the standard fundamental principles of Physics can be present and play a significant role. They can in particular modify cosmic-ray interactions, propagation or acceleration at very high energy. Thus, in a long-term program, UHECR data can hopefully be used to test relativity, quantum mechanics, energy and momentum conservation, vacuum properties... as well as the elementariness of standard particles. Data on cosmic rays at energies ≃ 10 20 eV may also be sensitive to new physics generated well beyond Planck scale. A typical example is provided by the search for possible signatures of a Lorentz symmetry violation (LSV) associated to a privileged local reference frame (the "vacuum rest frame", VRF). If a VRF exists, the internal structure of standard particles at ultra-high energy can undergo substantial modifications. Similarly, the conventional particle symmetries may cease to be valid at such energies instead of heading to a grand unification and the structure of vacuum may no longer be governed by standard quantum field theory. Then, the question whether the notion of Planck scale still makes sense clearly becomes relevant and the very grounds of Cosmology can undergo essential modifications. UHECR studies naturally interact with the interpretation of WMAP and Planck observations. Recent Planck data analyses tend to confirm the possible existence of a privileged space direction. If the observed phenomenon turns out to be a signature of the spinorial space-time (SST) we suggested in 1996-97, then conventional Particle Physics may correspond to the local properties of standard matter at low enough energy and large enough distances. This would clearly strengthen the cosmological relevance of UHECR phenomenology and weaken the status of the Planck scale hypothesis. Another crucial observation is that, already before incorporating standard matter and relativity, the SST geometry naturally yields a H t = 1 law where t is the age of the Universe and H the ratio between relative speeds and distances at cosmic scale. As standard cosmology is not required to get such a fundamental result, the need for a conventional Planck scale is far from obvious and the study of UHECR can potentially yield evidence for an alternative approach including new physics and new ultimate constituents of matter. UHECR may in particular allow to explore the possible indications of the existence of a transition scale at very high energy where the standard laws would start becoming less and less dominant and new physics would replace the conventional fundamental principles. We discuss prospects of searches for potential signatures of such a phenomenon.

Introduction

Recent measurements and analyses [START_REF] Letessier | Selvon for the Pierre Auger Collaboration[END_REF][START_REF] De Mello Neto For The | Measurements of cosmic rays at the highest energies with the Pierre Auger Observatory[END_REF] have not yet allowed to elucidate whether the observed UHECR flux suppression above ≃ 4 x 10 19 eV [START_REF] Pierre | Contributions to the 33rd International Cosmic Ray Conference (ICRC 2013)[END_REF][START_REF] Abu-Zayyad | Energy Spectrum of Ultra-High Energy Cosmic Rays Observed with the Telescope Array Using a Hybrid Technique[END_REF] is really a signature of the Greisen-Zatsepin-Kuzmin (GZK) cutoff [START_REF] Greisen | End to the Cosmic-Ray Spectrum?[END_REF][START_REF] Zatsepin | min, Upper Limit on the Spectrum of Cosmic Rays[END_REF] or corresponds to a flux-suppressing mechanism of a different dynamical or astrophysical origin. All possible situations can be sensitive to new physics modifying at very high energy the production of cosmic rays [START_REF] Gonzalez-Mestres | Superluminal Matter and High-Energy Cosmic Rays[END_REF][START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF], their propagation [START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Absence of Greisen-Zatsepin-Kuzmin Cutoff and Stability of Unstable Particles at Very High Energy, as a consequence of Lorentz Symmetry Violation[END_REF] or their acceleration at astrophysical sources [START_REF] Gonzalez-Mestres | Lorentz Symmetry Violation and Acceleration in Relativistic Shocks, Proceedings of the Heidelberg[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF]. Various violations of the standard basic principles of current particle physics (relativity, quantum mechanics and quantum field theory (QFT), energy and momentum conservation, vacuum homogeneity and "static" properties, effective space dimensions, symmetries, quark confinement, conventional elementariness of particles...) can be potentially involved ( [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF] and [START_REF] Gonzalez-Mestres | Superbradyons and some possible dark matter signatures[END_REF][START_REF] Gonzalez-Mestres | Lorentz symmetry violation, dark matter and dark energy[END_REF]). New physics may even be present in data above ≃ 10 16 eV [START_REF] Lokhtin | Ridge effect and alignment phenomenon[END_REF][START_REF] Gonzalez-Mestres | Lorentz violation, vacuum, cosmic rays, superbradyons and Pamir data[END_REF]. Data from UHECR experiments are in particular expected [START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF] to yield bounds on Lorentz symmetry violation (LSV) parameters in models incorporating a privileged local reference frame (the "vacuum rest frame", VRF [START_REF] Gonzalez-Mestres | Properties of a possible class of particles able to travel faster than light[END_REF][START_REF] Gonzalez-Mestres | Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light Proceedings of the TAUP 1995 Conference[END_REF]). But the phenomenological analysis of experimental data at ultra-high energy (UHE) will be a difficult task and requires a long-term experimental and theoretical effort [START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF][START_REF] Gonzalez-Mestres | Testing fundamental principles with high-energy cosmic rays[END_REF] having in mind all possible forms of LSV as well potential violations of other fundamental principles of standard physics and the necessary compatibility with low-energy bounds on LSV [START_REF] See | Relativity: Testing times in space[END_REF][START_REF] Hohensee | Limits on Violations of Lorentz Symmetry and the Einstein Equivalence Principle using Radio-Frequency Spectroscopy of Atomic Dysprosium[END_REF]. It can be readily checked, for instance, that LSV violations leading to spontaneous decays of UHECR may fake the GZK cutoff [START_REF] Gonzalez-Mestres | Lorentz symmetry violation and the results of the AUGER experiment[END_REF][START_REF] Gonzalez-Mestres | AUGER-HiRes results and models of Lorentz symmetry violation[END_REF]. A typical example was provided in [START_REF] Gonzalez-Mestres | Testing fundamental principles with high-energy cosmic rays[END_REF], where the GZK cutoff would be replaced by spontaneous emission of photons or e + e -pairs in a simple LSV pattern.

The Planck scale is often taken as a reference to parameterize possible deviations from standard physics at very high energy and at very small distances. But this choice does not necessarily make sense [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF], as new physics may become dominant at energies below Planck energy and at distances above the Planck distance scale [START_REF] Gonzalez-Mestres | Testing fundamental principles with high-energy cosmic rays[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, vacuum and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Planck, cosmic rays and unconventional cosmologies, contribution to the Planck 2011 Conference[END_REF]. Contrary to approaches to quantum gravity [START_REF] Kiefer | Conceptual Problems in Quantum Gravity and Quantum Cosmology[END_REF][START_REF] Ashtekar | Loop Quantum Gravity and the The Planck Regime of Cosmology[END_REF] where the role of LSV is basically reduced to deformations and extensions of existing algebras [START_REF] Amelino-Camelia | Relative locality: A deepening of the relativity principle[END_REF][START_REF] Smolin | General relativity as the equation of state of spin foam[END_REF], the approach based on the existence of a local VRF for each comoving observer explicitly breaks the associated symmetries that become low-energy limits just as the Lorentz-like symmetry of the kinematics of phonons in a lattice [START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Properties of a possible class of particles able to travel faster than light[END_REF]. Then, extrapolations from low-energy physics such as Grand Unification and the standard concept of gravitation do not necessarily remain valid at very high energy and very small distances. Similarly, the speed of light c may cease to dominate the natural relation between energy and momentum or between distance and time scales, and the Planck constant h may no longer be relevant to relate energy and time scales. In this case, two new scales of energy, momentum, distance and time become relevant: i) the fundamental scale (E a in terms of energy) at which a new physics is generated; ii) the transition scale E (energy) ≈ E trans at which the new physics starts becoming more and more significant in experimental data. New ultimate constituents of matter (e.g. superbradyons, superluminal preons) can also be present in such scenarios [START_REF] Gonzalez-Mestres | Properties of a possible class of particles able to travel faster than light[END_REF][START_REF] Gonzalez-Mestres | Observing air showers from cosmic superluminal particles[END_REF] where standard internal symmetries may cease to make sense at high enough energy [START_REF] Gonzalez-Mestres | Superbradyons and some possible dark matter signatures[END_REF][START_REF] Gonzalez-Mestres | Lorentz symmetry violation, dark matter and dark energy[END_REF] instead of becoming more exact as energy increases. Superbradyons can be part of the dark matter [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF][START_REF] Gonzalez-Mestres | Superbradyons and some possible dark matter signatures[END_REF].

Simultaneously, cosmological studies are expected to cast light on the same basic questions, in particular through WMAP [START_REF]Wilkinson Microwave Anisotropy Probe (WMAP[END_REF] and Planck [START_REF][END_REF] data [START_REF]Planck 2013 results. I. Overview of products and scientific results[END_REF][START_REF]Planck 2013 results. XVI. Cosmological parameters[END_REF] including the possible signature of a privileged space direction as the Planck analysis of maps of the cosmic microwave background (CMB) anisotropy [START_REF]Planck 2013 results. XXIII. Isotropy and statistics of the CMB[END_REF] seems to suggest. The existence of such a privileged space direction for each comoving observer is a natural prediction of the spinorial space-time (SST) geometry we suggested in 1996-97 [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF][START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF]. No standard matter or conventional relativity is required to get such a result. Thus, the SST pattern combined with parity violation can potentially account for Planck and WMAP observations [START_REF] Gonzalez-Mestres | Spinorial space-time and privileged space direction (I), mparc[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and Friedmann-like equations (I), mparc[END_REF]. Similarly, previous to any introduction of conventional matter and relativity, the SST leads by itself in an automatic way [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF] to the relation H t = 1 where t is the cosmic time (age of the Universe) and H the Lundmark-Lemaître-Hubble (LLH) constant describing the ratio between relative velocities and distances at cosmic scales. The H t = 1 law thus emerges as a natural asymptotic limit of the evolution of the Universe in a SST cosmic geometry [START_REF] Gonzalez-Mestres | Pre-Big Bang, space-time structure, asymptotic Universe[END_REF][START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF]. The fact that such results can be obtained without using standard physics clearly deserves a closer analysis [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, space-time structure, asymptotic Universe[END_REF]. In particular, the conventional principles of Physics obtained from particle experiments may happen to be valid only as a low-energy limit in the regions of the Universe where standard matter has nucleated. Such a scenario would confirm the picture already suggested in [START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Properties of a possible class of particles able to travel faster than light[END_REF] and in [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF][START_REF] Gonzalez-Mestres | Deformed Lorentz Symmetry and High-Energy Astrophysics (I), ICRC 1999 Evening Workshop Session talk[END_REF]. In particular, if a VRF associated to LSV exists, high-energy particles are obviously not the same objects as particles at rest. LSV and other new physics can then modify the internal structure, symmetries and interaction properties of particles at very high energy. Such a new physics can possibly be related to pre-Big Bang scenarios [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, space-time structure, asymptotic Universe[END_REF] and to new ultimate constituents of matter. Thus, Cosmology and UHECR studies should permanently interact in a common attempt to search for new signatures of new phenomena and hopefully understand the ultimate properties of matter, possible pre-matter and space-time.

In a pre-Big Bang approach with the spinorial space-time, if standard matter has nucleated on a pre-existing vacuum associated to the SST geometry, new cosmological forces can be generated by this transition. In particular, the gravitationally coupled matter can react to the pre-existing LLH expansion of the Universe and generate a new form of energy and cosmological force and pressure [START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF]. This reaction would become weaker as the density of the standard matter decreases with the Universe expansion and the relation H t = 1 would remain a suitable asymptotic limit at large t.

Another essential question is that of the role of high-energy accelerator experiments. Signatures of supersymmetry do not seem to have been found by now at LHC, and our analysis [START_REF] Gonzalez-Mestres | Comments on the recent result of the "Measurement of the neutrino velocity with the OPERA detector in the CNGS beam[END_REF][START_REF] Gonzalez-Mestres | Astrophysical consequences of the OPERA superluminal neutrino[END_REF] of the September 2011 OPERA claim [START_REF]Measurement of the neutrino velocity with the OPERA detector in the CNGS beam[END_REF] on the possible detection of a superluminal neutrino pointed out that such a comparatively strong effect could not be made consistent with existing and well-established cosmic-ray and astrophysical data. Even assuming that supersymmetry exists, the whole pattern can fail at higher energies it Lorentz symmetry is violated and a VRF exists. But finding a measurable signature of LSV at present accelerators can be a very hard task. The possibility to search for decays producing preon-like objects (superbradyons...), assuming they can exist as free particles in our Universe and be produced at accelerators, was considered in [START_REF] Gonzalez-Mestres | Superbradyons and some possible dark matter signatures[END_REF][START_REF] Gonzalez-Mestres | Lorentz Symmetry Violation and Superluminal Particles at Future Colliders EPS-HEP97[END_REF]. Astrophysical observations can also be relevant, as cosmic superbradyons are expected to spontaneously emit "Cherenkow" radiation in vacuum in the form of standard particles when moving at a speed larger than c or close to this value [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Properties of a possible class of particles able to travel faster than light[END_REF]. Superbradyon decays can even be a source of high-energy cosmic rays [START_REF] Gonzalez-Mestres | Superluminal Matter and High-Energy Cosmic Rays[END_REF][START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF].

We present here and updated discussion of the relevance of UHECR data for fundamental Particle Physics and Cosmology, emphasizing specific opportunities related to possible new physics and pre-Big Bang scenarios. The question of the relevance of the Planck scale is explicitly dealt with.

An illustrative example: LSV kinematics

Crucial for the possible existence of specific signatures of new physics in UHECR data is Lorentz symmetry violation with a vacuum rest frame. The Earth is then assumed to move slowly with respect to the local VRF. Then, the observable properties of UHECR can reflect new physics generated at a new fundamental scale E a , not only concerning space-time properties but more generally including the fate of particle symmetries and interactions or the properties of matter and pre-matter.

LSV patterns for kinematics can be used to illustrate possible descriptions of this new physics. Ways to deal with possible violations of standard quantum mechanics at high energy have been considered in [START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF][START_REF] Gonzalez-Mestres | Preon models, relativity, quantum mechanics and cosmology (I)[END_REF]. In most cases, the smallness of the mass term of a UHECR turns out to be a crucial property to make the UHECR properties and interactions sensitive enough to very small effects from unconventional physics impossible to observe with lower-energy particles.

A simple model and some estimates

An illustrative example, using LSV with a VRF, can be provided by the quadratically deformed relativistic kinematics (QDRK) defined by the relation [START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Absence of Greisen-Zatsepin-Kuzmin Cutoff and Stability of Unstable Particles at Very High Energy, as a consequence of Lorentz Symmetry Violation[END_REF]:

E ≃ p c + m 2 c 3 (2 p) -1 -p c α (p c E -1 a ) 2 /2 (1)
where p is the momentum assumed to be ≪ E a c -1 , m the mass, α a positive constant describing the strength of the deformation and E a the effective fundamental energy scale. Then, the negative deformation term:

∆E ≃ -p c α (p c E -1 a ) 2 /2 (2)
becomes larger than the positive mass term m 2 c 3 (2 p) -1 above a transition energy E trans :

E trans ≃ α -1/4 (E a m) 1/2 c ( 3
)
It immediately follows that for a proton at ultra-high energy (UHE) with α = 1, one would have E trans ≃ 10 20 eV if E a ≃ 10 3 E Planck where E Planck is the Planck energy. Therefore, a UHE proton would be potentially sensitive to new physics generated three orders of magnitude beyond the Planck energy scale [START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF][START_REF] Gonzalez-Mestres | Physics Opportunities Above the Greisen-Zatsepin-Kuzmin Cutoff: Lorentz Symmetry Violation at the Planck Scale[END_REF]. Other possible values of the LSV parameters have been considered in [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | AUGER-HiRes results and models of Lorentz symmetry violation[END_REF].

Furthermore, writing:

∆E ≃ -α E (E E -1 a ) 2 /2 (4)
E m ≃ E 2 rest E -1 /2 (5)
where E m is the kinematical mass term and E rest = m c 2 the rest energy of the particle, and taking α = 1, the relation

∆E E -1 m ≃ -E 4 (E a E rest ) -2 (6)
makes explicit the naturally strong energy dependence of the deformation at very high momentum.

The sharp rise of the ratio ∆E E -1 m thus obtained from QDRK can be the sign of a strong transition for E > E trans if the mass term E m is interpreted as an expression of the original internal structure of the particle considered. At the Grand Unification scale E ≃ 10 16 GeV, one would have ∆E E -1 m ≈ 10 20 for a proton with α = 1 and E a ≃ 10 3 E Planck . At the Planck energy scale, ∆E E -1 m ≈ 10 32 and ∆E E -1 ≈ 10 -6 . Assuming QDRK to be correct, such very large values of ∆E E -1 m would clearly suggest a possible change in the UHECR internal structure and dynamics at energies above E trans and already well below the grand unified and Planck energy scales.

Another possible parameterization, equivalent to QDRK, can be:

E 2 = p 2 c 2 + m 2 c 4 -(δE) 2 (7)
where δ E is assumed to vary proportionally to ≃ E 2 at high energy.

Then, if δE = m c 2 (defining the transition energy) at E ≃ 10 20 eV, one has δ E E -1 rest ≃ 10 10 at E ≃ 10 16 GeV and δE E -1 rest ≃ 10 16 at the Planck energy scale. Approaches based on and (1) or [START_REF] Gonzalez-Mestres | Superluminal Matter and High-Energy Cosmic Rays[END_REF] have also implications on the motion of the particle. Above E trans , the Hamiltonian velocity v = dE/d p does no longer tend to c as the energy increases. Instead, the positive difference c -v increases with energy. Similarly, the ratio E p -1 becomes smaller than Give the exact title of the conference c above E trans . This is substantially different from the standard relativistic scenario where the relative difference between c and v becomes as small as ≈ 10 -38 for a proton at the Planck energy scale. Instead, one would get from (1):

dE/d p ≃ c -m 2 c 3 (2 p 2 ) -1 -3 c α (p c E -1 a ) 2 /2 (8)
leading, for a proton with E trans ≃ 10 20 eV, to c -dE/d p ≃ 10 -6 c at the Planck energy scale.

Theoretical and phenomenological considerations

A natural alternative to QDRK would be to replace the deformation term ( 2) by an expression depending quadratically on p. But we rejected [START_REF] Gonzalez-Mestres | Deformed Lorentz Symmetry and High-Energy Astrophysics (I), ICRC 1999 Evening Workshop Session talk[END_REF] this linearly deformed relativistic kinematics (LDRK) because of its too strong predictions at low energy as compared to experimental bounds [START_REF] See | Relativity: Testing times in space[END_REF][START_REF] Hohensee | Limits on Violations of Lorentz Symmetry and the Einstein Equivalence Principle using Radio-Frequency Spectroscopy of Atomic Dysprosium[END_REF].

The value of α in the QDRK equation ( 1) is actually not expected to be necessarily the same for all standard particles [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | AUGER-HiRes results and models of Lorentz symmetry violation[END_REF]. Similarly, the universality of E a appears as a natural choice but the real dynamics can be more complex. Here, α and E a refer basically to effective values for each particle considered. Although Grand Unification may suggest taking the same values α and E a for all quarks and leptons, such a symmetry is not necessarily relevant at the fundamental energy scale where new physics can play a leading role. Contrary to conventional prejudices, low-energy symmetries can actually be a way to hide essential properties of the structure of standard particles at very high energy and very short distances where a new kind of matter or pre-matter possibly dominates.

According to conventional quantum mechanics and relativity, a ≈ 10 20 eV energy scale corresponds to a ≈ 10 -25 cm distance scale, 12 orders of magnitude below the proton radius (≃ 0.9 x 10 -13 cm with electric dipole moment ≃ 5.4 x 10 -24 e.cm where e is the elementary charge). Lower values of E trans are not excluded by data [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | AUGER-HiRes results and models of Lorentz symmetry violation[END_REF], in particular if the standard GZK cutoff is not the mechanism at work above ≃ 4 x 10 19 eV. For a proton, E trans ≃ 3 x 10 19 eV, just below the observed fall of the spectrum, would be obtained with E a ≃ 100 E Planck and α = 1. Taking E a ≃ 10 E Planck and α = 1 yields E trans ≃ 10 19 eV. In this case, one would have : i) ∆E E -1 m ≈ 10 24 and δ E E -1 rest ≃ 10 12 at the Grand Unification scale E ≃ 10 16 GeV ; ii) ∆E E -1 m ≈ 10 36 , ∆E E -1 ≈ 10 -4 and δE E -1 rest ≃ 10 18 (δE ≃ 0.1 E Planck ) at the Planck energy scale. For an electron, using (1) and (3) with α = 1 and E a ≃ 10 3 E Planck yields E trans ≃ 2 x 10 18 eV. Taking instead α = 10 -2 would lead to E trans ≃ 2 x 10 19 eV corresponding to a ≈ 10 -24 cm distance scale. The ACME bound [START_REF] Acme | Advanced Cold Molecule Electron EDM) Collaboration, Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron[END_REF] on the electric dipole moment of the electron is | d e | < 8.7 x 10 -29 e cm , where d e is the dipole moment. For a muon with α ≃ 1 and E a ≃ 10 3 E Planck , one gets E trans ≃ 3 x 10 19 eV. A bound on the muon electric dipole moment is [START_REF] Bennett | An Improved Limit on the Muon Electric Dipole Moment[END_REF] | d µ | < 1.9 x 10 -19 e cm, where d µ is the muon electric dipole moment.

If a nucleus is dealt with to a first approximation as an additive set of N nucleons, the associated effective value of E trans becomes N times larger than for protons [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | AUGER-HiRes results and models of Lorentz symmetry violation[END_REF]. This does not appears to be inconsistent with AUGER measurements [START_REF] Letessier | Selvon for the Pierre Auger Collaboration[END_REF][START_REF] Pierre | Contributions to the 33rd International Cosmic Ray Conference (ICRC 2013)[END_REF]. The detailed study of UHECR composition is a crucial ingredient of the test of basic principles through AUGER data. Another possibility to be considered is quark deconfinement at ultra-high energy, especially given the fact that the parton model cannot work at UHE [START_REF] Gonzalez-Mestres | High-Energy Nuclear Physics with Lorentz Symmetry Violation[END_REF][START_REF] Gonzalez-Mestres | Lorentz Symmetry Violation, Vacuum and Superluminal Particles International Conference on Relativistic Physics and some of its Applications[END_REF]. A cosmic free quark can satisfy equation ( 1) with an effective value of α well above that associated to the proton [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | AUGER-HiRes results and models of Lorentz symmetry violation[END_REF]. This property may help deconfinement. If E Q f ree is the minimum energy of a free quark, a quark with energy E > E Q f ree can release an energy ≃ E -E Q f ree in the atmosphere. After having released most of this energy, it can still produce significant effects through hadron mutations. For instance, a quark (antiquark) with charge 2/3 (-1/3) can replace a quark with charge -1/3 (2/3) inside a hadron, turning a neutron (proton) into a proton (neutron). The considered deconfinement at UHE is in principle a phase transition and the sequential absorption of nucleons by free quarks considered in [START_REF] De Rújula | Unconfined quarks and gluons[END_REF] would not necessarily be possible in the present approach, as the quark would itself become much slower when absorbing hadrons and nuclei.

Other possible UHE effects and tests of LSV have been discussed in [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF]. In the presence of LSV with a VRF, additional violations of standard basic principles can also produce observable signatures at ultra-high energy [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF]. It must be noticed that (1) and ( 2) include a possible deformation of the effective value of the speed of light through the ∆E term. For a photon with α = 1 , E a ≃ 10 3 E Planck and E = 10 19 eV, one would have E p -1c ≃ -3 x 10 -25 c and dE/d pc ≃ -10 -24 c . These values are much smaller than those considered at lower energies in connection with the OPERA claim of september 2011 [START_REF]Measurement of the neutrino velocity with the OPERA detector in the CNGS beam[END_REF]. On the other hand, the photon internal structure can be significantly modified by this apparently small deviation from the standard critical speed c. Similarly, specific models describing violations of other fundamental principles (quantum mechanics...) will also tacitly include an effective energy dependence of other fundamental constants (the Planck constant h ...). Coupling constants and interaction properties can undergo a stronger evolution as energy increases, due to the role of effective masses and particle internal structure. As a consequence, the validity of cosmic-ray energy estimations may become uncertain at UHE.

From a cosmological point of view, the values E ≃ 10 19 eV and E ≃ 10 20 eV correspond respectively to temperatures T ≃ 10 23 K and T ≃ 10 24 K in the history of the Universe. At a temperature scale of T ≃ 10 27 K or T ≃ 10 28 K usually associated to inflation, the ratios ∆E E -1 m and δE E -1 rest would already have large values. Then, the evolution of a pre-Big Bang scenario can reasonably replace the standard Big Bang and inflation. In pre-Big Bang patterns, the formation of standard matter can just correspond to a phase transition or be associated to a long evolution of the properties of matter and pre-matter. UHECR phenomenology is directly concerned by such fundamental questions and should be used to search for relevant cosmological signatures, including tracks of possible free UHE preonic objects such as superbradyons [START_REF] Gonzalez-Mestres | Superluminal Matter and High-Energy Cosmic Rays[END_REF][START_REF] Gonzalez-Mestres | Observing air showers from cosmic superluminal particles[END_REF].

In connection with UHE physics and cosmological observations of the early Universe [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, space-time structure, asymptotic Universe[END_REF], the internal structure of particles at very small distances is also a relevant subject including possible preonic (superbradyonic?) structures. The SST geometry, that may have successfully predicted [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and privileged space direction (I), mparc[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and Friedmann-like equations (I), mparc[END_REF] the existence of a privileged space direction as observed by Planck [START_REF]Planck 2013 results. XXIII. Isotropy and statistics of the CMB[END_REF], can hopefully be used to describe both the structure and evolution of the Universe at very large scale and the particle structure and dynamics at very small distances [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, space-time structure, asymptotic Universe[END_REF]. In the SST, the space-time position spinor ξ is made of two complex coordinates transforming under a SU(2) group associated to space rotations [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF]. The cosmic time t (age of the Universe) is given by the spinor modulus | ξ | or a function of this modulus. Taking t = | ξ | , the S 3 hypersphere formed by all spinors of modulus t defines the space at time t. If ξ 0 is the spinorial position of the observer at time t, the spatial coordinates of a point ξ on the same constant-time hypersphere are defined by the path from ξ 0 to ξ on the hypersphere. A position vector can thus be readily built using the relevant SU(2) transformation [START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, space-time structure, asymptotic Universe[END_REF]. However, it is also possible [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and privileged space direction (I), mparc[END_REF] to consider spinorial local coordinates if the requirement of constant cosmic time for the path between ξ 0 and ξ is abandoned. The difference ∆ξ = ξξ 0 , which changes sign under a 2π SU(2) rotation, provides a relative position spinor of ξ with respect to ξ 0 . The spinorial coordinates can be relevant at very small distances to describe the internal structure of a fermion in terms of its ultimate constituents. Trying to find a track of this spinorial internal structure in data on UHECR would be an important theoretical, experimental and phenomenological challenge.

While many questions remain open, the ratios ∆E E -1 m and δE E -1 rest obtained for E a ≃ 10 20 eV and E a ≃ 10 19 eV clearly suggest that the internal structure of standard particles, as well as their interaction properties, can considerably evolve between low energy and the Grand Unification scale or the Planck energy scale, and strongly deviate from conventional patterns above E trans . Quantum mechanics itself can be significantly modified or even cease to be valid. In such a situation, the conventional Grand Unified and Planck scales do not necessarily make sense. Similarly, the idea that low-energy symmetries should become more and more exact as energy increases has no reason to hold above E trans [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Testing fundamental principles with high-energy cosmic rays[END_REF]. Precise particle identification can therefore be an important ingredient of UHECR experiments and phenomenology. Another difficult task will be to take into account possible deviations from standard particle dynamics when analyzing UHECR interactions with the atmosphere.

UHECR internal structure

Equations ( 1) and ( 2) were obtained [START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Absence of Greisen-Zatsepin-Kuzmin Cutoff and Stability of Unstable Particles at Very High Energy, as a consequence of Lorentz Symmetry Violation[END_REF] from an analogy with the solid state Bravais lattice writing the standard equation for phonon kinematics:

d 2 /dt 2 [φ (n)] = -K [2 φ (n) -φ (n -1) -φ (n + 1)] -ω 2 rest φ (n) (9)
where n (integer) stands for the site under consideration, φ(n) is a complex order parameter, K an elastic constant and ω rest the frequency of the chain of oscillators in the zero-momentum limit. Then:

ω 2 (k) = 2 K [1 -cos (k a)] + ω 2 rest = 4 K sin 2 (ka/2) + ω 2 rest ( 10
)
ω being the frequency, k the wave vector and a the lattice spacing. QDRK is similar to the power expansion of (10) in terms of k a for k a ≪ 1 replacing the speed of sound by the speed of light and the lattice spacing by the fundamental length scale.

Another analogy was already introduced in our 1995 papers [START_REF] Gonzalez-Mestres | Properties of a possible class of particles able to travel faster than light[END_REF][START_REF] Gonzalez-Mestres | Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light Proceedings of the TAUP 1995 Conference[END_REF] comparing standard elementary particles to condensed-matter solitons. Then, the critical speed of the ultimate constituents of matter can be much larger than that of light, just as the speed of light is much larger than that of sound. The soliton analogy was further explored in [START_REF] Gonzalez-Mestres | High-Energy Nuclear Physics with Lorentz Symmetry Violation[END_REF][START_REF] Gonzalez-Mestres | Lorentz Symmetry Violation, Vacuum and Superluminal Particles International Conference on Relativistic Physics and some of its Applications[END_REF]. Similar to this analysis, one can consider the equation:

c -2 ∂ 2 Ψ/∂t 2 -∂ 2 Ψ/∂x 2 = 2 ∆ -2 Ψ (1 -2 Ψ) (11)
Ψ being a real function of x and t (x = space coordinate, t = time coordinate) and ∆ the distance scale that characterizes the soliton size at rest (soliton radius). The soliton solution to equation ( 11) is:

Ψ(x , t) = 1 -tanh 2 (λ 0 y) (12)
where y = x -v t , v is the speed of the soliton, λ 0 = ∆ -1 γ R , D is the rest radius of the soliton and γ R the standard relativistic Lorentz factor γ R = (1 -v 2 c -2 ) -1/2 . As conventional Lorentz contraction and time dilation are altered by LSV, this analogy allows in particular to explore the possible energy dependence of the effective particle size above the relevant transition energy. Adding to equation [START_REF] Gonzalez-Mestres | Lorentz Symmetry Violation and Acceleration in Relativistic Shocks, Proceedings of the Heidelberg[END_REF] a term proportional to a 2 ∂ 4 Ψ/∂ x 4 (a = fundamental length) can be considered as the equivalent of (1) and indeed leads to a similar kinematics [START_REF] Gonzalez-Mestres | High-Energy Nuclear Physics with Lorentz Symmetry Violation[END_REF][START_REF] Gonzalez-Mestres | Lorentz Symmetry Violation, Vacuum and Superluminal Particles International Conference on Relativistic Physics and some of its Applications[END_REF].

A simplified discussion of the effect of such a deformation of equations ( 11) and ( 12) can be as follows. Applying the operator a 2 ∂ 4 /∂ x 4 to the initial solution given by ( 12) yields a result proportional to a 2 γ 4 R D -4 that becomes much larger than D -2 for a γ 2 R ≫ D . A substantial modification of ( 11) and ( 12) is then required reproducing the maximum speed lower than c of QDRK. If the existence of a soliton can be preserved at UHE, the new equation should admit two branches of solutions (above and below the energy associated to the maximum of v). LSV will become dominant at ultra-high energy and introduce a contraction law different from the relativistic one. This result suggests once more the possible existence of a sharp transition in the UHECR internal structure and interaction properties.

For a proton with E ≃ 10 20 eV (γ R ≃ 10 11 ) and a ≃ 10 -36 cm ( ≃ 10 -3 l Planck , l Planck = Planck length), one would have a γ 2 R ≃ 10 -14 cm which is already significant as compared to the rest proton radius. Taking instead a = 0.1 l Planck would lead at E ≃ 10 20 eV to a γ 2 R ≃ 10 -12 cm that is larger than the rest proton radius. Therefore, the deformation of Lorentz contraction just considered can already be present in UHECR and produce observable signatures.

Conclusion

The highest-energy cosmic rays are a crucial experimental source of fundamental information for Particle Physics and Cosmology. They require careful exploration and study within a long-term program of Earth-based and satellite experiments, with particular attention to the first interactions of the incoming cosmic rays. If unconventional new physics exists, the fate of concepts such as the standard Planck scale and the grand unified scale appears very uncertain. UHECR experiments can then provide a unique way to help particle theory and cosmological research to uncover the actual structure and origin of matter and space-time, including the search for a possible correlation between the arrival directions of UHECR and the SST privileged space direction that Planck data may have disclosed.

Observable LSV signatures may come not only from deformations of kinematics [START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF][START_REF] Gonzalez-Mestres | Absence of Greisen-Zatsepin-Kuzmin Cutoff and Stability of Unstable Particles at Very High Energy, as a consequence of Lorentz Symmetry Violation[END_REF] but also from significant changes in the UHE particle internal structure [START_REF] Gonzalez-Mestres | High-Energy Nuclear Physics with Lorentz Symmetry Violation[END_REF][START_REF] Gonzalez-Mestres | Lorentz Symmetry Violation, Vacuum and Superluminal Particles International Conference on Relativistic Physics and some of its Applications[END_REF] expected to modify the interaction properties for nucleons, leptons and photons. But elucidating the nature of the actual dynamics that generates the observed data (including a possible absence of deviations from standard physics) will be a very difficult, long-term task. In parallel, the study of potential UHECR sources should be pursued as far as possible without forgetting nonstandard mechanisms (superbradyon decays [START_REF] Gonzalez-Mestres | Superluminal Matter and High-Energy Cosmic Rays[END_REF][START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF]...).

Post Scriptum (April 25, 2014)

After this paper was sent (on February 5) to the editors, the BICEP2 collaboration [START_REF]BICEP2 Collaboration, BICEP2 II: Experiment and Three-Year Data Set[END_REF] reported [START_REF]BICEP2 I: Detection Of B-mode Polarization at Degree Angular Scales[END_REF] a B-mode polarization of the cosmic microwave background (CMB) radiation. This result is often interpreted as a strong direct evidence for cosmic inflation and primordial gravitational waves, on the grounds that the detected B-modes of CMB cannot be generated primordially by density perturbations and that only gravitational waves originating from the inflationary expansion of the Universe can have produced such a signature. However, as pointed out in [START_REF] Gonzalez-Mestres | CMB B-modes, spinorial space-time and Pre-Big Bang (I), mparc[END_REF], this argument is tacitly based on the standard Big Bang approach where inflation is in any case required to get a potentially consistent description of the Universe. Pre-Big Bang models do not require any inflationary scenario or gravitational waves to produce CMB B-modes.

In particular, the spinorial space-time we introduced in [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF][START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF] naturally generates a privileged space direction for each comoving observer [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF]. Then, contrary to the standard cosmological scenario where only CMB E-modes associated to gradients can be present except for the B-modes due to inflationary gravitational waves, a cosmology based on the spinorial space-time can easily produce B-modes through rotations around the privileged space direction and vector products by this direction. This is just a conservative illustration of the new potentialities of unconventional space-time geometries in pre-Big Bang cosmologies. Pre-Big Bang patterns [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light Proceedings of the TAUP 1995 Conference[END_REF] can describe the origin of the Universe and its large-scale structure more naturally than ad hoc models based on cosmic inflation.

Thus, the BICEP2 result, if confirmed by further experimental work, can potentially provide an experimental evidence for the spinorial space-time we suggested in 1996-97 [START_REF] Gonzalez-Mestres | Spinorial space-time and privileged space direction (I), mparc[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and Friedmann-like equations (I), mparc[END_REF] and whose existence may also confirmed by Planck data [START_REF]Planck 2013 results. XXIII. Isotropy and statistics of the CMB[END_REF].

It must be emphasized, however, that the present experimental situation concerning the BICEP2 announcement remains unclear. It is even not yet excluded that the observed signal be due to galactic dust [START_REF] Liu | Fingerprints of Galactic Loop I on the Cosmic Microwave Background[END_REF]. Therefore, no theoretical assertion can yet be made on the grounds of such data.

In pre-Big Bang cosmologies, superbradyons and the SST with two complex cosmic coordinates can be related to the ultimate structure of matter and to a more fundamental space-time beyond that of standard relativity (the four real coordinates), as already felt in part by standard fermions. Such a new early cosmology would in particular be directly based on the ultimate internal structure of quarks, leptons and gauge bosons through an updated preon model [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF].

If a privileged space direction exists according to the SST geometry, the possibility that the propagation and interactions of fermionic high-energy cosmic rays be sensitive to this cosmic space anisotropy must be seriously considered. Identifying such effects in a reliable way requires a longterm experimental, phenomenological and theoretical effort. More generally, a close study seems necessary to identify the particle properties and interactions that can undergo (or may have undergone in the past) significant modifications due to such a fundamental cosmic anisotropy. These important issues will be discussed with more detail in a forthcoming paper.