Frederic Chevy

C Salomon

Strongly correlated Bose gases

Keywords: numbers: 05, 30, Jp, 51, 30, +i

published or not. The documents may come

Introduction

In 1995, the observation of the first dilute Bose-Einstein condensate confirming Einstein's textbook prediction for an ideal gas of bosons was saluted as one the major physics breakthroughs of the end of the XXth century. Paradoxically, it was quickly recognized that cold atoms could also provide a unique experimental platform for the study of strongly correlated quantum many-body systems and that the broad range of investigation tools available to atomic physicists could be harnessed to tackle some of the most complex open problems in modern physics [START_REF] Bloch | Many-body physics with ultracold gases[END_REF]. Following this quantum simulation program, several paradigms of condensed matter were implemented experimentally and tested quantitatively with great accuracy, from the Mott superfluid-insulator transition [START_REF] Greiner | Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms[END_REF] to the BEC-BCS crossover that connects within a unique framework Bose-Einstein Condensation (BEC) and Bardeen-Cooper-Schrieffer's (BCS) theory for superconductivity [START_REF] Zwerger | The BCS-BEC Crossover and the Unitary Fermi Gas[END_REF].

One of the most intriguing and most investigated systems in the past few years is the spin-1/2 Fermi gas with infinite scattering length, the so-called unitary Fermi gas [START_REF] Zwerger | The BCS-BEC Crossover and the Unitary Fermi Gas[END_REF]. In the unitary regime, the two-body scattering cross-section reaches the maximum value permitted by quantum mechanics, leading to strong many-body correlations challenging the most advanced theoretical methods. By contrast the macroscopic properties of this system are characterized by simple scale-invariance properties. For instance, at zero temperature, the chemical potential of the unitary Fermi gas µ is related to the density n by

µ = ξ 2 2m (6π 2 n) 2/3 = ξE F , (1)
where E F is the Fermi energy of an ideal gas and ξ 0.38 is a dimensionless number, the Bertsch parameter, that has been determined both experimentally and theoretically. Despite its apparently simple and fundamental nature, the strongly interacting Bose gas in free space is still a subject of both experimental [START_REF] Bs Rem | Lifetime of the Bose gas with resonant interactions[END_REF][START_REF] Richard | Stability of a unitary Bose gas[END_REF][START_REF] Makotyn | Universal dynamics of a degenerate unitary Bose gas[END_REF][START_REF] Eismann | Universal loss dynamics in a unitary Bose gas[END_REF][START_REF] Chang | Momentumresolved observation of quantum depletion in an interacting Bose gas[END_REF] and theoretical [START_REF] Basu | Stability of bosonic atomic and molecular condensates near a feshbach resonance[END_REF][START_REF] Zhou | Bose gases near resonance: Renormalized interactions in a condensate[END_REF][START_REF] Shao-Jian | Universal Bose gases near resonance: A rigorous solution[END_REF][START_REF] Sykes | Quenching to unitarity: Quantum dynamics in a three-dimensional Bose gas[END_REF][START_REF] Piatecki | Efimov-driven phase transitions of the unitary Bose gas[END_REF][START_REF] Liu | Equation of state and contact of a strongly interacting Bose gas in the normal state[END_REF][START_REF] Shao-Jian | Long-lived universal resonant Bose gases[END_REF] research that bridges the gap between the physics of liquid helium and ultracold vapours. The question of the existence, the stability and the scaling properties of an hypothetic unitary Bose gas is still open and this article reviews recent progress towards the experimental realization of such a system.

Mean-field Bose-Einstein condensates and beyond mean-field corrections

Thanks to the low density of ultracold vapours, the first experiments on alkali Bose-Einstein condensates could be described accurately using the mean-field approximation, where one assumes that all bosons occupy the same quantum state characterized by a macroscopic wave-function φ(r, t). For an ensemble of N spin-polarized bosons of mass m confined in an external potential V (r), this wave-function is solution of the Gross-Pitaevskii Equation

i ∂ t φ = - 2 2m ∇ 2 φ + V (r)φ + N 4π 2 a m |φ| 2 φ. ( 2
)
where a is the scattering length describing low-energy scattering properties of the atoms [START_REF] Gross | Structure of a quantized vortex in boson systems[END_REF][START_REF] Lp Pitaevskii | Vortex lines in an imperfect Bose gas[END_REF]. This equation was successful at describing the results of all early experiments [START_REF] Dalfovo | Theory of Bose-Einstein condensation in trapped gases[END_REF] from the density profile of the cloud for a > 0 [START_REF] Edwards | Numerical solution of the nonlinear schrödinger equation for small samples of trapped neutral atoms[END_REF][START_REF] Dalfovo | Bosons in anisotropic traps: Ground state and vortices[END_REF] and its instability for a < 0 [START_REF] Bradley | Bose-Einstein condensation of lithium: Observation of limited condensate number[END_REF], to the low-energy Bogoliubov spectrum [START_REF] Steinhauer | Bragg spectroscopy of the multibranch bogoliubov spectrum of elongated Bose-Einstein condensates[END_REF] or the properties of solitons [START_REF] Burger | Dark solitons in Bose-Einstein condensates[END_REF][START_REF] Denschlag | Generating solitons by phase engineering of a Bose-Einstein condensate[END_REF][START_REF] Khaykovich | Formation of a matter-wave bright soliton[END_REF][START_REF] Strecker | Formation and propagation of matter-wave soliton trains[END_REF] and vortices [START_REF] Michael | Vortices in a Bose-Einstein condensate[END_REF][START_REF] Kw Madison | Vortex formation in a stirred Bose-Einstein condensate[END_REF][START_REF] Jr Abo-Shaeer | Observation of vortex lattices in Bose-Einstein condensates[END_REF]. The mean-field approximation is valid only at low temperature and in the dilute limit where the density n of bosons satisfies the condition na 3 1. When temperature or interactions are increased, thermal and quantum fluctuations of the low-energy Bogoliubov modes deplete the condensate and lead to a modification of the equation of state of the BEC [START_REF] Proukakis | Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics[END_REF]. At zero-temperature, the first beyond-mean field corrections were calculated in the 50's by Lee, Huang and Yang who showed that for a Bose-Einstein condensate with positive scattering length the zero-temperature equation of state relating the chemical potential µ to the density n could be expanded as

µ(n) = gn 1 + 32 3 √ π √ na 3 + ... . (3)
with g = 4π 2 a/m [START_REF] Lee | Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties[END_REF][START_REF] Lee | Many-body problem in quantum mechanics and quantum statistical mechanics[END_REF]. The leading order term µ = gn corresponds to the mean-field contribution and can be derived from Eq. ( 2) using the uniform solution φ(r, t) = n/N exp(-iµt/ ). The Lee-Huang-Yang beyond mean-field correction follow a √ na 3 scaling, similar to that of the quantum depletion. In ultracold atoms, the value of the scattering length can be modified arbitrarily using an external magnetic field, owing to the existence of Feshbach resonances [START_REF] Courteille | Observation of a feshbach resonance in cold atom scattering[END_REF][START_REF] Chin | Feshbach resonances in ultracold gases[END_REF]. Moreover, since Eq. ( 3) depends only on the value of the scattering length, and not the actual shape of the interparticle potential, its validity can be tested on any bosonic species. The first quantitative test of the Lee-Huang-Yang correction was obtained using composite bosons made of pairs of atomic fermions in the molecular side of the BEC-BCS crossover [START_REF] Altmeyer | Precision Measurements of Collective Oscillations in the BEC-BCS Crossover[END_REF][START_REF] Navon | The equation of state of a low-temperature fermi gas with tunable interactions[END_REF]. As shown in [START_REF] Leyronas | Superfluid equation of state of dilute composite bosons[END_REF], Eq. ( 3) should still apply to this system, replacing the scattering length by the dimer-dimer scattering length a dd = 0.6a [START_REF] Petrov | Weakly bound dimers of fermionic atoms[END_REF][START_REF] Brodsky | Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles[END_REF]. In [START_REF] Altmeyer | Precision Measurements of Collective Oscillations in the BEC-BCS Crossover[END_REF], the Lee-Huang-Yang correction was tested by the study of the breathing mode of an elongated fermionic superfluid, under the assumption of an hydrodynamics flow [START_REF] Ge Astrakharchik | Equation of state and collective frequencies of a trapped fermi gas along the bec-unitarity crossover[END_REF]. By contrast, a more straightforward thermodynamical method was proposed in [START_REF] Ho | Obtaining phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases[END_REF] and implemented in [START_REF] Navon | The equation of state of a low-temperature fermi gas with tunable interactions[END_REF] to extract the equation of state of an homogeneous system from the analysis of density profile in a harmonic trap. This scheme is based on the Local Density Approximation, where the global chemical potential µ 0 satisfies at all positions the thermodynamical equilibrium condition µ 0 = µ(n(r))+V (r). Assuming a harmonic potential and using the constant-temperature Gibbs-Duhem Relation n = ∂ µ P , one readily gets that the doubly-integrated density distribution n(z) = dxdyn(x, y, z) is given by

P (µ 0 -mω 2 z z 2 /2, T ) = mω 2 ⊥ 2π n(z), (4)
where ω z and ω2 ⊥ = ω x ω y are the axial and mean transverse-trapping frequencies. According to this equation, measuring the doubly-integrated density profile is thus equivalent to measuring the grand-canonical equation of state P (µ, T ). This scheme, and similar ones based on thermodynamical identities, yield a high-precision, model-free experimental determination of the equation of state of the system and was used first to study properties of strongly correlated fermionic systems, both spin-balanced and imbalanced, at zero and finite temperature [START_REF] Nascimbène | The equation of state of ultracold Bose and Fermi gases: a few examples[END_REF][START_REF] Ku | Revealing the superfluid lambda transition in the universal thermodynamics of a unitary fermi gas[END_REF]. It was then applied to bosonic 7 Li atoms for which the Lee-Huang-Yang corrections were confirmed with a 10% accuracy [START_REF] Navon | Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas[END_REF]. Combining the equation of state for composite and atomic bosons (Fig. 1) shows that the Lee-Huang-Yang corrections are indeed universal and depend only on the scattering length and not on the internal structure of the particles.

An independent test of the Lee-Huang-Yang correction was provided by the analysis of the response of a cloud of bosons to radiofrequency (rf) pulses [START_REF] Wild | Measurements of tans contact in an atomic Bose-Einstein condensate[END_REF]. The experiment was based on Tan's contact parameter C 2 [START_REF] Tan | Energetics of a strongly correlated fermi gas[END_REF][START_REF] Braaten | Universal relations for identical bosons from three-body physics[END_REF], a physical quantity relating shortrange correlations to thermodynamical quantities for systems with contact interactions. For such systems, the momentum distribution n(k) scales as 1/k 4 for large k and the twobody contact is simply defined as C 2 = lim k→∞ k 4 n(k). By definition, C 2 characterizes the short range physics the system. The contact can be measured by a time of flight in the absence of interaction [START_REF] Chang | Momentumresolved observation of quantum depletion in an interacting Bose gas[END_REF], but can also be obtained by the response Γ(ω) of the cloud 7 Li (green squares) and of a gas of composite dimers of 6 Li (red discs). Here h S = P/2P 0 is the pressure of the cloud normalized to the pressure P 0 of an ideal Fermi gas and the interaction parameter δ is defined as δ = / √ 2mμa, where μ is the chemical potential shifted by the two-body bound-state energy for the fermions. Picture from [START_REF] Navon | Thermodynamics of ultracold Bose and Fermi gases[END_REF].

to a radio-frequency excitation at frequency ω. For large detuning, we have indeed

Γ(ω) C 2 4π 2 ω 3/2 m . (5)
One of Tan's remarkable results is that, while up to now C 2 has been restricted only to the high energy/high momentum behaviour of the system, it can be also related to its static and macroscopic properties owing to the so-called adiabatic sweep theorem which states that

dF d1/a = - 2 C 2 4πm , ( 6
)
where F is the free energy of the cloud. Combined to Eq. ( 5), this expression shows that the equation of state of the cloud can be obtained from the high-frequency tail of its response to high frequency rf pulses. This scheme was applied in [START_REF] Wild | Measurements of tans contact in an atomic Bose-Einstein condensate[END_REF] to the case of a strongly interacting Bose gas. This measurement demonstrated a significant departure from the mean-field prediction that could be attributed to the Lee-Huang-Yang corrections. Finally, beyond-mean field physics was demonstrated in dipolar gases where the stability of superfluid droplets can only be understood by the role of quantum fluctuation that modify the equation of state of the superfluid [START_REF] Ferrier-Barbut | Observation of quantum droplets in a strongly dipolar Bose gas[END_REF].

The previous discussion was restricted to the low temperature properties of the Bose gas but as aforementioned, the condensate can also be depleted by thermal fluctuations. At finite temperature, Ginzburg's argument predicts that for a threedimensional superfluid, thermal fluctuations dominate close to the critical temperature T c , leading to a breakdown of the mean-field approximation, even in the dilute limit. For a homogeneous system, interactions treated within the mean-field approximation just create a global shift of the single particle spectrum and as such do not modify the value of the critical temperature for a given atomic density. To leading order, the shift ∆T c of the critical temperature is therefore only due to non-perturbative effects. Its value, and even its sign, have been the subject of a longstanding debate [START_REF] Baym | Bose-Einstein transition in a dilute interacting gas[END_REF][START_REF] Arnold | Bec transition temperature of a dilute homogeneous imperfect Bose gas[END_REF][START_REF] Va Kashurnikov | Critical temperature shift in weakly interacting Bose gas[END_REF][START_REF] Holzmann | Bose-Einstein transition temperature in a dilute repulsive gas[END_REF][START_REF] Andersen | Theory of the weakly interacting Bose gas[END_REF] but it is now thought to be given by ∆T c

T (0) c 1.3 3 √ na 3 (7)
However, most experiments so far are performed in a trap that modifies significantly this scenario. Firstly, the inhomogeneity of the density profile confines the critical fluctuations to a narrow region of the cloud and reduces their global influence. Second, within LDA, the critical temperature is given to leading order by the same condition n(r = 0)λ 3 th = ζ(3/2) that one would obtain for a homogeneous Bose gas of density n(0) and where ζ is Riemann's function and λ th = /2πmk B T is the thermal wavelength. Since mean-field interactions modify the cloud profile, it means that, contrary to the homogeneous system where the density is fixed, the critical temperature of a trapped gas is shifted even at the mean-field level by

∆T (MF) c T (0) c -1.33N 1/6 a a ho , (8)
with a ho = /mω is the typical extension of the ground-state wavefunction in a harmonic potential of frequency ω [START_REF] Giorgini | Condensate fraction and critical temperature of a trapped interacting Bose gas[END_REF]. Eq. ( 8) was quantitatively tested in [START_REF] Gerbier | Critical temperature of a trapped, weakly interacting Bose gas[END_REF][START_REF] Meppelink | Thermodynamics of Bose-Einstein-condensed clouds using phase-contrast imaging[END_REF] in a regime of weak interactions where the mean-field shift amounts to a few percent of the critical temperature of the ideal gas. Using a Feshbach resonance in 39 K it was possible to extend the parameter range accessible to experiments, and the observation of a beyond mean-field corrections of T c and on the thermal fraction were reported in [START_REF] Robert | Effects of interactions on the critical temperature of a trapped Bose gas[END_REF][START_REF] Robert P Smith | Condensed fraction of an atomic Bose gas induced by critical correlations[END_REF]. More recently, the breakdown of the mean-field approximation close to the critical temperature was demonstrated in the dynamics of a uniform gas of Bosons after a temperature quench across the phase transition and the characterization of the exponents of the Kibble-Zurek mechanism of topological defect formation [START_REF] Navon | Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas[END_REF].

Non-universality and Efimov physics

Going one step further in the expansion of the chemical potential with the dilution parameter na 3 yields the following result [START_REF] Hugenholtz | Ground-state energy and excitation spectrum of a system of interacting bosons[END_REF][START_REF] Wu | Ground state of a Bose system of hard spheres[END_REF][START_REF] Braaten | Quantum corrections to the energy density of a homogeneous Bose gas[END_REF]

µ(n) = gn 1 + 32 3 √ π √ na 3 + 4(4π -3 √ 3) ln(na 3 ) + C na 3 + ... (9)
The numerical constant C entering Eq. ( 9) is non-universal and depends on the shape of the interatomic potential, more specifically on three-body interactions between bosons.

It can indeed be shown that the three-body problem for bosons with contact interactions is incomplete and that its mathematical consistency requires the introduction of an adhoc boundary condition when the three particles get close to each other. The hermiticity of the contact Hamiltonian therefore constrains the three-body wavefunction to obey the following boundary condition [START_REF] Gs Danilov | On the three-body problem with short-range forces[END_REF]

ψ(R) 1 R 5/2 e is 0 ln(R/R * ) + e -is 0 ln(R/R * ) , ( 10
)
where R is the relative hyper-radius of the three particles, R * is the so-called three-body parameter characterizing the dephasing of the wave-funtion after a three body scattering event and s 0 1.00624 is solution of the transcendental equation

s 0 cosh(πs 0 /2) = 8 √ 3 sinh(πs 0 /6). (11)
This boundary condition is invariant under the scaling transformation group R * → R * λ k , with k an arbitrary integer number and λ = exp(π/s 0 ) 22.7. As a consequence, all physical quantities involving the three-body parameter are a logperiodic function of R * . In the special case of the parameter C appearing in Eq. ( 9), the amplitude of these oscillations is quite small, yielding an actual value of C fairly independent of the three-body interactions, with C 7 [START_REF] Braaten | Nonuniversal effects in the homogeneous Bose gas[END_REF][START_REF] Braaten | Dilute Bose-Einstein condensate with large scattering length[END_REF].

A dramatic consequence of the scaling invariance associated with the three-body boundary condition [START_REF] Zhou | Bose gases near resonance: Renormalized interactions in a condensate[END_REF] is the existence of a universal family of bound states when the scattering length is brought to infinity. Indeed, from a general dimensional argument, the energy of a three-body bound state takes the form

E n = - 2 mR * 2 f n (R * /a) (12)
Using the scaling invariance, the function f n obeys the scaling

f n (x) = f 0 (x/λ n )/λ 2n , (13)
with x = R * /a. The generic energy spectrum of the trimers is displayed in Fig. 2. The end points of each energy branch correspond to their intersection a n with the single body continuum on the negative a side of the Feshbach resonance, and in a n with the molecular state on the positive side. According to the general scaling [START_REF] Piatecki | Efimov-driven phase transitions of the unitary Bose gas[END_REF], the value a n and a n both follow a geometric progression with the same scaling factor λ.

Let's consider finally the case of an infinite scattering length system. In this case, we have R * /a = 0, hence the scaling f n (0) = f 0 (0)/λ 2n . Defining c = f 0 (0), we readily obtain the geometric spectrum [START_REF] Efimov | Energy levels arising from resonant two-body forces in a three-body system[END_REF]

E n (a = ∞) = -c 2 2mR * 2 λ 2n (14)
This infinite series of bound state was predicted first by V. Efimov for nucleons. One remarkable feature of these three-body bound states is that they exist in parameter regimes where the two-body problem does not present any bound state. For this reason,

2 *2 [ / ] E mR  * / R a * / n R a * / ' n R a
Figure 2. Sketch of the Efimov spectrum. The red (blue) solid lines represent the energy of the Efimov trimers (molecular two-body bound state). The shaded area corresponds to the regions where Efimovian physics no longer applies. For large |a| the isolated trimer approximation is no longer valid, while for small |a| the zero range approximation breaks-down.

Efimov trimers are sometimes dubbed Borromean states by analogy with the Borromean Rings represented in the coat-of-arms of italian Borromeo family.

In practice the number of observable Efimov resonances is energetically limited both from above and below. At short scale, Efimov scenario is valid only as long as the zero-range approximation is valid and breaks down when their size becomes comparable with the range r e of the interatomic potential. On the other hand, when the size of the trimers become comparable with the interparticle distance , the surrounding bosonic gas cannot be neglected anymore and the overlap of the trimer wavefunction with the background atoms has to be included. Since the trimer size expands geometrically, the maximum number of observable trimers is ln( /r e )/ ln(λ). Taking 1 µm and r e 1 nm, we obtain n 3.

Evidence for Efimov physics was revealed by the analysis of three-body losses close to a Feshbach resonance. Ultracold vapours are indeed metastable and when three atoms get close to each other, they can form deeply bound dimers. The released energy being in general much higher than the trap depth, such a three-body recombination event expels the three particles out of the trap. These losses are modeled by a phenomenological loss rate L 3 such that the trapped-atom number obeys the evolution equation

Ṅ = -L 3 n 2 N, (15)
where • represents the spatial average over the density profile of the cloud. Three-body recombination can be incorporated into the microscopic description by adding to Eq. 10 a loss term to the pure dephasing of the three-body wavefunction characterized by R * .

We note e 2η the probability for three atoms to recombine at short distance (for alkali, η 0.1). Taking η as a new physical parameter, we see that on a dimensional ground, L 3 must follow the scaling

L 3 = a 4 m G η (a/R * ), (16)
where as before G η is a log-periodic function of a/R * . The general trend is therefore given by the a 4 behaviour of the prefactor [START_REF] Weber | Three-body recombination at large scattering lengths in an ultracold atomic gas[END_REF] and leads to a dramatic reduction of the lifetime of a Bose gas close to a Feshbach resonance, as reported in [START_REF] Inouye | Observation of feshbach resonances in a Bose-Einstein condensate[END_REF][START_REF] Jacob L Roberts | Magnetic field dependence of ultracold inelastic collisions near a feshbach resonance[END_REF]. On the other hand, Efimov physics is revealed by the modulation around the a 4 -scaling.

The first evidence of Efimov resonances on 133 Cs was presented in [START_REF] Kraemer | Evidence for Efimov quantum states in an ultracold gas of caesium atoms[END_REF][START_REF] Ferlaino | Efimov resonances in ultracold quantum gases[END_REF] where the end points of the energy spectra of the Efimov trimers correspond to enhancement and suppression of three-body losses on the negative and positive sides of the Feshbach resonances respectively (see Fig. 3). This seminal experiment was later on extended to other atomic species and the scaling factor λ 22.7 was tested owing to the observation of multiple resonances in [START_REF] Gross | Observation of universality in ultracold li 7 three-body recombination[END_REF][START_REF] Scott E Pollack | Universality in three-and four-body bound states of ultracold atoms[END_REF][START_REF] Zaccanti | Observation of an efimov spectrum in an atomic system[END_REF].

The universal scaling L 3 ∝ a 4 /m sets a fundamental constraint on the prospect of reaching a strongly correlated regime with bosons. Indeed, a BEC being a metastable object, equilibrium statistical physics can be used for its description only if the lifetime of the cloud is larger than the characteristic equilibration time of the system. For a Bose-Einstein condensate, the time scale of the microscopic evolution is given by the chemical potential. The metastability criterion is thus L 3 n 2 µ/ , hence na 3 1. In other words, a low-temperature Bose gas can be considered at thermal equilibrium only in the dilute limit.

Metastability of the Finite Temperature Unitary Bose Gas

Eq. ( 16) predicts that on resonance, the three-body loss rate should become infinite. This paradox is lifted by noting that the previous discussion neglects temperature and is only valid for very cold samples. By contrast, a finite temperature T introduces a new length-scale λ th = 2π 2 /mk B T which modifies the dimensional argument leading to Eq. ( 16). Assuming that the loss rate remains indeed finite at unitarity, L 3 should therefore scale as [START_REF] D'incao | Limits on universality in ultracold three-boson recombination[END_REF][START_REF] Jp D'incao | The short-range three-body phase and other issues impacting the observation of efimov physics in ultracold quantum gases[END_REF][START_REF] Li | Bose gases near unitarity[END_REF]]

L 3 = λ 4 th m H η (λ th /R * ), ( 17
)
where as before H η is a log-periodic function of λ th /R * . Eq. ( 17) was derived rigorously from the resolution of the three-body problem with contact interactions in [START_REF] Werner | General relations for quantum gases in two and three dimensions. ii. bosons and mixtures[END_REF][START_REF] Bs Rem | Lifetime of the Bose gas with resonant interactions[END_REF]. In the case of identical bosons, the oscillation amplitude of H η is negligible and one obtains the approximate result

L 3 5 m 3 36 √ 3π 2 1 -e -4η (k B T ) 2 . ( 18
)
We see that the three-body loss rate follows a universal 1/T 2 behaviour, with a prefactor set by the loss parameter η. This scaling has been verified experimentally for 7 Li, 39 K Figure 3. Three-body losses in 7 Li (data from [START_REF] Gross | Observation of universality in ultracold li 7 three-body recombination[END_REF]). Due to Efimovian physics, losses are enhanced (a > 0) and reduced (a < 0) with respect to the a 4 scaling -dashed line.)

and 133 Cs [START_REF] Bs Rem | Lifetime of the Bose gas with resonant interactions[END_REF][START_REF] Richard | Stability of a unitary Bose gas[END_REF][START_REF] Eismann | Universal loss dynamics in a unitary Bose gas[END_REF], see Fig. 4. According to this scaling, the unitary Bose gas is stable at high temperature where L 3 vanishes. As before, the metastability region is obtained by the comparison between the loss rate L 3 n 2 and the elastic equilibration rate γ el . At high temperature, the cloud is classical and the relaxation time is γ el ∝ nσv, where σ is the scattering cross-section and v ∝ k B T /m ∝ /mλ th is the typical thermal velocity of the cloud. From the comparison between these two scalings, we see that a unitary Bose gas is stable when the phase space density nλ 3 th is small. In other words, many-body losses dominate in the quantum degenerate regime.

The Unitary Bose Gas

The (meta)-stability diagram of the Bose gas is summarized in Fig. 5. It shows that the existence of the unstable region at low temperature and large scattering length is to some extend a fundamental property of bosonic systems emerging from three-body physics.

The existence of an unstable regime does not necessarily precludes the possibility of observing experimentally a strongly correlated Bose gas. Firstly, the size of the unstable region depends on η and shrinks for vanishingly small values of the three-body loss parameter. Second, when a and λ th become very large, Eq. ( 16) and ( 17) both predict an unphysical, infinitely large loss rate. It has been conjectured that, as before, the paradox is cured by noting that in this regime, the only remaining physical length scale is the interparticle distance, yielding to the scaling

L 3 mn 4/3 K η (R * n 1/3 ). ( 19
)
The density dependence of the loss-rate means that Ṅ does not scale as n 2 , but rather as n 2/3 . This modification of the scaling exponent is the signature that threebody correlations are no longer sufficient to describe molecular recombination and that

m 3 T in 2 in m H 3 ΜK 2 L 3 in cm 6 s Figure 4
. Three body loss rate for two Bose gases. Data from [START_REF] Eismann | Universal loss dynamics in a unitary Bose gas[END_REF]. Red (blue) points correspond to measurements performed on 7 Li ( 133 Cs). The red and blue solid lines correspond to Eq. ( 18) with η = 0.21 and η = 0.098 respectively. m H stands for the hydrogen mass and the dashed line correspond to the maximum value for the three-body loss rate obtained for η = ∞.

higher-order correlations are required. A consequence of Eq. ( 19) is that in the saturated regime, the inelastic vs elastic collision rate ratio is only a function of η and does not depend on temperature or density. This conjecture paves the way to the experimental study of a strongly correlated Bose gas since for small values of η this ratio may become small enough to allow for the quasi-thermalization of the cloud before it recombines towards deeply bound molecular states. The properties of this state have been the subject of several theoretical investigations. The role of molecular state and the transition between atomic and molecular BECs have been considered in [START_REF] Basu | Stability of bosonic atomic and molecular condensates near a feshbach resonance[END_REF][START_REF] Zhou | Bose gases near resonance: Renormalized interactions in a condensate[END_REF], universality and the role of Efimovian physics was discussed in dimension 4- [START_REF] Shao-Jian | Universal Bose gases near resonance: A rigorous solution[END_REF]; large N -expansion was performed in [START_REF] Liu | Equation of state and contact of a strongly interacting Bose gas in the normal state[END_REF]; Efimovian liquids have been predicted in Monte-Carlo simulations [START_REF] Piatecki | Efimov-driven phase transitions of the unitary Bose gas[END_REF].

Several experiments have tried to probe quantitatively the properties of this hypothetical unitary Bose gas. Since in real systems inelastic losses are always present, these measurements are always performed after a magnetic field ramp to resonance, starting from a value of the scattering length where the cloud is stable and well thermalized. This scheme leads to an experimental dilemma since one must compromise between adiabaticity that requires a slow ramp and three-body recombination that will be negligible only for fast ramps.

A first set of experiments was based on the measurement of the radius of a 7 Li cloud during a "slow" magnetic field ramp to favor adiabaticity at the expense of a large atom Metastability of an attractive Bose gas. Thermalisation rate is faster than three-body recombination as long as the gas is dilute or is not quantum degenerate. In this regime, a quasi-equilibrium state can be defined and thermodynamical properties of the system can be obtained using equilibrium statistical physics. The blue area corresponds to the unstable region where three-body recombination prevents thermalization. Note that the frontier between the metastable and unstable regimes depends on the three-body loss parameter η. loss during the ramp (about half the atoms are lost during the transfer to resonance) [START_REF] Navon | Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas[END_REF]. Starting from a weakly interacting BEC at almost zero-temperature, and assuming that a well defined many-body state exists from low and positive a to resonance, the equation of state at resonance should be given by

µ(n) = 2 2m (6π 2 n) 2/3 ξ(n 1/3 R * ), (20)
where the coefficient (6π 2 ) was introduced in analogy to the chemical potential of an ideal Fermi gas and ξ is as usual a log-periodic function. Assuming that i) increasing the value of a only inflates the size of the cloud and ii) due to the finite duration of the ramp the actual radius of the cloud is smaller than its equilibrium value, the measurement presented in [START_REF] Navon | Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas[END_REF] provides a lower bound for ξ. The experimental bound ξ ≥ 0.44( 8) is compatible with previous theoretical estimates ξ = 0.66 [START_REF] Lee | Universality and stability for a dilute Bose gas with a feshbach resonance[END_REF] and the variational bound ξ ≤ 0.80 [START_REF] Song | Ground state properties of cold bosonic atoms at large scattering lengths[END_REF], or ξ ≤ 2.93 [START_REF] Cowell | Cold Bose gases with large scattering lengths[END_REF]. More recently, [START_REF] Makotyn | Universal dynamics of a degenerate unitary Bose gas[END_REF] reported a measurement of the momentum distribution of a resonant Bose gas after a variable waiting time at resonance. In this second set of experiment, the ramp duration is very short. Compared to [START_REF] Navon | Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas[END_REF], losses are completely negligible during the ramp. The density profile of the cloud at arrival on resonance is identical to that of the initial weakly interacting BEC and is not affected by the modification of the scattering length. The main results of this work are i) the lifetime of the cloud is much larger than one would expect from Eq. ( 18) for the initial temperature of the BEC ii) the momentum distribution equilibrates on a time-scale shorter than the measured three-body recombination time and iii) the quasi-equilibrium momentum distribution is a universal function of k/k n , with k n ∝ (6π 2 n) 1/3 . The interpretation of this experiment is still debated and the role of the fast ramp has been discussed in several articles [START_REF] Sykes | Quenching to unitarity: Quantum dynamics in a three-dimensional Bose gas[END_REF][START_REF] Rançon | Equilibrating dynamics in quenched Bose gases: characterizing multiple time regimes[END_REF][START_REF] Kain | Nonequilibrium states of a quenched Bose gas[END_REF][START_REF] John | Bound-state signatures in quenched Bose-Einstein condensates[END_REF][START_REF] Ancilotto | Quenched dynamics of the momentum distribution of the unitary Bose gas[END_REF][START_REF] Kira | Coherent quantum depletion of an interacting atom condensate[END_REF]. Indeed since the ramp is very fast, energy is pumped into the system and the distance of the final state from the actual hypothetical universal ground-state is unknown. Assuming that it is quite close then (i) would be the signature of the saturation of the lifetime conjectured above, allowing for a metastability timewindow. Assuming that, as often, the modulation of the momentum density associated with the three-body parameter is weak, (iii) is thus a demonstration of the universal scaling characterizing unitary systems. Alternatively, if the energy transferred by the ramp is large, then the final state is thermal. Assuming that initial interactions can be neglected, the final temperature then follows a scaling T T n where k B T n = 2 k 2 n /2m is defined in analogy to the Fermi temperature of an ideal Fermi gas. In this case, virial expansion including two-body [START_REF] Laurent | Momentum distribution of a dilute unitary Bose gas with three-body losses[END_REF] as well as three-body [START_REF] Castin | Le troisième coefficient du viriel du gaz de Bose unitaire[END_REF][START_REF] Barth | Efimov correlations in strongly interacting Bose gases[END_REF] correlations have been proposed, and they suggest that the final fugacity of the cloud is z 0.4.

Non-universal feature can nevertheless appear as a consequence of Efimovian physics. The three-body parameter indeed implies the existence of a three body contact defined as [START_REF] Braaten | Universal relations for identical bosons from three-body physics[END_REF]

C 3 = mR * 2 2 ∂F ∂R * . (21)
C 3 provides the next-to-leading order contribution to the momentum distribution that now scales as

n(k) C 2 k 4 + C 3 k 5 Λ(R * k) + ... ( 22
)
where Λ is log-periodic function that breaks the universal k/k F dependence of n(k) at unitarity and low temperature [START_REF] Braaten | Universal relations for identical bosons from three-body physics[END_REF]. Assuming that the measured momentum distribution reaches a regime where this asymptotic behavior is valid, C 2 and C 3 can be obtained from experimental data [START_REF] Smith | Two-body and three-body contacts for identical bosons near unitarity[END_REF].

Conclusion: open questions and prospects

Even if it is stable against small perturbations, the Bose-Einstein condensate with positive (and large) scattering length is intrinsically metastable due to the existence of universal molecular states. As a consequence, this system belongs to the broader class of the so-called "upper-branch physics", where the lower branch refers to the dimer states. Another member of this class of physical systems is the repulsive Fermi gas, for which the existence of Stoner's instability towards an itinerant ferromagnetic state is still an open question [START_REF] Stoner | Atomic moments in ferromagnetic metals and alloys with nonferromagnetic elements[END_REF]. Just like repulsive Bose systems, its experimental investigation is strongly affected by recombination towards the molecular states of the "lower branch" [START_REF] Jo | Itinerant ferromagnetism in a Fermi gas of ultracold atoms[END_REF][START_REF] Pekker | Competition between pairing and ferromagnetic instabilities in ultracold fermi gases near feshbach resonances[END_REF][START_REF] Sanner | Correlations and pair formation in a repulsively interacting fermi gas[END_REF][START_REF] Valtolina | Evidence for ferromagnetic instability in a repulsive Fermi gas of ultracold atoms[END_REF].

Compared to spin 1/2 fermions where Pauli Principle prevents three or more particles to be at the same location, the physics of strongly correlated bosons is strongly influenced by three-body phenomena and Efimov physics that enrich the phase diagram of these systems [START_REF] Piatecki | Efimov-driven phase transitions of the unitary Bose gas[END_REF]. However, the existence of enhanced three-body losses towards deeply-bound, non-universal molecular states remains a major experimental challenge to the understanding of this system. A promising path to get quantitative insights is radio-frequency spectroscopy which has proven a powerful tool to address unstable many-body systems, as for instance the Fermi [START_REF] Kohstall | Metastability and coherence of repulsive polarons in a strongly interacting fermi mixture[END_REF] and Bose polarons [START_REF] Hu | Bose polarons in the strongly interacting regime[END_REF][START_REF] Nils B Jørgensen | Observation of attractive and repulsive polarons in a Bose-Einstein condensate[END_REF]. A measurement of C 3 based on Ramsey spectroscopy has recently been presented in [START_REF] Fletcher | Two and three-body contacts in the unitary Bose gas[END_REF].

To alleviate the effect of recombination losses, this experiment was performed at high temperature on the experimental point of view, but it nevertheless paves the way to further spectroscopic studies in the quantum degenerate regime. Furthermore the recent development of box potentials where the density inhomogeneity caused by the trap is absent [START_REF] Alexander L Gaunt | Bose-Einstein condensation of atoms in a uniform potential[END_REF] and the three-body decay rate is homogeneous across the sample should enable direct measurements of the Virial coefficients of the unitary Bose gas [START_REF] Castin | Third virial coefficient of the unitary Bose gas[END_REF][START_REF] Barth | Efimov correlations in strongly interacting Bose gases[END_REF].

It is also possible to consider stabilizing the system by tailoring a modified dispersion relation for the kinetic energy [START_REF] Shao-Jian | Long-lived universal resonant Bose gases[END_REF], or by preventing atoms to get close to each other by extending to three-body recombination [START_REF] Jose | Optical control of efimov state properties via blue-shielding[END_REF] laser induced blue-shielding demonstrated for two-body interaction [START_REF] Suominen | Optical shielding of cold collisions[END_REF][START_REF] Weiner | Experiments and theory in cold and ultracold collisions[END_REF][START_REF] Av Gorshkov | Suppression of inelastic collisions between polar molecules with a repulsive shield[END_REF]. Another intriguing, and rather unexplored, consequence of strong interactions in bosonic systems is the existence of a back-bending in the dispersion relation of the Bogoliubov modes, that leads to the so-called roton minimum proposed by Landau to explain the anomalously small critical velocity in superfluid Helium 4. Evidence for a modification of the dispersion relation in the strongly interacting regime was reported in [START_REF] Papp | Bragg spectroscopy of a strongly interacting 85 Rb Bose-Einstein condensate[END_REF] but a comprehensive and quantitative understanding of this problem is still missing both theoretically and experimentally [START_REF] Jj Kinnunen | Bragg spectroscopy of a strongly interacting Bose-Einstein condensate[END_REF][START_REF] Shai | The dispersion relation of a Bose gas in the intermediate-and high-momentum regimes[END_REF][START_REF] Sahlberg | Dynamic effects of a feshbach resonance on Bragg scattering from a Bose-Einstein condensate[END_REF].

Figure 1 .

 1 Figure1. Universality of the Lee-Huang-Yang corrections: Dimensionless groundstate equation of state P (µ) of an atomic Bose gas of7 Li (green squares) and of a gas of composite dimers of6 Li (red discs). Here h S = P/2P 0 is the pressure of the cloud normalized to the pressure P 0 of an ideal Fermi gas and the interaction parameter δ is defined as δ = / √ 2mμa, where μ is the chemical potential shifted by the two-body bound-state energy for the fermions. Picture from[START_REF] Navon | Thermodynamics of ultracold Bose and Fermi gases[END_REF].

  Figure 5. Metastability of an attractive Bose gas. Thermalisation rate is faster than three-body recombination as long as the gas is dilute or is not quantum degenerate. In this regime, a quasi-equilibrium state can be defined and thermodynamical properties of the system can be obtained using equilibrium statistical physics. The blue area corresponds to the unstable region where three-body recombination prevents thermalization. Note that the frontier between the metastable and unstable regimes depends on the three-body loss parameter η.

Figure 6 .

 6 Figure 6. Momentum distribution of a Bose gas after a fast ramp towards resonance. Momentum is scaled in units of k n = (6π 2 n ) 1/3 , where n is the mean atomic density. The solid blue (red) line correspond to experimental measurement for n = 5.5 × 10 12 /cm 3 ( n = 1.2 × 10 12 /cm 3 )[START_REF] Makotyn | Universal dynamics of a degenerate unitary Bose gas[END_REF]. The dashed line corresponds to the second order virial expansion of the momentum distribution for a fugacity z = 0.4[START_REF] Laurent | Momentum distribution of a dilute unitary Bose gas with three-body losses[END_REF].

Acknowledgments

The authors thank C. Chin, L. Khaykovich, W. Krauth, X. Leyronas, D. Petrov as well as past and present members of the ENS ultracold Fermi group for useful discussions. They acknowledge support from Région Ile de France (DIM IFRAF/NanoK) and European Union (ERC Grant ThermoDynaMix).


Nicolas Poupart
email: nicolas.poupart@yahoo.fr

The Galactic Dark Matter As Relativistic Necessity

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

Introduction

Since the assumption of the existence of galactic dark matter (dark mass) by Vera Rubin to explain the flatness of galactic rotation curves [START_REF] Cooper Rubin | Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions[END_REF][START_REF] Cooper Rubin | Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc)[END_REF][START_REF] Cooper Rubin | Rotation Velocities of 16 Sa Galaxies and a Comparison of Sa, Sb, and Sc Rotation Properties[END_REF] , no convincing explanation about the nature of this dark mass has been made. Attempts to explain this missing mass by an invisible form of ordinary baryonic matter was largely refuted by the programs MACHO [START_REF] Alcock | The MACHO Project: Microlensing Results from 5.7 Years of LMC Observations[END_REF] , EROS [START_REF] Tisserand | Limits on the Macho Content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds[END_REF] and AGAPE [START_REF] Gondolo | AGAPE: a Microlensing Search for Dark Matter by Monitoring Pixels[END_REF] , it is the same for explanations using ordinary non-baryonic matter. Numerous detection attempts of exotic particle that could explain the missing mass, have also all been unsuccessful. Similarly, the new CERN accelerator appears to confirm that the physics is limited to the standard model and the existence of an exotic particle is less and less probable [START_REF] Wolchover | Supersymmetry Fails Test, Forcing Physics to Seek New Ideas[END_REF][START_REF] Akerib | Results from a search for dark matter in the complete LUX exposure[END_REF][START_REF] Mcgaugh | The Radial Acceleration Relation in Rotationally Supported Galaxies[END_REF] .

It also seems extremely difficult, even impossible, to explain this phenomenon with the current theory of gravitation either the Newtonian gravity 10,[START_REF] Bekenstein | Relativistic gravitation theory for the modified Newtonian dynamics paradigm[END_REF] (NG) or general relativity (GR). An alternative is to modify gravity so as to adapt it to regime change at galactic level. On the contrary, such a project is exposed to the prodigious adequacy of the GR to the phenomenological reality [START_REF] Zhang | Probing Gravity at Cosmological Scales by Measurements which Test the Relationship between Gravitational Lensing and Matter Overdensity[END_REF][START_REF] Reyes | Confirmation of general relativity on large scales from weak lensing and galaxy velocities[END_REF] and the physical existence of dark matter [START_REF] Mavromatos | Can TeVeS avoid Dark Matter on galactic scales?[END_REF][START_REF] Clowe | A Direct Empirical Proof of the Existence of Dark Matter[END_REF] .

The explanation proposed in this article is of an entirely different nature. The dark mass is not some form of actual matter, it is also not an epiphenomenon caused by gravitation. The dark mass is actually a necessary consequence of relativistic mechanics (RM), that is to say, the combination of classical mechanics (CM) and special relativity (SR). It will be demonstrated that this relativistic mass is necessary if a body like a galaxy can physically collapse into a compact ball. The explanation proposed ignores the forces of physics and therefore is a purely mechanical explanation.

The existence of a compact ball state

The theorem of the Schwarzschild radius limit can be simply derived from NG and the postulate of the light speed as a maximum speed. Indeed, simply pose that escape velocity V = (2GM/Rs) is equal to c which gives Rs = 2GM/c 2 . It is also known that the same equation can be derived with GR.

The fundamental axiom of our demonstration is that a galaxy can contract in a compact ball with a radius proportional to the mass thus Ra = aM. The only constraint is that Ra radius is much smaller than that of the compacted galaxy, which limits the choice of a. To simplify calculations, we'll set b = a c 2 /2 and so Ra = 2bM/c 2 [D1].

We consider two models for the dynamics of this compact ball: 1) the rigid ball in relativistic rotation at constant angular velocity v(r) = r and 2) the homogeneous energy ball at constant linear velocity (r) = v/r. The rigid ball requires the existence of a bonding force, non existent in RM only, while a simple force of friction would produce a homogenization of speeds.

In the case of the rigid ball, the spin is defined as a = / max and in the case of the homogeneous energy ball a = (r)/ max (r). The spin value is a constant and by posing  = (R) and  max =  max (R) then in both cases a = (r)/ max (r) = (R)/ max (R) = / max .

1/6

The energy and angular momentum of the ball in relativistic rotation First calculate the mass M of a rigid ball of radius R in relativistic rotation around a central axis. (r) is the relativistic density at the distance r from the axis of rotation and  0 the inert density on the axis of rotation (no relativistic expansion of the mass). The relativistic density is (r) =  0 /[1-v(r) 2 /c 2 ]. The height of the basic cylinder at the distance r from the axis of rotation is h(r) = 2[R 2 -r 2 ] and its volume is given by V(r) = 2rh(r) dr.

Therefore, taking into account the relativistic mass change by integrating from 0 to R :

M =  (r) V(r) =  (r) 2rh(r) dr = 2 0 R 2  2[R 2 -r 2 ]/[1-v(r) 2 /c 2 ] d[r 2 /2R 2 ] = 2 0 R 3  [1-r 2 /R 2 ]/[1- v(r) 2 /c 2 ] d[r 2 /R 2 ].
1) In the case where the ball is rigid and rotates at constant angular speed  max = c/R (maximum speed at the equator) we have

M = 2 0 R 3  [1-r 2 /R 2 ]/[1-(r max ) 2 /c 2 ] d[r 2 /R 2 ] = 2 0 R 3  [1-r 2 /R 2 ]/[1- c 2 r 2 /c 2 R 2 ] d[r 2 /R 2 ] = 2 0 R 3  d[r 2 /R 2 ] = 2 0 R 3 . So M = 2 0 R 3 [L1]. Because M 0 = 4 0 R 3 /3 (density by the volume of the ball) therefore M = 3M 0 /2. The energy gain M r = M-M 0 when a = 1 is maximum M r = M 0 /2 [T1].
2) In the case where the ball has a homogeneous energy and rotates at the constant linear velocity

v = r(r) we have M = 2 0 R 3  [1-r 2 /R 2 ]/[1-(r(r)) 2 /(r max (r)) 2 ] d[r 2 /R 2 ] = 2 0 R 3 /(1-a 2 )  (1- r 2 /R 2 ) d[r 2 /R 2 ] = 4 0 R 3 /(1-a 2 )  r(1-r 2 /R 2 )/R 2 dr = 4 0 R 3 /3(1-a 2 ) [L2]. Since that M 0 = 4 0 R 3 /3 (density by the volume of the ball) then M = M 0 /(1-a 2 ). The energy gain M h = M-M 0 = M 0 (1/(1-a 2 )-1)
[T2] tends to infinity with the spin of the ball (a = 1) and is zero for a nul spin (a = 0).

It is now possible to calculate the relativistic moment of inertia of the ball :

I =  r 2 dm =  r 2 (r) V(r) =  r 2 (r) 2rh(r) dr = 2 0 R 4  2[R 2 -r 2 ]/[1-v(r) 2 /c 2 ] d[r 4 /4R 4 ] =  0 R 5  [1-r 2 /R 2 ]/[1-v(r) 2 /c 2 ] d[r 4 /R 4 ].
1) In the case where the ball is rigid and rotates at constant angular speed  max = c/R (maximum speed at the equator) we have

I =  0 R 5  [1-r 2 /R 2 ]/[1-(r max ) 2 /c 2 ] d[r 4 /R 4 ] =  0 R 5  [1-r 2 /R 2 ]/[1-c 2 r 2 /c 2 R 2 ] d[r 4 /R 4 ] =  0 R 5 . Thus I =  0 R 5 and by [L1] I r = ½MR 2 .
2) In the case where the ball has a homogeneous energy and rotates at the constant linear velocity v = r(r) we have

I =  0 R 5  [1-r 2 /R 2 ]/[1-(r(r)) 2 /(r max (r)) 2 ] d[r 4 /R 4 ] =  0 R 5 /(1-a 2 )  (1-r 2 /R 2 ) d[r 4 /R 4 ] =  0 R 5 /(1-a 2 )  4r 3 (1-r 2 /R 2 )/R 4 = 8 0 R 5 /15(1-a 2 ). So I = 8 0 R 5 /15(1-a 2 ) and by [L2] I r = 2MR 2 /5.
The relativistic angular momentum of the maximum speed rotating rigid ball is given by J r = I r  = ½MR 2  [T3], which is not very different from the non-relativistic angular momentum J = 3MR 2 /5. This calculation is not exact because J r should vary from 3MR 2 /5 for a spin of a = 0 to ½MR 2  for a spin of a = 1. The relativistic angular momentum of the rigid ball of homogenous energy is

J h = I h  = 2MR 2 /5 [T4],
this calculation is, on the other hand, exact.

The dark mass production

2/6

For virtually any rigid body I = jMR 2 such as j is a constant characterizing the shape of the body : j = 1 for the ring, j = 1/2 for the disc, j = 2/3 for the sphere, and j = 3/5 for the ball. Since

 = V/R, it follows that Jj = I = (jMR 2 )(V/R) = jMRV [T7].
If such a body is compacted into a ball with a radius Ra then by [T5] J f = abM 2 /c and by [T7] J i = jMRV. Therefore, by the conservation of angular momentum J i = J f and so a = jRVc/bM. Yet, if the values of the Milky Way are used 17,18 R = 3.310 [START_REF] Torres-Flores | GHASP: an Hα kinematic survey of spiral and irregular galaxies --IX. The NIR, stellar and baryonic Tully-Fisher relations[END_REF] , V = 2.210 [START_REF] Tisserand | Limits on the Macho Content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds[END_REF] , M = 210 42 and j = 1 and b = G we get a spin a = 262, which is not an admissible value because a  [0,1]. In fact, we must modify of one order of magnitude at least three parameters to get a possible value. Worse, the mass M should be multiplied by a factor to approach a valid value but it is already the total mass including the dark mass.

Our calculation is incorrect because the final ball and the initial Galaxy are not in the same Galilean reference frame. Indeed, the mass of J i is actually only the baryonic mass whereas the mass of J f is the total mass expanded by relativistic velocity produced by the rotation. Therefore, to maintain the same mass-energy of the initial to the final state, it is necessary to contract the mass-energy of the final state. This is exactly a change of relativistic reference frame requiring the introduction of a Lorentz factor. Since the unit of angular momentum is kgm²/s then  2 = J i /J f because  2 = (1/)kg(²)m²/(1/)s This allows to write  2 = J i /J f = jRVc/abM [T8] and get by setting j = a = 1 and b = G  = M/M 0  16. What is much more than the maximum dilatation of M = 3M 0 /2 by [T1], we must conclude that the compact ball is not rigid.

Using the same calculation for the homogeneous energy ball it is obtained by [T6] and [T7]  2 = J i /J f = 5jRVc/4abM [T9] and posing j = a = 1 and b = G for the Milky Way one obtains  = M/M 0  18. That which is, here, an allowed energy value by [T2].

To get a better approximation, it is possible to model a spiral galaxy by a disk of homogeneous surface density  0 of mass M =  0 R 2 . It is subsequently possible to divide the disk in rings of masses m(r) =  0 2r dr, of moments of inertia I(r) = m(r) r 2 and angular momenta J(r) = I(r) (r) with (r) = V/r so a constant speed of rotation for the entire disc. Integrating J(r) from 0 to R we obtain J = 2MRV/3, thus by [T7] j = 2/3 which gives for the values of the Milky Way applied to [T8]  = M/M 0  13 or  = M/M 0  15 by using [T9].

If we compare the equation of the rigid ball a 2 = jRVc/bM with that of the homogeneous energy ball a 2 = 5jRVc/4bM we realize that it is practically the same equation with a small multiplicative factor difference. Since the equation of the angular momentum of the rigid ball is identical to that obtained using the Kerr black hole, it will be favored. However, to not contradict the value of energies by the boundary of the rigid ball, we will use the energy of the homogeneous ball, a 2 = jRVc/bM and

 =M/M 0 = 1/(1-a 2 ) [T10].

The baryonic Tully-Fisher relation

Using [T10] and b = G then a 2 = jRVc/GM, the radius R being the maximum galactic radius, V the linear speed at the galactic circumference and M the total galactic mass, it is possible to apply the virial theorem 19 M = 2V 2 R/G. By posing a  1, it is thus possible to obtain  2 = (M/M 0 ) 2  jRVc/GM then 8V 5 R 2 /M 0 2  jcG 2 by applying the virial and thus V 5  jcG 2 (M 0 2 /8R 2 ). By simply modeling the galaxy as a disk with a thickness e and a homogeneous density  0 , we obtain M 0 =  0 eR 2 so R 2 = M 0 / 0 e therefore V 5  M 0 ( 0 e)(jcG 2 /8) which implies a relation of the type M 0  V 5 . However, this calculation ignores the coefficient ( 0 e) which lacks only the multiplication by R 2 to get M 0 . Let . Since  0 /b is on the order of 10 -2 and R on the order of 10 20 we can approximate k  1/3 and therefore V 5  M 0 4/3 (jcG 2 /8) which implies a relation of the type M 0  V 3.75 . This relationship is in perfect agreement with the baryonic Tully-Fisher law 20 V 3.5  M 0  V 4 and it is therefore possible to derive this law without changing Newton's gravitation [START_REF] Mcgaugh | The Baryonic Tully-Fisher Relation of Gas-Rich Galaxies as a Test of ΛCDM and MOND[END_REF] or the general relativity.

Discussion on the mechanical determinism

Since a galaxy can contract into compact ball with a great amount of kinetic energy and this energy can be found, by relativity, as mass-energy, one must ask where this energy comes from. By the conservation of energy, if that energy does not come from a source outside the system, it must pre-exist in the galaxy before the compactification. So it must be a tremendous amount of energy in the form of mass-energy in the galaxy. If the cause of the compactification is gravitation, because it is an internal force to the system, this energy must lie somewhere in the gravitational energy field.

The problem in our explanation is that the mass energy being found in a system depends on a potential destiny. If the destiny of a system is not to collapse into a black hole because it does not have the critical mass [START_REF] Chandrasekhar | The Maximum Mass of Ideal White Dwarfs[END_REF] of Chandrachekar, it should produce much less dark mass. How the other forces interact with gravity to control this mass production is, for now, a mystery.

Theoretical prediction

Since a galaxy is found to be a Galilean reference frame, comparable to a body at a constant speed. Thus, the classical Lorentz transformation of the velocities composition must consequently be used. This consequently causes a shift of the radiation frequency from a transmitting galaxy to a receiving galaxy of this radiation. This intrinsic frequency shift is distinct from the gravitational shift.

Thus, from a radiation emitting galaxy 1/ e = M 0e /M e to a galaxy receiving the radiation 1/ r = M 0r /M r , it is possible to get the velocities equivalents v e = c(1-1/ e 2 ) and v r = c(1-1/ r 2 ) which may be composed as v = (v r -v e )/[1+(v r v e /c 2 )] which gives a shift z = [(1+v/c)/(1-v/c)] -1. This can lead, for the Milky Way, to intrinsic shifts between z  -1% and z  8%. This shift should affect the redshift but also the measurement of regular phenomena as pulsars (expansion and contraction of the time).

The Tully-Fisher relation is the most important secondary measure of the distance measurements of a broad set of spiral galaxies, it has a significant influence on the conventional calculation (through the creation of distances scales) of the Hubble constant, yet the errors persist inexplicably [START_REF] Conover | Debate accelerates on universe's expansion speed[END_REF] . Russell (2015) [START_REF] Russell | Intrinsic Redshifts and the Tully-Fisher Distance Scale[END_REF] after a comprehensive analysis concluded at the existence of an intrinsic redshift which can exceed 5000 km/s and a clear tendency for the intrinsic redshifts to be more important than the intrinsic blueshifts. The result of this analysis is in perfect agreement with our theory.

Conclusion

The results obtained are mere consequences of the application of relativistic mechanics (RM) to the problem of contraction of a galactic sized body into a ball with the Schwarzschild radius, no other hypothesis on the nature of gravitation was used. A first surprising result is that the relativistic angular momentum of a rigid ball, with the Schwarzschild radius, is strictly that calculated by Kerr using general relativity (GR). We must therefore conclude that, from a strictly mechanical point of view, the moment of inertia of a black hole is identical to that of a rigid ball rotating at maximum speed and its energy is probably that of a homogeneous energy ball.

The second result is that the kinetic energy of rotation of a ball resulting from the compactification of a galactic sized system is enormous, more than thirteen times its own mass in the case of the Milky Way. If this compactification can indeed occur by the mere internal force of gravity, without any external energy input to the system then, by the law of the conservation of energy, this energy does exist somewhere in the system before the compactification. Since this surplus energy can't be stored mechanically in the galaxy, it is necessarily stored in the gravitational energy field. This energy surplus enables the derivation of the Tully-Fisher relation and correlates well with the amounts of galactic dark matter (mass). All this evidence suggest that this necessary mechanical consequence is the cause of the galactic dark matter.

The third result is a consequence of the fact that without an external energy input, the system is in the same inertial reference frame before and after the compactification. Therefore, an intrinsic redshift phenomenon should occurs between galaxies transmitting and receiving radiation. This new experimental prediction, which permits to accurately calculate the intrinsic redshift from the galactic dark mass, offers a possibility of experimental refutation of this theory. Note that if this theory is proved false, this will strongly question the consistency of relativistic mechanics.

5/6

  Consider a rigid ball in relativistic rotation with a radius Ra, thus by [T3] J = I = ½MRa 2  [L3]. Since a = / max and  max = c/Ra then  = ac/Ra and by [L3] J = ½acMRa [L4]. By [D1] and [L4] we obtain J br = abM 2 /c [T5], which is exactly the calculated angular momentum 16 by Kerr if b = G. Thus, the moment of inertia of the Kerr black hole, regardless of its spin, is that of a rigid ball of which the surface rotates at the speed of light. Using the same calculation with the ball of homogeneous energy is obtained by [T4] and [D1] J bh = 4abM 2 /5c = 4J br /5 [T6].

3 / 6 e

 36 = R/b thus M 0 =  0 (R 3 /b) [L5] and so ln(M 0 ) = ln( 0 /b)+3ln(R) [L6], therefore, by posing M 0 k =  0 (R/b) then k = [ln( 0 /b) + ln(R)]/ln(M 0 ) and consequently k = [ln( 0 /b) + ln(R)]/ [ln( 0 /b) + 3ln(R)] by [L5] and [L6]