I Bediaga

M Zatarain

J Muñoa

R Lizarralde

L N López De Lacalle

Application of continuous spindle speed variation for chatter avoidance in milling. Optimisation of speed variation parameters

Keywords: Chatter, Milling, Continuous Spindle Speed Variation, Semidiscretization

This paper investigates the effectiveness of Continuous Spindle Speed Variation (CSSV) technique as a chatter suppression strategy in milling process. On this purpose, a twofold study was carried out: on the one hand, a simulation analysis, and on the other, experimental tests. First, Semidiscretization method has been extended to n-mode systems in any spatial directions and modified to include simulation capabilities concerning spindle speed variation with different waveforms. . A new kind of stability map for different amplitude and frequency variations is proposed in order to optimise the parameter selection when machining. Finally, the effectiveness of CSSV technique is verified through experimental tests.

INTRODUCTION

The productivity of milling operations is often constrained by the regenerative chatter. This occurs when relative vibrations between the cutting tool and the workpiece feed back the cutting force, leading to self-excited vibrations. Depending on cutting conditions and dynamic properties, the vibration feed back can lead to a decreasing vibration (stable cut) or to an increasing vibration resulting in chatter (unstable cut). This affects the quality of the machined surface, along with excessive tool wear or tool breakage. Thereby the material removal rate is limited. In order to increase the material removal rate a chatter suppression method based on Continuous Spindle Speed Variation (CSSV) is analysed. This technique consists in a continuous variation of the tooth passing frequency, yielding a continuous variation of the phase between the tool-workpiece relative displacement and the tooth passing cycle. This is an 'electronic' or software way to continuously change the phase lag, comparable to the 'mechanic' or hardware way of using uneven pitch mills.

STATE OF THE ART

Knowledge of chatter vibration has been progressing since the first papers by Tobias [START_REF] Tobias | Theory of Regenerative Machine Tool Chatter[END_REF] and Tlusty [START_REF] Tlusty | Beispiele der behandlung der selbsterregten Schwingung der Werkzeugmaschinen FoKoMa[END_REF]. The basic theory did not advance significantly until the research by Altintas and Budak [START_REF] Altintas | Analytical Prediction of Stability Lobes in Milling[END_REF], were the general single frequency multimodal solution was developed [START_REF] Muñoa | Stability study of the milling process using an exponential forcé model in frequency domain[END_REF], and Budak and Altintas [START_REF] Budak | Analytical Prediction of Chatter Stability Conditions for Multidegree of Freedom Systems in Milling. Part I: General Formulation, Part II: Application of the General formulation to Common Milling Systems[END_REF] were the general solution for milling was obtained by using multifrequency [START_REF] Zatarain | Estudio comparativo de los modelos matemáticos de chatter en fresado: monofrecuencia, multifrecuencia y simulación en el tiempo[END_REF]. Later, some new ways of obtaining the lobe diagram for milling were proposed. Fundamental papers were those of Davies et al. [START_REF] Davies | The stability of low radial immersion milling[END_REF], Insperger et al. [START_REF] Insperger | Stability of High Speed Milling[END_REF], and Baily et al. [START_REF] Bayly | Stability of Interrupted Cutting by Temporal Finite Element Analysis[END_REF]. In these papers the uprising of a new chatter situation in milling was shown: that of period doubling, which appears mainly at high cutting speeds and low radial immersions. Merdol and Altintas [START_REF] Merdol | Multi Frequency Solution of Chatter Stability for Low Immersion Milling[END_REF] proved that multifrequency theory also explains the emergence of this chatter mechanism. Variable pitch mills have been used since many years ago to increase the stable limit depth of cut. Probably a paper by Slavicek [START_REF] Slavicek | The Effect of Irregular Tooth Pitch on Stability of Milling[END_REF] is the oldest one showing this effect.

Outstanding papers in this direction are those by Stone [START_REF] Stone | The effect on the chatter behaviour of machine tools of cutters with different helix angles on adjacent teeth[END_REF], Altintas et al. [START_REF] Altintas | Analytical Stability Prediction and Design of Variable Pitch Cutters[END_REF], and Budak [START_REF] Budak | Improvement of Productivity and Part Quality in Milling of Titanium Based Impellers by Chatter Suppression and Force Control[END_REF]. Continuous speed variation might be considered as an equivalent for variable pitch mills in single cutting edge processes, like turning. In milling, speed variation provides a more flexible solution than variable pitch mills, as proposed by Stone [START_REF] Stone | The effect of time varying parameters on the built-up of chatter in turning[END_REF]. Inamura and Sata [START_REF] Inamura | Stability analysis of cutting under varying spindle speed[END_REF] produced a simple function for the study of the stability with variable spindle speed for the turning process. Their experimental results were not as good as those calculated by their theory. Takemura et al. [START_REF] Takemura | Active suppression of chatter by programmed variation of spindle speed[END_REF] analysed the stability of turning with variable speed by means of an energy balance. The analysis was performed for triangular, rectangular and sinusoidal speed variations. Sexton and Stone [START_REF] Sexton | The Stability of machining with continuously varying spindle speed[END_REF], disagreed with previous works, like [START_REF] Inamura | Stability analysis of cutting under varying spindle speed[END_REF], showing that the results for variable speed machining are not spectacular. The theoretically good results of previous works were due to oversimplified models of the system. De Canniere et al. [START_REF] De Canniere | A contribution to the mathematical analysis of variable spindle speed machining[END_REF] used the perturbation analysis for the determination of the stability in turning with CSSV. This analysis is only valid for small speed and frequency variations. With these conditions, they showed that the speed modulation is equivalent to the modulation of the time lag. This fact was also used by Altintas and Chan [START_REF] Altintas | In-process detection and suppression of chatter in milling[END_REF], who developed a system based on the modulation of the speed for stabilisation of chatter vibration. Tsao et al. [START_REF] Tsao | A new approach to stability analysis of variable speed machining systems[END_REF] used the angle domain to analyse variable speed machining. It was shown that this technique is very sensitive to the number of samples used in the discretization of the time system, and the conclusions were not trustworthy. Jayaram et al. [START_REF] Jayaram | Analytical stability analysis of variable spindle speed machining[END_REF] developed a Fourier expansion of the equations of the process of turning. They used an approximation of the delayed vibration and obtained a system of an infinite number of frequencies, which could be solved approximately by truncation to a number of frequencies close to the main frequency only. The mathematics behind this approach resemble the method of Multifrequency [START_REF] Budak | Analytical Prediction of Chatter Stability Conditions for Multidegree of Freedom Systems in Milling. Part I: General Formulation, Part II: Application of the General formulation to Common Milling Systems[END_REF] for constant speed milling. This development was extended by Sastry et al. [START_REF] Sastry | Floquet theory based approach for stability analysis of the variable speed face-milling process[END_REF] for the process of face milling. Again, an infinite order system is found, which has to be truncated to be solved. This is the most advanced approach, up to now, to the application of CSSV in milling, but it still applies some simplifications to the problem. Zatarain et al. [START_REF] Zatarain | Stability of Milling Processes with Continuous Spindle Speed Variation: Analysis in the Frequency and Time Domains, and Experimental Correlation[END_REF] presented the general theory for analysis of milling with CSSV in the frequency domain and for any speed variation strategy. The results were compared with those obtained by Semidiscretization and time integration, as well as with those obtained by experiments. Insperger and Stépán [START_REF] Insperger | Comparison of analytical and numerical simulations for variable spindle speed turning[END_REF] used Semidiscretization to obtain lobe diagrams for turning processes with CSSV. Three kinds of speed variation schemes are investigated: Sinusoidal, increasing saw teeth, and decreasing saw teeth. They concluded that the best solution was that of sinusoidal speed variation.

CONTENT OF THE PAPER

Semidiscretization method has been well documented for systems with a single modes in X and Y directions. This paper will present the way to enhance the Semidiscretization method to n-mode systems in any spatial directions. Also, CSSV will be considered with different spindle speed variation strategies, and results of experimental tests that have been carried out for verification of the effectiveness of CSSV technique will be shown. The arrangement of this paper is as follows. In section 4, machining conditions used for illustration of the theoretical developments and for the experiments are summarised. In section 5 the theoretical development of CSSV using extended Semidiscretization method is included. In section 6 CSSV parameter optimisation charts are shown. In section 7, the evolution of eigenvalues during CSSV simulation is analysed. Afterwards in section 8, experimental tests are detailed to demonstrate the utility of the SSV in milling.

PARAMETERS OF SYSTEM USED.

The developments that will be presented in this paper will be applied to a system with a flexible part support. The system has a clearly dominant mode in Y direction (perpendicular to feed), with parameters ωn=95.6 Hz, stiffness=49N/μm, ζ=0.6%. Cutting conditions are: Tool diameter 80 mm, 3 flutes, lead angle 45º, half immersion, down milling. Specific cutting coefficients: Tangential 1889 MPa, Radial 776 MPa, Axial 364 MPa.

THEORETICAL DEVELOPMENT USING SEMIDISCRETIZATION

The Semidiscretization method is a well-known technique in the analysis of rigid solids, or in the fluid mechanics. The application of this theory to stability modelling of the cutting process was carried out by Insperger and Stépán [START_REF] Insperger | Semi-discretization of delayed dynamical systems[END_REF], and fully detailed for milling and turning processes with a single mode in X and Y directions by Insperger [START_REF] Insperger | Stability analysis of periodic delay-differential equations modelling machine tool chatter[END_REF]. In this section the Semidiscretization method is enhanced to n-mode systems in any spatial directions and the CSSV with different waveforms is proposed.

Semidiscretization for n-modes of excitation

The first step to develop Semidiscretization method consists in the definition of the time interval [ti, ti+1) of Δt increments for i=0,1…n. So that the time increment at step can be expressed as Δt = τ / m where τ is the time delay and m is the approximation parameter of the Semidiscretization method. The mechanical model of the milling process can be seen in the Figure 1, where the dynamical parameters are defined as m mass of the system, c damping coefficient, and k spring stiffness. The angular spindle speed is expressed as Ω in rad/s, and the cutting force in modal direction as Fm. The cutting angle is divided into sufficiently small arcs, according to the defined time increment. So that, the cutting angle achieves from φ0 to φn, then the arcs are φ0, φ1, φ2, φ3… φn. At each of these positions the displacement vectors of the tool at the previous pass are stored [START_REF] Tobias | Theory of Regenerative Machine Tool Chatter[END_REF]. These would represent the trajectory of tool tip in the previous pass.

  T n n q q   3 2 1 0     (1)
Starting with the tool tip at the last angle, it is needed to know the state at that position, that is, the modal displacement and velocity for all the modes. For that position, these variables will be stored instead of the displacements, because the displacements do not contain all the necessary information of the state. With position and speed defined by the parameters with sub index 'n', the state when the tool tip reaches the angle '0' is obtained again by calculating the free trajectory defined by modal coefficients, Time from 'n' to '0':

     n    0 2 Free decay from 'n' to '0', n m m n n m m q q q q A A A A q q q q                                                               1 1 0 0 1 1 0 22 21 12 11 q A q (2)
Where A11 and A12 represent the damped free oscillation from 'n' to '0' position. And A22 correspond to its derivate. Now, at angle '0' chip width is calculated as,

                                                                 0 0 1 1 1 0 0 δ q  zm z ym y xm x T cos cos sin sin sin h            (3) Cutting force at state q0 is,               0 0 0 δ q Φ                                   T a r t a r t cos cos sin sin sin k k k b f f f      (4)
Projection of force into the mode is,

                                 a r t T f f f cos sin sin cos b f 1 0 0 0 0      Φ (5)
If a more accurate solution was needed, the position between 0 and 1 could be obtained by including the velocity at position 0 for interpolation. Similarly, excess material from previous pass could be obtained by linear interpolation between 0 and 1. Now the new state at position 1 can be calculated,

                 
A A A A q q A δ C q B A f R q A q   (6)
Where A11 and A12 represent the damped free oscillation from one position to the next one. And A21, A22 correspond to its derivate. The term B01 of equation ( 6) is calculated as,

                      Φ Φ Bf Bf Bf B T a r t T cos cos sin sin sin k k k cos sin sin cos b                                                                  1 0 0 0 0 0 0 2 6 2 2 3 2 01 (7)
In the same way, the term C01 of equation ( 6) is expressed as,

                    T a r t T cos cos sin sin sin k k k cos sin sin cos b                                                     1 0 0 0 0 2 2 01 Φ Cf Cf C (8)
Now, the new displacements of the tool/part will be,

    1 1 1 0 0 0 0 0 Tq q Φ Tq q Φ       ˆ ( 9
)
Writing down the complete vector for position 0,

  n n n n n n n ˆq A q q δ δ TA q TA Tq δ 0 0 1 0 0 0 0 0 0 0                      (10)
And for position 1,

                                       n n n n n n ˆq δ δ A TD TC TA δ δ δ C q A D δ C q B q q A q 1 0 0 01 01 0 1 0 0 01 0 01 0 01 0 01 0 0 01 1 0 0 0 0 0   (11)
The last step consists in determining the state transition matrix Φ over the principal period T= k•Δt. The transition matrix links the last displacement to the first one, δk to δ0, in the form, δk= Φ•δ0.

                                                                                           n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n ˆˆˆq A D C C D C D C D A TD TC C TD C TD C TD TDDDA TC TDC TDDC TDDA TC TDC TDA TC TA q δ C q D δ C q B q A q 1 3 2 1 0 0 3 2 1 0 1 4 3 2 0 0 0 0 1 3 2 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0                         (11)
The stability analysis of the system reduces to obtain the eigenvalue of maximum amplitude of the transition matrix [START_REF] Stone | The effect on the chatter behaviour of machine tools of cutters with different helix angles on adjacent teeth[END_REF]. If any eigenvalue has a magnitude larger than 1, then the system is unstable, whereas when all magnitudes are lower than 1 the system is stable.

Development for different continuous spindle speed variation waveforms

The sinusoidal spindle speed variation method consists in adding a sinusoidal component to the constant spindle speed according to equation [START_REF] Altintas | Analytical Stability Prediction and Design of Variable Pitch Cutters[END_REF]. For constant speed machining the time delay τ takes a constant value, but for variable speed cutting this time lag is time dependent: τ=τ(t). Calling Ω the instantaneous speed, a sinusoidal speed variation can be represented by,

  t sin       0 (5)
where Ω 0 represents the average speed, Δ is the maximum speed variation, and γ is the speed variation frequency. The adaptation of Semidiscretization to continuous spindle speed variation is straightforward, with the condition that the period of the speed variation is an integer multiple of the tooth pass period: It suffices to calculate the transition matrix over the speed variation period, that is, an integer number of tooth pass periods. Again, the magnitude of the eigenvalues of this matrix will define the stability of the system. Another speed variation law can be considered to analyse CSSV strategy to eliminate chatter: trapezoidal waveform. This law permits obtaining particular cases like triangular and square variations, but also many more possibilities. In this paper only sinusoidal and triangular variation are considered, although other variation laws can be analysed in the same way.

OPTIMIZATION OF SSV PARAMETERS

In fact, if an operator dials the speed override knob back and forth, chatter sometimes vanishes. But not all kind of variations permit avoiding chatter. The effectiveness of the spindle speed variation method depends on the forcing input signal, and on its parameters (amplitude and frequency). Continuous Spindle Speed Variation can be considered as a way of avoiding chatter vibration when chip load is too large and cutting velocity corresponds to a high order lobe. In this case it is more interesting to optimise speed variation parameters rather than using lobe diagrams for selecting an optimal constant speed. Figure 2 shows the limit depth of cut for different frequencies and amplitudes of the sinusoidal speed variation at the speed 525 r/min. The figure has been obtained by the analysis in the frequency domain [START_REF] Zatarain | Stability of Milling Processes with Continuous Spindle Speed Variation: Analysis in the Frequency and Time Domains, and Experimental Correlation[END_REF]. Considering a plane (in yellow) at 4mm axial depth of cut that sections the map of limit depth of cut, an excellent view of unstable areas (in dark blue) below the yellow plane could be obtained. Semidiscretization calculates a stability level for each depth of cut. In order to obtain a map similar to the one shown in Figure 2, Semidiscretization simulation would require to evaluate a very large amount of levels along axial depth of cut. This is not a practical approach due to the large time of computation required. Therefore, a new approach is proposed: representation of the stability level for different frequencies and amplitudes of the SSV at a defined axial depth of cut. The stability level is defined by the magnitude of the maximum eigenvalue for the whole transition matrix. Figure 3 shows inverted eigenvalues to permit an easier comparison with the map of depth of cut in Figure 2. Considering that the stability limit is denoted by the unit (λ=1), a plane (in yellow) will divide up stable areas (in red) from unstable areas (in dark blue). Comparing figures 2 and 3 it can be deducted that both of them give very similar information. The system becomes stable from amplitude variations of 10%. In addition, some unstable areas appear for amplitude variations from 40% to 50%. It becomes clear that speed variation ratio is a more influent parameter than frequency. As a general rule, stability grows with the amplitude of speed variation, although there are some areas of local optimum solutions. In this particular example, optimum values of speed variation ratio are at 15% and 30%. In the practice, 30% variation should already be considered too large, and also it should be analysed whether the spindle is able to get the acceleration required.

Figure 4 shows the limit depth of cut for different frequencies and amplitudes of the triangular speed variation at the speed 525 r/min. The figure has been obtained by the analysis in the frequency domain [START_REF] Zatarain | Stability of Milling Processes with Continuous Spindle Speed Variation: Analysis in the Frequency and Time Domains, and Experimental Correlation[END_REF]. Again, a yellow plane at 4mm axial depth of cut sections the map, offering a better view of unstable areas (in dark blue) below the yellow plane. As explained before for the sinusoidal variation, Figure 5 shows Semidiscretization map of the degree of stability for triangular speed variation. It can be appreciate that the process becomes stable for amplitude variations larger than 15%, and, opposite to Figure 3, no unstable areas appear at higher amplitude and frequencies. From these graphs it is possible to select variation parameters that are adequate for the cutting conditions required. A consequence obtained from comparing Figure 3 and Figure 5 is that sinusoidal speed variation is more performing than triangular speed variation, as stable area is obtained with lower amplitude and frequency variations, at least for small amplitude variations.

EVOLUTION OF STABILITY

A system that varies with time can be stable globally while being unstable at some portion of the time. The requirement is that any vibration growth in the unstable period is damped out during the stable part of the cycle. This phenomenon was presented by Zatarain et al. [START_REF] Zatarain | Stability of Milling Processes with Continuous Spindle Speed Variation: Analysis in the Frequency and Time Domains, and Experimental Correlation[END_REF] and can be analysed by the method of Semidiscretization. As explained in section 5, this method produces a transition matrix, whose eigenvalues define the stability of the system. This matrix is computed along several tooth pass periods, up to reaching the speed variation period. The evolution of this matrix can be used to analyse the progression of stability during the speed variation cycle, which means, that it is possible to analyse the evolution of the amplitude of the eigenvalues at different moments of the cycle.

Figure 7: Experimental results with 15% variation at 3Hz.

Whenever the speed variation period is not an exact multiple of the tooth pass period, the analysis should be done on a longer time, a time that is and exact multiple of both the speed variation period and the tooth pass period. This can produce a beating with this new, longer period. Figure 6 shows this effect, obtained with a velocity variation frequency of 3 Hz and spindle velocity 525 r/min. Tooth pass period is 0.0381s, while velocity change period is 0.333s. producing a repetition period of 1.333 seconds (= 4x0.333 = 35x0.0381).

COMPARISON WITH EXPERIMENTAL RESULTS

The industrial implementation of the CSSV method inside the kernel of the SORALUCE SV6000 milling machine working with a Sinumerik 840D CNC was presented in [START_REF] Bediaga | Reducción de la inestabilidad en cortes interrumpidos en fresado a alta velocidad mediante variación de la velocidad del husillo[END_REF][START_REF] Bediaga | Chatter Avoidance Method for Milling Process based on Sinusoidal Spindle Speed Variation Method: Simulation and Experimental Results, 10 th[END_REF]. The experimental results are shown in Figure 6, where the severity of the vibration is measured instead of the amplitude of the chatter peak in the spectrum. The reason is that the measurement of chatter peak could lead to a mistaken conclusion, since the CSSV excites more frequencies than the normal machining but with less energy. Experience has shown that the overall vibration measured over a fixed range (i.e. 2 to 500 Hz) gives the best indication of vibration level. A probable explanation is that a given velocity level corresponds to a given energy level, so that vibration at low and high frequencies are equally weighted from a vibration energy point of view. Figure 8 represents the vibration severity level for each amplitude and frequency of the CSSV. It is evident that for variation amplitudes larger than 10%, the vibration level is highly reduced. The same conclusion could be obtained from the Semidiscretization map -Figure 3-and frequency domain map -Figure 2-. It is remarkable that there is an optimum area of vibration reduction in the range of 15% amplitude variation of spindle speed, as was foreseen by the results of frequency domain analysis shown in Figure 2, and the Semidiscretization map in Figure 3. Although, the further improvement of behavior at higher speed variation amplitudes foreseen in Figure 2 and Figure 3 were not found in experimental results, the general tendency agrees.

CONCLUSIONS

Continuous Spindle Speed Variation is a successful technique to reduce the vibration level in milling when the system is unstable. It is an appropriate strategy at a high order lobe. And although certain vibration level might be maintained, the surface quality obtained at the work part is improved.

Modelling milling processes with CSSV with the Semidiscretization method gives a good orientation on the optimum parameters of the speed variation. For that, a new chart representing the degree of stability for a range of amplitude and frequency of speed variation is proposed.

In the simulations done so far, sinusoidal variation shows to be more performing than triangular variation.

In addition, Semidiscretization method has been extended to n-mode systems in any spatial directions.

Figure 1 :

 1 Figure 1: Mechanical model of milling process.

Figure 2 :

 2 Figure 2: Map of limit depth of cut for sinusoidal spindle speed variation.

Figure 3 :

 3 Figure 3: Semidiscretization map of the degree of stability for sinusoidal speed variation.

Figure 4 :

 4 Figure 4: Map of limit depth of cut for triangular spindle speed variation.

Figure 4 :

 4 Figure 4: Semidiscretization map of the degree of stability for triangular speed variation.

Figure 8 :

 8 Figure 8: Vibration severity obtained in experimental tests.

ACKNOWLEDGMENTS

The authors gratefully appreciate the help provided by Mr. Gorka Arrate, Mr. Ruben Merino and Mr. Juan Pablo Rodriguez for their contributions in the machining tests.


Roberto Bernal

Chantal Tassius

Francisco Melo

Jean-Christophe Géminard

Mechanical characterization of elastic membranes: Cell mechanics applications

Wrinkled patterns produced by cells crawling onto elastic membranes provide a useful tool to test living cells' locomotion. The technique however requires a characterization of the intrinsicproperties and tension of the membrane as well as an accurate determination of the deformation induced by the cell. Here, we present a capillary-type technique to estimate the elastic modulus of such membranes, describe an indentation technique to measure the initial tension, and develop a simple interferometric method to measure the amplitude field of the wrinkled pattern produced, for instance, by the action of living cell.

1

Wrinkling in elastic films under stress is ubiquitous in nature [1][2][3]. Common systems exhibiting this phenomenon are our skin [4], textiles and more specifically thin coatings subjected to stresses mismatch [5,[START_REF] Hobart | Special Issue on Alternative Substrate Technology -Foreword[END_REF]. Apart from a basic interest, wrinkled membranes have found unexpected applications recently. For instance, in modern technology, pre-wrinkled conductive films are proposed as stretchable electrical contacts required in large-area electronics [7]. In medicine, surgery has always involved cutting, thickening contraction and in many cases wrinkling of the skin. Potential applications of wrinkle mechanics to improve wound healing have been recently summarized in Ref. [4]. In turn, in cell biomechanics, wrinkled patterns produced by cells crawling onto elastic membranes have provided a useful tool to test living cells' locomotion. With this method, in which membranes are obtained by vulcanization of the free surface of polymeric fluids, the wrinkle length has been correlated to the force applied by the cell cytoskeleton [1,[START_REF] Burton | [END_REF]. In a recent article [10], we reported an experimental and theoretical study of wrinkles formed in an initially flat elastic membrane, locally subjected to an axi-symmetric tangential force. We show that wrinkles appear as a consequence of a supercritical instability : both the length and amplitude of the wrinkles increase as the imposed force is increased beyond a well defined critical value which is a function of the membrane tension. Beyond the instability threshold, the number of nucleated wrinkles is nearly independent of the applied force, varies very slowly with membrane tension, and is a linear function of the size of the object responsible for the wrinkling i.e., the typical dimension of the region (relative to the membrane thickness) in which the cell attaches to the substrate. By contrast, the wrinkle length and amplitude proved to be very sensitive to the membrane tension, both quantities decreasing as the tension is increased.

Moreover, the length resulted a slowly varying function of applied force, which challenges the sensitivity of cellular force measurements based on the direct measurement of the length.

We suggested thus that a considerable more sensitive measurement can be obtained if the force is correlated directly to the amplitude instead. However, such a program requires the development of simple techniques to carry out film characterization and wrinkle topography measurements. To overcome these difficulties, here we present a protocol to characterize elastic membranes used in cell locomotion measurements and we develop an interferometric method to account for the wrinkle amplitude.

The membranes are prepared by vulcanization, as described in Refs. [1,[START_REF] Burton | [END_REF], of a polymeric fluid, either PMP (phenylmethyl polysiloxane, Dow Corning 710) or PDMS (poly dimethyl siloxane, Sylgard 184). First, a thin fluid film is formed by spin coating on a clean piece of silicium wafer (Si: B[111], P/E) : A thickness of about 20 µm is achieved after spinning at 4500 rpm during 2 min. Then, an elastic thin membrane, a few nanometers in thickness, is produced at the free surface by the direct effect of a homogeneous flame from a Bunzer-type burner. In order to insure the final membrane homogeneity, the silicium sample is displaced at constant speed by means of a motorized mount, the contact time of the flame with the fluid surface being of about 1 s.

The first relevant quantity necessary for the interpretation of the wrinkled pattern is the membrane thickness, h. However, in our experimental configuration, nondestructive measurements are difficult. For instance, optical methods do not provide enough contrast as the membrane floats on top of a fluid having similar refraction-index. Hence, we propose to make use of an atomic force microscope. However, such a technique necessitates to separate the membrane from the liquid, which we propose to achieve by transferring, by mechanical contact, onto a clean piece of silicium wafer (Fig. 1). The fluid, which may remain at the new created surface, is removed by spinning the whole. The cleaning process is improved by adding a small drop of an appropriate solvent (isopropilic ethanol). Immediately after spinning in order to avoid contamination, we measure the difference in height between two regions across the film edge by means of an atomic force microscope (AFM) in the contact mode (Fig. 1). The preparation protocol described above is found to lead to PMP (resp.

PDMS) membranes having reproducible thicknesses

h = (18±1) nm [resp. h = (60±2) nm].
The second relevant quantity is the 2D elastic modulus B ≡ Eh/(1 -σ 2 ) associated with the stretching of the membrane (We denote σ the Poisson ratio and E the Young modulus):

In principle, provided that the membrane is thick enough, B can be determined from the force-displacement relation or Hertz law [START_REF] Landau | Theory of Elasticity[END_REF],

F (δ) = 4 3 E 1-σ 2 √ rδ 3 2 = 4 3 B √ r h δ 3 2
, observed during the indentation of the dry membrane lying on the solid substrate with the AFM tip (Fig. 1c, r stands for the radius of curvature of the AFM tip). However, the relation holds true only when the indenter is pushed against an elastic half-space: In our experimental configuration, the membranes are so thin that the measurements are inaccurate and altered by finite size effects (the thickness h compares to r). We propose thus the capillarity-type method sketched in Fig. 2 instead. By using the vulcanization protocol described above, an initially flat membrane is prepared at the open end of a tube (radius r 0 = 1 mm) filled with the polymeric fluid (Fig. 2a). We then impose a pressure difference, ∆P , across the membrane and, imaging the system from the side, we measure the resulting curvature C of the free surface. In the case of a liquid interface, the Laplace law would apply and ∆P = 2ΓC with C the curvature of the interface cross-section and Γ the surface tension (or free energy). In the case of an elastic membrane, the tension of the interface is not a thermodynamical quantity, which we shall denote T 0 , in order to avoid confusion. For a membrane, the minimization of the elastic energy [11] leads to a non-linear relation between the pressure difference ∆P and the curvature C,

∆P ≈ 2T 0 C + 2.66B arcsin(CR) -CR R , (1)
where, the elastic term, proportional to B, accounts for the increase in the tension resulting from the increase in the surface. Thus, the measurement of the non-linear term provides us with an estimate of the elastic modulus B (Fig. 2b). We get B = (240 ± 10) mN/m The third relevant quantity is the tension T 0 of the membrane. As T 0 depends on the preparation of the system, the measurements from the capillary-type method are not applicable for the samples used in the biology experiment, and we propose here in situ measurements, instead. We choose to push vertically on the flat membrane with the suitable tip (Fig. 2c). In this case, the force-displacement relation F (δ) satisfies [11],

F ≈ ξB 1-ν T ν 0 δ, ( 2
)
where ξ = 1.143 ± 0.002 and ν = 0.881 ± 0.003 have been obtained by numerically solving the elasticity problem. As an example, we made use of a microneedle to measure the force-displacement relation in the same experimental condition as in Fig. 2c for the PDMS membrane. The slope indicates T 0 64 mN/m, in fairly good agreement with the value T 0 60.5 mN/m obtained from the pressure-curvature relation (Fig. 2b).

Finally, in the biology experiment, the geometry of the wrinkle-pattern can be characterized by using a Michelson type interferometer in which the sample acts as one of the mirrors.

We made use of two objective lenses (Carl Zeiss CP Acromat, 10 × /0.25), a dichroic cube of 25 mm side, a discharge Na-source, and a video camera (Nikon DXM 1200, 12 Megapixels).

The sample consists of fibroblast-TYPE cells, cultured following the standard procedure described in [START_REF] Burton | [END_REF], attached to a PMP membrane prepared as described above on a silicon

wafer. An O-ring around the substrate and a cover glass form a narrow container, about 0.5 mm depth, for the culture medium and fibroblasts. By imposing a small tilt of the second mirror, one produces regularly-spaced parallel-fringes which are distorted when the membrane wrinkles (Fig. 3a). The contrast is increased by locating a drop of optical fluid on the second mirror of the interferometer and the wrinkles are better visualized when they are perpendicular to the fringes. The profile of each of the wrinkle, in a plane perpendicular to the sample plane, can be reconstructed from the local displacement of the fringes (Fig. 3b&c). The vertical resolution achieved is about 25 nm whereas the spatial resolution is better than 5 µm.

In order to demonstrate the applicability of the method, let us now interpret, in the framework of our previous theoretical analysis of the problem [10], the wrinkle pattern described in Fig. 3, which results from the contraction of a single fibroblast. We expect from the theory the wrinkle maximum-amplitude A √ hr 0 3 τ -T 0 B , where τ stands for the tension imposed by the cell in the adhesion region of typical size r 0 . From the experimental A and r 0 (Fig. 3), we obtain τ 40 mN/m or forces, τ r 0 200 nN, which are equivalent to those reported on [START_REF] Burton | [END_REF]. However, it should be noticed that the dynamical range for force measurement is limited. First, radial forces smaller than T 0 can not be detected and second for a force twice T 0 the cell requires to contract by r 0 /3 since B/T 0 is of the order of unity. As described in [START_REF] Burton | [END_REF] this difficulty is overcome by tuning the membrane elastic properties by either increasing B (thicker membranes) or by weakening the membrane by the use of U V light.

In conclusion, the protocol, which we proposed for the preparation and characterization of the membrane, provides us with the thickness h 20 nm, the tension T 0 27 mN/m and the elastic modulus B 20 mN/m, which together with interferometry methods allows us to measure, in situ, the field amplitude of a wrinkled pattern produced by living cells on a silicone substrate. Our optical method, in addition, might prove useful for a variety of related applications, for instance, the study of the evolution of the cell body itself.

[T 0

 0 = (60.5 ± 0.5) mN/m] for the PDMS membrane, and B = (20 ± 10) mN/m [T 0 = (27 ± 1) mN/m] for the PMP membrane. We mention that we can estimate from the values of B the Young modulus E 3 MPa (resp. E 0.8 MPa) of the PDMS (resp. PMP) membranes.

FIG. 1 :FIG. 3 :

 13 FIG.1:In order to obtain a clean membrane deposited on a silicon wafer, we approach the free surface of the sample from the solid surface (a), which results in a piece of membrane underneath a thin fluid layer (b). The remaining fluid is removed by spinning at large velocity (c). The thickness h [here (17.8 ± 0.2) nm] is measured at one edge of the membrane [image in the optical microscope (d)] which vertical profile (e) is obtained by using the AFM in the contact mode.

Acknowledgments

This work was supported by Conicyt-Chile under Fondap Program N o 11980002.


Slavkov Dragan

Hajdukovic
email: dragan.hajdukovic@physac.me

Quantum vacuum as the cause of the phenomena usually attributed to dark matter

We show that if quantum vacuum fluctuations are virtual gravitational dipoles, then the phenomena usually attributed to hypothetical dark matter, may be considered as a consequence of the gravitational polarization of the quantum vacuum by the immersed baryonic matter; apparently, at least mathematically, the galactic halo of dark matter can be replaced by the halo of the polarized quantum vacuum. The eventual gravitational effects of the quantum vacuum "enriched" with virtual gravitational dipoles, can be revealed by the study of orbits of tiny satellites in trans-Neptunian binaries (for instance UX 25 and Eris-Dysnomia).

According to the Standard Model of Particles and Fields everything is made from quarks and leptons interacting through the exchange of gauge bosons. So far, the Standard Model is the most successful and the best tested theory of all time; together with General Relativity (our best theory of gravitation) it is one of the two cornerstones of our fundamental knowledge.

One of the greatest mysteries in contemporary physics is that in galaxies and clusters of galaxies, the gravitational field is much stronger than it should be according to our theory of gravitation and the existing amount of the Standard Model matter (note that astrophysicists use "baryonic matter" as a synonym for the Standard Model matter). There are two schools of thinking related to this mystery. According to the first school, our theory of gravitation is correct, but the content of the Universe proposed by the Standard Model is highly incomplete; additional sources of gravitational field are halos of hypothetical dark matter in which galaxies are immersed. For the second school the content of the Universe proposed by the Standard Model is correct but our theory of gravity must be modified.

In fact, both schools have in common the incorrect assumption that the matter of the Universe exists in the classical, non-quantum vacuum; hence, quantum vacuum [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] is neglected as the content of galaxies and clusters of galaxies (but also as the content of our Solar System). If quantum vacuum is the source of the gravitational field, any theory that neglects the existence of the quantum vacuum is blind to some crucial gravitational phenomena, and, as a compensation for the lost phenomena, must invoke some artificial mechanisms. The inevitable question is if dark matter and dark energy are such "artificial stuff" which only mimics the phenomena which are in fact caused by the quantum vacuum.

We point out (and it may be the emergence of a third school of thinking) that, if we adopt the working hypothesis that, by their nature, quantum vacuum fluctuations are virtual gravitational dipoles, the phenomena usually attributed to hypothetical dark matter, may be considered as a consequence of the gravitational polarization of the quantum vacuum [START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF][START_REF] Hajdukovic | The signatures of new physics, astrophysics and cosmology?[END_REF][START_REF] Hajdukovic | A new model of dark matter distribution in galaxies[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF] by the immersed baryonic matter within it. An analogous result for dark energy is presented in a separate paper in these proceedings. Let us note that the simplest and the most elegant realization of this hypothesis is, if particles and antiparticles have the gravitational charge of the opposite sign; of course nature may surprise us with a different realization of the gravitational dipoles-like behavior of the quantum vacuum.

It is illuminating to remember that in quantum electrodynamics quantum vacuum behaves [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] as a fluid of virtual electric dipoles. For instance, quantum vacuum, as "ocean" of virtual electric dipoles has a tiny impact (but impact!) on the "orbits" of electrons in atoms. It is known as the Lamb shift [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF]. Another ______________________________________________________________________________ 28th Texas Symposium on Relativistic Astrophysics Geneva, Switzerland --December 13-18, 2015 important example: An electron "immersed" in the quantum vacuum produces around itself a halo of nonrandom oriented virtual electric dipoles, i.e. a halo of the polarized quantum vacuum. This halo screens the "bare" charge of an electron; what we measure is the effective electric charge which decreases [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF][START_REF]L3 Collaboration 2000 Measurement of the running of the fine-structure constant[END_REF] with distance! These phenomena well established within the quantum electrodynamics make more plausible the effects of the eventual existence of virtual gravitational dipoles.

According to our hypothesis we consider a quantum vacuum fluctuation as a system of two gravitational charges of the opposite sign; consequently the total gravitational charge of a vacuum fluctuation is zero but it has a non-zero gravitational dipole moment g p :

c m g    g g p d p , (1)
Here, m g denotes the magnitude of the gravitational charge, while, by definition, the vector d directed from the antiparticle to the particle, and has a magnitude d equal to the distance between them. The inequality in (1) follows from the fact that the size d of the virtual pair must be smaller than the reduced Compton wavelength. Consequently, the gravitational polarization density

  

. In addition, following the previous results [START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | The signatures of new physics, astrophysics and cosmology?[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF], in calculations concerning dipoles, we use the mass of a pion  m (which is a typical mass in the physical vacuum of quantum chromodynamics).

In the limit of zero external gravitational field quantum vacuum may be considered as a fluid of randomly oriented gravitational dipoles (Figure 1); both the total gravitational charge and g P are zero.

Random orientation of virtual dipoles may be broken by the immersed Standard Model matter. Massive bodies (stars, black holes …) but also multi-body systems as galaxies are surrounded by an invisible halo of the gravitationally polarized quantum vacuum, i.e. a region of non-random orientation of virtual gravitational dipoles (Figure 2). This halo of the polarized quantum vacuum acts as an effective gravitational charge. Namely, the spatial variation of the gravitational polarization density g P , generates a gravitational bound charge density [START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF] of the quantum vacuum

g P    qv  (2)
which may explain phenomena usually attributed to dark matter.

In the simplest case of spherical symmetry, the equation ( 2) reduces to

        g P   r P r P r dr d r r g g qv ; 1 2 2  (3)
The most robust solution corresponds to the region of saturation (a sphere with a characteristic radius R ); the effective gravitational charge of the quantum vacuum is given [START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF] by

       m M R r r P r M b sat g qv    ; 4 2 max (4)
It is obvious that in the region of saturation, equation ( 4 According to equation ( 4), in the region of saturation, the gravitational polarization of the quantum vacuum produces an additional anomalous constant gravitational field oriented towards the centre. If such acceleration exists it causes a retrograde precession of the perihelion of the satellite in a binary; the anomalous acceleration and precession per orbit in radians are respectively [START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF]

    2 max max 8 , 4 a P GP g g qv g qv     (5)
where a and  are respectively the semi-major axis of the orbit and the total mass of the binary.

Transneptunian binaries (mini planets with miniscule satellites, as for instance UX25) are the best natural laboratory [START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF] to test the prediction (5) that quantum vacuum causes an anomalous perihelion precession (in fact a gravitational version of the Lamb shift!). Astronomers have already started [START_REF] Gai | Astrometric detection feasibility of gravitational effects of quantum vacuum[END_REF] the preparation of the future tests.

Considering quantum vacuum as an ideal system of non-interacting gravitational dipoles in an external gravitational field (analogous to polarization of a dielectric in external electric field, or a paramagnetic in an external magnetic field) leads [START_REF] Hajdukovic | A new model of dark matter distribution in galaxies[END_REF] to the following approximation

         r R r P r M sat g qv tanh 4 2 max  (6)
valid also for the region sat R r  which is the most relevant for "dark matter" phenomena. It is encouraging that such a simplified model for a galaxy (spherical symmetry and ideal system of non-interacting gravitational dipoles) gives good results. Let us give a few examples. First, our theory of the gravitational effects of the quantum vacuum leads [START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF] to the Tully-Fisher relation which is one of the most robust empirical results, unexplained by "dark matter". Let us note that at this point (Tully-Fisher relation) MOND is more successful than "dark matter" theory. The significant success of MOND is a sign that there is something special about their acceleration

gP

  , i.e. the gravitational dipole moment per unit volume, may be attributed to the quantum vacuum. It is obvious that the magnitude of the gravitational polarization density P satisfies the inequality to the random orientations of dipoles, while the maximal magnitude max g P corresponds to the case of saturation (when all dipoles are aligned with the external field). The most plausible theoretical estimate [

.

  ) predicts a universal surface density In fact, such a surprising surface density (nearly the same for all galaxies!) was ______________________________________________________________________________ 28th Texas Symposium on Relativistic Astrophysics Geneva, Switzerland --December 13-18, 2015 observed[START_REF] Kormendy | Scaling laws for dark matter halos in late-type and dwarf spheroidal galaxies[END_REF][START_REF] Donato | A constant dark matter halo surface density in galaxies Mon[END_REF] and presents a surprise and mystery for dark matter paradigm.

Figure 1 .Figure 2 .

 12 Figure 1. Randomly oriented gravitational dipoles (in absence of an external gravitational field)

  model there is not any modification of the Newton's law, as proposed by MOND, for gravitational fields weaker than 0 a . In our model, 0 a is rather a transition point, from saturation in stronger fields to non-saturated polarization in weaker fields. Second, according to observations[START_REF] Boylan-Kolchin | [END_REF] the median Milky Way mass within 260kpc is

Acknowledgments: We thank Hipotekarna Banka in Montenegro for sponsorship of our participation on the 28 th Texas Symposium on Relativistic Astrophysics.

. Third, apparently the best estimate [13] for the local dark matter density (an average over a small volume, typically a few hundred parsecs around the Sun) is Let us end with three important notes-clarifications. First, it is obvious that gravitational field can align only quantum vacuum fluctuations which are gravitational dipoles but not electric dipoles. Second, only a weak interaction such as gravity can polarize large volumes of quantum vacuum and create large galactic halos! Namely, the gravitational acceleration produced by a pion (roughly a typical mass in the physical vacuum of quantum chromodynamics) at the distance of its Compton wavelength is . The mean distance between two dipoles which are first neighbours is one Compton wavelength; hence, the small value of this acceleration indicates that the gravitational polarization can be caused by a very weak gravitational field. Third, and the most important, there is a maximum size of the halo for each massive body, galaxy or cluster of galaxies; simply, after a characteristic size the random orientation of dipoles dominates again. A halo of the maximum size can be formed only if other bodies are sufficiently far; hence galactic halos, and consequently the effective gravitational charge of the Universe, increase with the expansion of the Universe asymptotically reaching a maximum size. The ratio of the effective gravitational charge and the baryonic mass of the Universe is not a constant, contrary to the ratio of the quantity of hypothetical dark matter and baryonic matter.


Slavkov Dragan

Hajdukovic
email: dragan.hajdukovic@physac.me

What if quantum vacuum fluctuations are virtual gravitational dipoles?

The hypothesis stated in the title might be the basis for a new model of the Universe; we give a pedagogical overview of the new model. According to the new model, the only content of the Universe is the known Standard Model matter immersed in the quantum vacuum "enriched" with virtual gravitational dipoles. Apparently, what we call dark matter and dark energy can be explained as the local and global effects of the gravitational polarization of the quantum vacuum by the immersed baryonic matter. Further, the hypothesis leads to a cyclic model of the Universe with cycles alternatively dominated by matter and antimatter; with each cycle beginning with a macroscopic size and the accelerated expansion. Consequently, there is no singularity, no need for inflation field, and there is an elegant explanation of the matterantimatter asymmetry in the universe: our universe is dominated by matter because the previous cycle of the Universe was dominated by antimatter. The forthcoming experiments (AEGIS, ALPHA, GBAR …) will reveal if particles and antiparticles have gravitational charge of the opposite sign, while study of orbits of tiny satellites in trans-Neptunian binaries (e.g. UX25) can be a reasonable test of some astronomical predictions of the theory.

Introduction Contemporary physics has two cornerstones.

The first cornerstone is the Standard Model of Particles and Fields. According to the Standard Model everything is made from apparently structureless fermions (quarks and leptons) which interact through the exchange of gauge bosons (photons for electromagnetic, gluons for strong, and W + , W - and Z for weak interactions).

So far, the Standard Model is the most successful and the best tested theory of all time. The recent LHC experiments at CERN have been a new triumph for the Standard Model contrary to the mainstream conviction that experiments will be a triumph of supersymmetric theories.

The second cornerstone is Einstein's General Relativity, i.e. our best theory of gravitation. The Standard Model tell us what is the content of the Universe, while, for a given content, General Relativity can tell us how the Universe evolves in time.

The cosmological principle (i.e. the assumption of the large-scale homogeneity and isotropy of the Universe) determines the Friedman-Lemaitre-Robertson-Walker (FLRW) metric (see for instance [START_REF] Hobson | General Relativity an Introduction for Physicists[END_REF]): The dynamics of the above space-time geometry is entirely characterised by the scale factor R(t) which is solution of the Einstein equation

              
     T c G G 4 8   (2)
Einstein tensor  G is determined by FLRW, but in order to solve the Einstein equation we must know Energy-momentum tensor  T . Key point: Energy-momentum tensor is approximated by energy-momentum tensor of a perfect fluid characterised at each point by its proper density  and pressure p .

If cosmological fluid consists of several distinct components denoted by n, the final results are cosmological field equations [START_REF] Hobson | General Relativity an Introduction for Physicists[END_REF]:

          n n n c p R G R 2 3 3 4     (3) kc R G R n n      2 2 3 8  (4)
The cosmological field equations can be solved only if we know the content of the Universe. According to the Standard Model the content of the Universe are three cosmological fluids:

Non-relativistic Standard Model matter (usually called pressureless matter or dust)

0 , 3 0 0         m m m p R R   (5)
Relativistic Standard Model matter (usually called radiation)

2 4 0 3 1 , c p R R r r o r r            (6) Quantum vacuum 2 constant, c p qv qv qv      (7)
Note that index 0 denotes the present day value. The first two fluids are well established content of the Universe, while the cosmological constant problem (a nice name for the worst prediction in the history of physics) prevents the use of the Standard Model vacuum as the content of the Universe [START_REF] Weinberg | The cosmological constant problem[END_REF].

A Standard Model physicist can calculate the mass-energy density ve  of the quantum vacuum.

The result of calculations [START_REF] Weinberg | The cosmological constant problem[END_REF] is given by the first of equations ( 8):

R M M M c Mc Mc c qv Mc c c ve         3 3 4 2 2 2 ; 2 16 1           (8)
For instance, for vacuum energy density in quantum chromodynamics the cut-off mass M c is roughly mass of a pion and the density is about 5.6×10 13 Instead of nearly 10 14 kg/m 3 you will get about 10 -27 kg/m 3 in agreement with observations! It might be a miraculous numerical coincidence but it might be the true nature of quantum vacuum as well.

At the end of this introduction let us remember that astronomical observations have revealed a series of phenomena which are a complete surprise and mystery for contemporary theoretical physics.

First, in galaxies and clusters of galaxies, the gravitational field is much stronger than it should be according to our theory of gravitation and the existing amount of the Standard Model matter (note that astrophysicists use "baryonic matter" as a synonym for Standard Model matter). The mainstream "solution" is ad hoc introduction of dark matter of unknown nature.

Second, in certain periods of its history the expansion of the Universe is accelerated. The mainstream "explanation" is dark energy of unknown nature.

Third, our Universe is dominated by matter. Apparently, in the primordial Universe, something has forced the matter-antimatter asymmetry. The mainstream "solution": A kind of CP violation, but please do not be misled by the known CP violation; it must be an unknown type of CP violation, many orders of magnitude stronger than the known one! Fourth, the cosmological constant problem already mentioned above.

In addition the Big Bang model has two inherent troubles, the initial singularity and conflict with observations. For instance, the old Big-Bang theory predicts the existence of the cosmic microwave background (CMB), but contradicts its major characteristics: high level of homogeneity and isotropy. The current mainstream "solution" is cosmic inflation, within the first 10 -30 seconds. The first content of the Universe was not matter but a mysterious hypothetical inflation field; the creation of matter has happened after inflation, at a macroscopic size, when the energy concentrated in the inflation field was converted into particle-antiparticle pairs.

We are faced with so many fundamental problems and presumably the fall of supersymmetries; apparently physics goes through the greatest crisis in its history.

Quantum vacuum and virtual gravitational dipoles: the common solution to all problems?

Quantum vacuum is an essential part [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] of the Standard Model of Particles and Fields.

"Nothing is plenty" as nicely said by Aitchison in his classical review [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF]. Physical vacuum is plenty of quantum vacuum fluctuations, or, in more popular wording, of short-living virtual particleantiparticle pairs (for instance, the lifetime of a virtual electron-positron pair is only about 10 -21 s) which in permanence appear and disappear (as is allowed by time-energy uncertainty relation). Quantum vacuum is an omnipresent state of matter-energy apparently as real as the familiar states: gas, liquid, solid, plasma in stars, quark-gluon plasma… And, quantum vacuum is a state with perfect symmetry between matter and antimatter; a particle always appears in pair with its antiparticle.

Let us briefly consider a few illuminating phenomena in quantum electrodynamics (QED).

In QED, quantum vacuum can be considered as an omnipresent "ocean" of virtual electric dipoles with random orientation; the random orientation of virtual dipoles can be perturbed only by a very strong electric field. For example, the electric field of an electron at the distance of its Compton wavelength is of the order of 10 14 V/m. Such a strong field perturbs the random orientation. Hence, roughly speaking, an electron "immersed" in the quantum vacuum produces around itself a halo of non-random oriented virtual electric dipoles, i.e. a halo of the polarized quantum vacuum. The size of a halo is roughly of the order of the Compton wavelength of an electron; as far as we know, there are no sufficiently strong electric fields in nature to produce halos of the macroscopic size. Thus, the observed charge of an electron, or equivalently the fine structure constant should be position dependent; today, this theoretical prediction is a confirmed reality [START_REF]L3 Collaboration 2000 Measurement of the running of the fine-structure constant[END_REF].

Quantum vacuum, as "ocean" of virtual electric dipoles has a tiny impact (but impact!) on the "orbits" of electrons in atoms [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF]. It is known as the Lamb shift.

In QED, vacuum fluctuations can be converted into real particles. For instance, a virtual electronpositron pair can be converted to a real one by an external field which, during their short lifetime, can separate particle and antiparticle to a distance of about one reduced Compton wavelength; this is known as the Schwinger mechanism [START_REF] Schwinger | On gauge invariance and vacuum polarization[END_REF]. The Schwinger mechanism is not the only way to create something from the quantum vacuum; for instance virtual photons can be converted into directly observable real photons [START_REF] Wilson | Observation of the dynamical Casimir effect in a superconducting circuit[END_REF].

If quantum vacuum fluctuations are virtual gravitational dipoles, we may expect the gravitational version of the above phenomena. For instance the gravitational version of the Lamb shift would be if quantum vacuum produces a tiny perturbation of orbits of planets and satellites! Hence, it is intriguing to study the astrophysical and cosmological consequences of the working hypothesis that, by their nature, quantum vacuum fluctuations are virtual gravitational dipoles.

The simplest and the most elegant realization of this hypothesis is, if particles and antiparticles have the gravitational charge of the opposite sign; of course nature may surprise us with a different realization of the gravitational dipoles like behavior of the quantum vacuum.

Preliminary results published in a series of papers [START_REF] Hajdukovic | What Would Be Outcome of a Big Crunch[END_REF][START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Do we live in the universe successively dominated by matter and antimatter[END_REF][START_REF] Hajdukovic | Can the New Neutrino Telescopes Reveal the Gravitational Properties of Antimatter Advances in Astronomy[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | Quantum vacuum and virtual gravitational dipoles: the solution to the dark energy problem?[END_REF][START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF][START_REF] Hajdukovic | The signatures of new physics, astrophysics and cosmology?[END_REF][START_REF] Hajdukovic | A new model of dark matter distribution in galaxies[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF] suggest that what we call dark matter and dark energy may be result of the gravitational polarization of the quantum vacuum by the immersed Standard Model matter. This part of the theory can be understood in a simple intuitive way.

According to our hypothesis we consider a quantum vacuum fluctuation as a system of two gravitational charges of the opposite sign; consequently the total gravitational charge of a vacuum fluctuation is zero but it has a non-zero gravitational dipole moment g

p c m g    g g p d p , (9)
Here, m g denotes the magnitude of the gravitational charge, while, by definition, the vector d is directed from the antiparticle to the particle, and has a magnitude d equal to the distance between them. The inequality in [START_REF] Hajdukovic | Do we live in the universe successively dominated by matter and antimatter[END_REF] follows from the fact that the size d of the virtual pair must be smaller than the reduced Compton wavelength (for a larger separation a virtual pair becomes real). Consequently, the gravitational polarization density g P , i.e. the gravitational dipole moment per unit volume, may be attributed to the quantum vacuum. It is obvious that the magnitude of the gravitational polarization density In the limit of zero external gravitational field quantum vacuum may be considered as a fluid of randomly oriented gravitational dipoles (Figure 1); both the total gravitational charge and g P are zero.

Random orientation of virtual dipoles may be broken by the immersed Standard Model matter. Massive bodies (stars, black holes …) but also multi-body systems as galaxies are surrounded by an invisible halo of the gravitationally polarized quantum vacuum, i.e. a region of non-random orientation of virtual gravitational dipoles (Figure 2). This halo of the polarized quantum vacuum acts as an effective gravitational charge. Namely, the spatial variation of the gravitational polarization density g P , generates a gravitational bound charge density [8, 11, 16] of the quantum vacuum g P    qv  [START_REF] Hajdukovic | Can the New Neutrino Telescopes Reveal the Gravitational Properties of Antimatter Advances in Astronomy[END_REF] what may explain phenomena usually attributed to dark matter.

It is obvious that gravitational field can align only quantum vacuum fluctuations which are gravitational dipoles but not electric dipoles; for instance random orientation of electron-positron pairs cannot be broken by a gravitational field, while neutrino-antineutrino pairs and gluon fluctuations might be aligned.

Dark matter as the local effect of gravitational vacuum polarization

In the simplest case of spherical symmetry, the equation ( 10) reduces to

        g P   r P r P r dr d r r g g qv ; 1 2 2  (11)
The most robust solution corresponds to the region of saturation (a sphere with a characteristic radius sat R ); the effective gravitational charge of the quantum vacuum is given by [8, 11 and 16]

       m M R r r P r M b sat g qv    ; 4 2 max (12)
It is obvious that in the region of saturation, equation ( 12) predicts a universal surface density

  max 2 4 g q P r r M    .
In fact, such a surprising surface density (which is the same for all galaxies)

was observed [START_REF] Kormendy | Scaling laws for dark matter halos in late-type and dwarf spheroidal galaxies[END_REF][START_REF] Donato | A constant dark matter halo surface density in galaxies Mon[END_REF] and presents a surprise and mystery for dark matter paradigm. According to equation [START_REF] Hajdukovic | Quantum vacuum and virtual gravitational dipoles: the solution to the dark energy problem?[END_REF], in the region of saturation, the gravitational polarization of the quantum vacuum produces an additional anomalous constant gravitational field oriented towards the centre. If such acceleration exists it causes a retrograde precession of the perihelion of the satellite in a binary; the anomalous acceleration and precession per orbit in radians are respectively [START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF]

    2 max max 8 , 4 a P GP g g qv g qv     (13)
where a and  are respectively the semi-major axis of the orbit and the total mass of the binary.

Transneptunian binaries (mini planets with miniscule satellites, as for instance UX25) are the best natural laboratory [START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF] to test the prediction (13) that quantum vacuum causes an anomalous perihelion precession (in fact a gravitational version of the Lamb shift!). Astronomers have already started [START_REF] Gai | Astrometric detection feasibility of gravitational effects of quantum vacuum[END_REF] the preparation of the future tests (see also the paper of Gai and Vecchiato on the Project QVADIS in the present Proceedings). Considering quantum vacuum as an ideal system of non-interacting gravitational dipoles in an external gravitational field (analogous to polarization of a dielectric in external electric field, or a paramagnetic in an external magnetic field) leads to the following approximation

         r R r P r M sat g qv tanh 4 2 max  (14)
valid also for the region sat R r  which is the most relevant for "dark matter" phenomena.

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3rd International Workshop on Antimatter and Gravity (WAG2015) University College London (UK) --4-7 August 2015 expansion, which means that the polarized quantum vacuum behaves as a fluid with negative pressure, and, according to the cosmological equation (3) it is exactly what is necessary for the accelerated expansion of the Universe. Namely, the cosmological field equation (3) tells us that the expansion of the Universe can be accelerated (

0  R  
) only if there is a cosmological fluid with sufficiently big negative pressure, so that the sum in the equation (3) is negative.

A cyclic Universe alternatively dominated by matter and antimatter

During the expansion of the Universe quantum vacuum converts from a cosmological fluid with negative pressure to nearly pressureless fluid. According to the cosmological field equation ( 3) it means that the accelerated expansion converts to the decelerated one.

The eventual collapse of the Universe cannot end in singularity. There is an ultimate mechanism to prevent it: the gravitational version of the Schwinger mechanism [START_REF] Hajdukovic | What Would Be Outcome of a Big Crunch[END_REF][START_REF] Hajdukovic | Do we live in the universe successively dominated by matter and antimatter[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF] i.e. conversion of quantum vacuum fluctuations into real particles, by an extremely strong gravitational field.

For a constant acceleration g the particle creation rate per unit volume and time, can be written as

m cr m cr n cr m m m c g mc g g n n g g c V t N     2 1 2 2 4 , ; exp 1 d d d                         (15)
An extremely strong gravitational field (stronger than the critical acceleration cr g in the equation ( 15)) would create a huge number of particle-antiparticle pairs from the physical vacuum; with the additional feature that matter tends to reach toward the eventual singularity while antimatter is violently ejected farther and farther from singularity. The amount of created antimatter is equal to the decrease in the mass of the collapsing matter Universe. Hence, the quantity of matter decreases while the quantity of antimatter increases by the same amount; the final result might be conversion of nearly all matter into antimatter. If the process of conversion is very fast, it may look like a Big Bang but it is not a Big Bang: it starts with a macroscopic initial size without singularity and without need for inflation field of unknown nature. In addition, there is an elegant explanation of the matter-antimatter asymmetry in the universe: our cycle of the universe is dominated by matter because the previous cycle was dominated by antimatter (and the next cycle would be dominated by antimatter again).

Just for an illustration of the above qualitative picture, let us limit to the radiation dominated Universe. According to the cosmological field equation (3) With such an enormous conversion rate the matter of our Universe can be transformed into antimatter of the next cycle of the universe in a tiny fraction of second!

3 4 0 0 3 8 R R G R r       (

Other topics

If our hypothesis is correct a black hole made from matter is in fact a white hole for antimatter [START_REF] Hajdukovic | What Would Be Outcome of a Big Crunch[END_REF][START_REF] Hajdukovic | Can the New Neutrino Telescopes Reveal the Gravitational Properties of Antimatter Advances in Astronomy[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF]. Hawking radiation (and the information loss paradox) based on a different model of the vacuum cannot exist. Supermassive black holes in the centre of Andromeda and Milky Way, through the gravitational version of the Schwinger mechanism, may emit a huge quantity of antineutrinos [START_REF] Hajdukovic | Can the New Neutrino Telescopes Reveal the Gravitational Properties of Antimatter Advances in Astronomy[END_REF] which might be detectable with a future generation of neutrino telescopes. Also, supermassive black ______________________________________________________________________________ 3rd International Workshop on Antimatter and Gravity (WAG2015) University College London (UK) --4-7 August 2015 holes may contribute to the high energy component of antimatter in cosmic rays; namely antiparticles created in collisions of infalling matter deep inside the horizon would be violently ejected. Astronomical tests would be complemented with experiments in our laboratories; three competing experiment in CERN will reveal if atoms of antihydogen fall down or fall up…

The 21 st century may be the century of antimatter gravity.

  where k=+1, k=-1 and k=0 correspond respectively to closed, open and flat Universe.______________________________________________________________________________3rd International Workshop on Antimatter and Gravity (WAG2015) University College London (UK) --4-7 August 2015

  ______________________________________________________________________________3rd International Workshop on Antimatter and Gravity (WAG2015) University College London (UK) --4-7 August 2015

  to the random orientations of dipoles, while the maximal magnitude max g P corresponds to the case of saturation (when all dipoles are aligned with the external field). The most plausible theoretical estimate for the previous results[START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | Quantum vacuum and virtual gravitational dipoles: the solution to the dark energy problem?[END_REF][START_REF] Hajdukovic | The signatures of new physics, astrophysics and cosmology?[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF], in calculations concerning dipoles, we use the mass of a pion  m (which is a typical mass in the physical vacuum of quantum chromodynamics).

Figure 1 .Figure 2 .

 12 Figure 1.Randomly oriented gravitational dipoles (in absence of an external gravitational field)

  e. when universe has the size of the Sun), R   has tremendous value of the order of of magnitude greater than the critical acceleration needed for creation of let us say neutron-antineutron pairs. Using the equation (15), the corresponding creation rate for neutronantineutron pairs is of the order of

Acknowledgments: We thank "The Petrovic Njegos Foundation" for sponsorship of our participation on the 3 rd International Workshop on Antimatter and Gravity.

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3rd International Workshop on Antimatter and Gravity (WAG2015) University College London (UK) --4-7 August 2015

In the absence of physical understanding of the phenomenon, the distribution of (real or effective) dark matter in a galaxy is described by empirical laws (NFW profile, Einasto profile, Burkert profile . . .). Contrary to it, distribution ( 14) is not empirical one but comes from fundamental principles. It is encouraging that such a simplified model for a galaxy (spherical symmetry and ideal system of noninteracting gravitational dipoles) gives good results. Let us give a few examples. First, our theory of the gravitational effects of the quantum vacuum leads [START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe Physics of the Dark Universe[END_REF] to the Tully-Fisher relation which is one of the most robust empirical results, unexplained by "dark matter". Second, according to observations [START_REF] Boylan-Kolchin | [END_REF]

Dark energy as the global effect of gravitational vacuum polarization

Roughly speaking there is a maximum size of the halo for each massive body, galaxy or cluster of galaxies. Simply, after a characteristic size the random orientation of dipoles dominates again. However, because of proximity of other bodies, many halos are smaller than the maximum size. A halo of the maximum size can be formed only if other bodies (competing in the polarization of the quantum vacuum) are sufficiently far. This behaviour is schematically presented in Figure 3. Consequently, with the expansion of the universe, galactic halos must increase in size and the content of the effective gravitational charge caused by the polarization of the quantum vacuum. However, it is not an endless increase; once the halos have reached their maximum size they remain unchanged with the further expansion.

If quantum vacuum fluctuations are virtual gravitational dipole, the quantum vacuum in the Universe is a cosmological fluid with the sum of all gravitational charges equal to zero; however, because of the existence of the halos of non-random oriented dipoles, it is a fluid with a large effective gravitational charge. Now, the key point is that there is a period in which the size of the individual galactic halos and the total effective gravitational charge of the quantum vacuum increase with the