Christelle Stodel
email: stodel@ganil.fr

Methods of Targets' Characterization

The paper reviews the characterization's techniques for solid targets used in nuclear physics with special emphasis on actinide targets. The determination of the thickness, isotopic and chemical composition are described for actinide targets during their fabrication process. Accent is given on their monitoring during irradiation.

Introduction

In nuclear physics, targets are an essential ingredient for the success of experiments. Precise knowledge of target properties like e.g., the thickness, its distribution (or homogeneity) over the surface and its purity either isotopically or chemically is irrevocable for proper analysis of experimental data. Indeed, knowledge of the kind and level of target impurities is required for evaluating side reactions competitive to the one under study and in estimating the background level. In addition, the characterization of actinide targets is important as the base material consists often of various radioactive isotopes, including decay products. Although some nuclides are present in a very low amount, they can induce high activity due to their short half-life but also they might interfere the measurements. For instance, in the thermal-neutron induced fission reaction of 238 U target cited in [1], the main contributions (at 37.3% and 5.3 % respectively) in the measured cross-section comes from the high fissility of 235 U present at only 12 ppm in the raw material and from 239 Pu formed after neutron capture in 238 U. The areal density (thickness) of targets and stripper foils is also a crucial parameter for nuclear physics experiments. As a matter of fact, it is taken into account in the kinematical characteristics of the produced nucleus of interest, which will be used to tune correctly the spectrometer in order to optimize its transmission. Moreover, the thickness of the targets influence directly the results of cross-sections measurements which are often required with uncertainties in the level of percent. P. Schillebeeckx et al. [2] and D. Sapundjiev et al. [3] report that the improvement in the quality of cross-section data in various fields (astrophysics, neutron-induced reaction for nuclear power production, fuel cycle) relies mainly in the high quality targets with known properties at any time. Another aspect to be considered is the variation of target properties during particle irradiation due to physical and chemical instability of the material, then the monitoring of various target performance parameters all along the experiments is essential. This paper does not pretend to be exhaustive and read as a textbook, but recalls some general principles and results of the used techniques for which details are provided in the quoted bibliography. As mentioned, only active material and monitoring under irradiation are under consideration, additional information on target characterization can be found in [START_REF] Lommel | Targets for Particle Accelerators in Encyclopedia of Applied High Energy and Particle Physics[END_REF][START_REF] Stolarz | [END_REF][6] and references therein. The section 2 describes specific techniques for thickness, homogeneity and purity characterization applied to active targets during their fabrication process. The section 3 reports on methods performed during the irradiation of the material in order to monitor their thickness.

Characterization of actinide targets

Thickness and homogeneity

For radioactive targets, their activity at a certain time, 𝐴(𝑡), determines directly the number of atoms, 𝑁(𝑡), of a sample through its definition as 𝐴(𝑡) =  𝑁(𝑡) knowing the decay constant, , of the radio-nuclide under consideration. The obtained areal density given in atoms/cm² is then straightforward applicable in the cross-sections equation.

Low geometry -particle counting (LGA)

Counting at a defined solid angle is the most accurate method for activity measurements of alpha emitters, and is also suited for x-ray emitters. The method consists of measuring with a silicon detector, the α-particle recoiling out of the active sample. Namely, the counting particles move undisturbed along a straight line from the source to the detector, other particles, i.e. those out of the opening solid angle or scattered ones are absorbed in the diaphragm placed just before the detector or in side materials (e.g. threads or baffles). All geometric parameters (distance source-detector, knife-edge aperture of the diaphragm, sample size) define precisely the solid angle and the silicon background is as low as possible. With LGA method the uncertainties as low as 0.02 % are feasible [7,8]. As reported in [9], two different set-ups can be used according to the activity of the samples, either with a variable or a fixed source-detector distance (up to 20 cm or 1.6 m under glove box respectively) where uncertainties of 0.3-0.5 % are reached with tens of spectra acquired in about 40 hours. The distribution of the activity over the sample area is achieved by introducing an additional diaphragm other the source and moving it [10].

Radiographic Imaging (RI)

For large area targets (10-30 cm² area), Radiographic Imaging is well suited for activity measurements. Indeed, the active target is exposed in front of a phosphor screen called imaging plate (IP) which traps and stores the radiation energy. Then the IP is scanned with a laser beam which releases the energy as luminescence. This "photo-stimulated luminescence" (PSL) phenomenon is thus proportional to the amount of radiation exposed enabling to investigate the homogeneity of the target layer. The resulting spatial resolutions depend on the radioactive decay properties of the material: less than 200 µm for 198 Au (E = 961 keV, E = 412 keV) [11,12], 42 µm was achieved for 238 U (E = 4151 keV, E = 50 keV) [START_REF] Sibbens | AIP Conf. Proc[END_REF]. In order to determine the absolute target activity a calibration with a reference sample is needed [12] and a thickness standard deviation of ±5.2% is obtained. This process requires about 6 hours per target, it can be applied before and after irradiation as shown in [START_REF] Liebe | Annual Report[END_REF]. As mentioned in [START_REF] Vascon | [END_REF], when test experiments are performed with homologous of the actinide elements, radiotracers are used enabling the application of RI for layer homogeneity measurements.

Isotopic Abundances

Thermal Ionization Mass Spectroscopy (TIMS)

Thermal Ionization Mass Spectrometry (TIMS) is a highly sensitive technique to measure isotope ratio of nuclear material. It combines the thermal ionization effect of the sample to be analysed with a separation corresponding to the mass-tocharge-ratio of each isotope as performed with an electromagnet [16]. This method is applied to an aliquot of the active solution used for the electro-deposition. The atomic abundances of the solution are precisely measured with a dynamic range of the ratios up to 10 6 . The activity fractions of each isotope are then deduced and when applied to the total activity measured with LGA technique, the areal density of each active element in the target is deduced [9,10].

Neutron Activation Analysis (NAA)

Neutron Activation Analysis (NAA) is an analytical technique based on gamma spectroscopy measurements of samples irradiated with neutron in nuclear reactors. This method is based on the identification by measuring the gamma emission of the reaction products 𝐴 𝑍 𝑁+1 of a neutron-induced reaction 𝐴 𝑍 𝑁 (𝑛, 𝛾) 𝐴 𝑍 𝑁+1 , with 𝐴 𝑍 𝑁 the isotope present in the supernatant electrolyte solution to be identified. In order to be unbiased by the total neutron flux, a reference sample can be coirradiated under similar conditions to compare the activities of the unknown sample (𝐴 𝑥 ) and a known standard (𝐴 𝑠𝑡 ), the element mass (m) is then obtained by : 𝑚 𝑥 = 𝑚 𝑠𝑡 (𝐴 𝑥 𝐴 𝑠𝑡 ⁄ ) [17]. The significant advantage of NAA is simultaneous determination of up to 30-40 elements with remarkably low detection limits, especially particular lanthanides and actinides can be evaluated in the range of 0.1-10 ppb due to their high cross-section for thermal neutron capture [18]. The process can be considered as time consuming (6 -24h) compared to other analytical techniques [12] but fits well in cases when a direct radiometric measurement of the target is not possible since (i) the activity is far too high, (ii) in case of αparticle measurements the layer is too thick, or (iii) lateral dimensions of the target do not allow the introduction of the sample into a counter. The NAA technique is applied at Mainz to qualify actinide targets. Fig. 1 displays the -ray spectrum of 1 ml of 244 Pu supernatant solution (extracted in the deposition cell after deposition) irradiated for 2 hours with a thermal neutron beam from TRIGA Mainz reactor at a flux of 10 11 cm -2 s -1 . After a cooling time of 0.5 hour, the ray measurements are performed for 10 minutes, main -rays correspond to 245 Am and 245 Pu.

Chemical identification

Rutherford Back Scattering (RBS)

The thickness of the deposited material and the backing as well as the composition and concentration of the materials can be assigned with the Rutherford Back Scattering (RBS) method [20,21]. A beam of particles or light ions with energy below the Coulomb barrier is impinging on targets, then scattered projectiles are registered by detectors placed at various angles to the incident beam. In order to estimate the homogeneity of the target, the measurement can be performed at different position on the surface. As shown in figure 3 of [22], the spectrum reflects the homogeneity of the deposit (long tail of the uranium low energy peak) and the presence of oxygen.

Energy Dispersive X-Ray Spectroscopy (EDS)

The chemical composition of deposited layer is obtained with energy dispersive X-ray spectrometry (EDS) using an electron beam and X-ray detectors measuring the LM-transition line of present elements [12]. This method is often coupled with a Scanning Electron Microscope in order to investigate the morphology of the targets [START_REF] Sibbens | AIP Conf. Proc[END_REF].

X-Ray Photoelectron spectroscopy (XPS)

More detailed analysis of the composition and binding of the deposition is performed with X-ray photoelectron spectroscopy (XPS). It consists of scanning the target's surface with micro-focused mono-energetics X-ray source and of analysing the emitted electrons. From the shape, position and intensity of photoelectric peaks, one can deduce the chemical state and quantify the element [START_REF] Vascon | [END_REF][START_REF] Sibbens | AIP Conf. Proc[END_REF]. Impurities in the range of percent are analysed with the EDS and XPS methods.

Monitoring of targets under irradiation

Electron attenuation

By comparing the emitted current of an electron beam of 20 or 30 kV with the scattered one passing through a material, a thickness measurement of targets is obtained [START_REF] Mann | Patent DE 10242962 A1 GSI[END_REF][START_REF] Mann | [END_REF]. When targets are mounting on a wheel, the relative thickness information over the target area is achieved by deflecting the electron beam in the radial direction (with a magnetic deflector) across the rotation, as depicted in Fig. 2. Fig. 2. Scheme of the electron gun with associated electronics and target wheel [START_REF] Kallunkathariyil | AIP Conf. Proc[END_REF].

Due to the narrow width of the electron beam, the position resolution is about 0.5 mm and the accuracy in the determination of the relative thickness is ±2%. As the electron beam, delivered by an electron gun, is parallel to the heavy ion beam used for the nuclear reaction, the targets can be analysed during irradiation, and the thickness mapping lasts at least 100 times the period of the wheel in order to achieve a spatial accuracy of 0.5 mm for targets' height of 40 mm. This method enables first to clearly identify the position of the beam on the target and secondly to evaluate the process of pinholes or surface cracks formation according to heavy ion beam dose. With this information, the heavy ion beam position and time structure on the targets can be corrected on-line, moreover it helps in the decision of replacing the targets due to material losses [START_REF] Kallunkathariyil | AIP Conf. Proc[END_REF].

Monitoring with particle emission

Energy loss

For targets thin enough to slow down alpha particles emitted by a source, the measurement of the alpha energy loss is well suited to deduce the effective thickness of the target whose composition and stopping power are known [START_REF] Stolarz | [END_REF], [START_REF] Bimbot | [END_REF][27][START_REF] Hofmann | AASI -Program for simulating energy spectra in alpha spectrometry[END_REF]. This technique can be implemented in vacuum reaction chambers of rotating targets. By placing the wheel between the source and the detector away from the heavy ion beam direction, an on-line evaluation (depending on the source activity) of the targets thickness as a function of their length is achievable with accuracies of ±5%.

 activity

For actinide targets, when the beam is stopped and the spectrometer is tuned to the  emission velocity, the  activity of targets is measured at the focal plane. As described in [29,[START_REF] Brewer | [END_REF], the  spectra measured before and time to time during the experiment show various energy distributions according to the heavy ion beam dose. The position and width of the peaks reveal the changes in the targets' structure at each step of the conditioning process. Indeed, the narrowing of the low energy tail (see figure 3 of [29]) indicates that volatile contaminants from the targets, originated from rests of solvents, resulting from the production process involving electrochemical deposition, evaporates. The saturation of the shape of the spectrum reveals that the targets' structure changes into a homogeneous transparent glass-like layer during irradiation. As plotted in figure 6 of [START_REF] Brewer | [END_REF], a gradually growth of the low-energy part suggested that the target was covered by a layer of some material which was confirmed by its visual inspection. In addition, the losses of material due to irradiation can be checked by two methods: firstly with an  source moved into the beam position in front of the target, where the counting rate of the  particles from the target is compared to the one from source; or secondly by analysing the charge equilibration foil which collects the scattered target material [29]. The -spectra obtained at the focal plane during experiment reveals relatively large shift and broadening of the  lines compared to the ones measuring the scattered target material collected in the charge equilibration foil after experiment. The biggest contributions (estimated to be ≈60%) to these differences are due to the energy loss of the  particles in the materials (targets and detectors before the focal plane detector), nevertheless a further contribution arises to the mechanical change of the targets which become dented during irradiation [29]. Fig. 3. Scheme of the a-activity method for active targets.

Rutherford scattering

One common technique to monitor the target thickness at the beam impact during irradiation is to register the elastically scattered projectiles and normalize their count rate to the beam current. The detection is performed with scintillation detectors mounted at opposite angles (± 30° in fusion-evaporation reaction) relative to the beam axis. The accuracy of the measurements depends on the precise knowledge of the solid angle of the detector.

Temperature

Due to the increase of beam intensities, targets' heating is a major issue. The target temperature can be monitored on-line by means of pyrometer [31] or infra-red devices (camera, thermo-viewer or fiberscope) [32][33][34]. As the targets are in a high-radiation environment, the thermal systems have to be shielded and often deported on large distance, which requires precise adjustment of the measuring position and efficient optical mirror and lenses. With the system of [33], a position resolution on targets of 0.5 mm is achieved. An essential parameter in temperature measurements is the overall emissivity on the target surface. It can be determined with heated calibrators and/or part of the surface to be analysed with known emissivity, e.g. heat-resistant black paint covering the material [34]. Complementary devices, such as CDD camera or endoscopes are used to monitor the beam spot position [33].

Summary

Due to the increasing demand for more accurate data and to the upgrade of accelerators, high quality and stable targets with precisely known properties are requested. As listed above, techniques are on common use to characterize actinide targets in order to improve the manufacture processes and to monitor and control them during irradiation in order to maintain their integrity. These techniques are also essential for the development and evaluation of new target production techniques.

Fig. 1 .

 1 Fig. 1. -ray spectrum of a 244 Pu solution irradiated at the research reactor TRIGA Mainz [19].

Fig. 4 .

 4 Fig. 4. Principle of the Rutherford scattering method to monitor targets' thickness.

Acknowledgments.

For their help in achieving this review and for giving me materials for the oral presentation, I want to express my gratitude to my colleagues Goedele Sibbens, Klaus Eberhardt, Bettina Lommel and Anna Solarz from European commission, Joint Research Center, Geel; Universität Mainz, Institut fur kernchemie; GSI Helmholtzcentrum für Schwerionenforshung, Darmstadt and Heavy Ion Laboratory, University Warsaw respectively.


Slavkov Dragan

Hajdukovic
email: dragan.hajdukovic@cern.ch

Antimatter gravity and the Universe

Keywords: antimatter gravity experiments, antihydrogen, muonium, positronium, Dirac-Milne cosmology, Lattice Universe, virtual gravitational dipoles, dark matter and dark energy, cyclic universe

The aim of this brief review is twofold. First, we give an overview of the unprecedented experimental efforts to measure the gravitational acceleration of antimatter; with antihydrogen, in three competing experiments at CERN (AEGIS, ALPHA and GBAR), and with muonium and positronium in other laboratories in the world. Second, we present the 21 st Century's attempts to develop a new model of the Universe with the assumed gravitational repulsion between matter and antimatter; so far, three radically different and incompatible theoretical paradigms have been proposed. Two of these 3 models, Dirac-Milne Cosmology (that incorporates CPT violation) and the Lattice Universe (based on CPT symmetry), assume a symmetric Universe composed of equal amounts of matter and antimatter, with antimatter somehow "hidden" in cosmic voids; this hypothesis produced encouraging preliminary results. The heart of the third model is the hypothesis that quantum vacuum fluctuations are virtual gravitational dipoles; for the first time, this hypothesis makes possible and inevitable to include the quantum vacuum as a source of gravity. Standard Model matter is considered as the only content of the Universe, while phenomena usually attributed to dark matter and dark energy are explained as the local and global effects of the gravitational polarization of the quantum vacuum by the immersed baryonic matter. An additional feature is that we might live in a cyclic Universe alternatively dominated by matter and antimatter. In about three years, we will know if there is gravitational repulsion between matter and antimatter; a discovery that can forever change our understanding of the Universe.

Prelude

Nine decades after the discovery of antimatter, we don't know the answer to the simplest question: In which direction would an anti-apple fall in the gravitational field of the Earth, down or up? We all know that an apple falls down, but, no one knows if an anti-apple would also fall down or would fall up. The aim of this review is: (1) a basic description of experiments that would answer this fundamental question and (2) to provide an elementary understanding of theoretical speculations about the possible impact (in astrophysics and cosmology) of eventual affirmation of gravitational repulsion between matter and antimatter.

During the first few decades after the discovery of antimatter (roughly 1930-1960), which are also the first decades of modern cosmology, gravitational repulsion between matter and antimatter was considered as a serious possibility. However, everything changed during the 7 th decade of 20 th Century, when, further thinking about a negative gravitational charge (gravitational mass) of antimatter was suppressed by purely theoretical arguments that antimatter must fall in the same way as matter. In brief, despite the absence of experimental evidence, the gravitational attraction between matter and antimatter was imposed as an absolute truth; any questioning of this prescribed truth was highly damaging to the scientific reputation of rare scientists that had the courage to oppose group-thinking. It is fair to say that the authors of arguments, great scientists and deep thinkers, are not responsible for the later dark period of suppression of alternative thinking. Our understanding of antimatter gravity in the 20 th Century is well reviewed [START_REF] Nieto | The arguments against "antigravity" and the gravitational acceleration of antimatter[END_REF] in the article "The arguments against "antigravity" and the gravitational acceleration of antimatter".

The most beautiful and ingenious of 3 major arguments against "antigravity" was given by Good 2 ; roughly speaking, if "antigravity" exists, the stationary states of a neutral kaon system (composed of a quark and an antiquark) would be perturbed with an inevitable violation of CP symmetry. It was before the discovery of CP violation when a huge majority of physicists (too huge to be right!) was convinced that CP is an exact symmetry of nature; consequently, the presumed non-existence of CP violation was taken as an argument against gravitational repulsion between matter and antimatter.

Perhaps, the most serious arguments against "antigravity" are improved versions (for a brief overview see Ref. 3, Section 5.5) of the initial Schiff's argument [START_REF] Schiff | Gravitational properties of antimatter[END_REF] . The essence of these arguments is that virtual particle-antiparticle pairs (i.e. the quantum vacuum) dominate the mass of nucleons and hence atoms. Consequently, gravitational charges of the opposite sign should produce observable violation of the universality of free fall of macroscopic bodies made of different materials. Of course, this argument seems very plausible. However, even if we neglect already known shortcomings of Schiff's calculations [START_REF] Nieto | The arguments against "antigravity" and the gravitational acceleration of antimatter[END_REF] , and recent contra-arguments [START_REF] Chardin | Gravity, antimatter and the Dirac-Milne universe[END_REF] , my question is why we should trust this theoretical prediction after the theoretical debacle called the cosmological constant problem [START_REF] Weinberg | The cosmological constant problem[END_REF] ; the essence of the cosmological constant problem is that the gravitational impact of the quantum vacuum is at least forty orders of magnitude larger than permitted by empirical evidence. How we can trust our calculation of the gravitational impact of the quantum vacuum in one case if it so dramatically wrong in the other case? Let us note (thank you to the excellent Reviewer 1 for this remark) that this argument is not critical in theories that do not use quantum vacuum as a source of gravity (i.e. Dirac-Milne Cosmology and the Lattice Universe, presented in Section 3.1).

In any case this review is not about already reviewed [START_REF] Nieto | The arguments against "antigravity" and the gravitational acceleration of antimatter[END_REF][START_REF] Chardin | Gravity, antimatter and the Dirac-Milne universe[END_REF] theoretical arguments. This review is exclusively about antimatter gravity experiments and cosmological theories that start with the hypothesis of surprising outcomes of these experiments (i.e. gravitational repulsion between matter and antimatter).

Before we continue, let us clarify the relation between the fundamental CPT symmetry and the gravitational repulsion between matter and antimatter. CPT is violated by gravity only if antimatterantimatter interaction is different from matter-matter interaction. Pictorially speaking, according to CPT, an anti-apple must fall in the gravitational field of an anti-Earth in the same way as an apple falls in the gravitational field of the Earth. However, the predictive power of CPT is not enough to impose constraints on matter-antimatter interaction; from the point of view of CPT, both the gravitational attraction and the gravitational repulsion between matter and antimatter are possible.

In principle, a theory of gravity can be compatible, but also can be incompatible with CPT symmetry; at the present stage of our knowledge CPT is not a criterium of validity of a theory of gravity. An intriguing and plausible argument was given by Villata [START_REF] Villata | CPT symmetry and antimatter gravity in General Relativity[END_REF] that General Relativity and CPT are compatible only if there is gravitational repulsion between matter and antimatter.

After suppression during the 20 th Century, in the beginning of 21 st Century we witness a strong, both experimental [START_REF] Amole | Description and first application of a new technique to measure the gravitational charge of antihydrogen[END_REF][START_REF] Bertsche | Prospects for comparison of matter and antimatter gravitation with ALPHA-g[END_REF][START_REF] Brusa | The AEGIS experiment at CERN: measuring antihydrogen free-fall in earth's gravitational field[END_REF][START_REF] Perez | The GBAR antimatter gravity experiment[END_REF][START_REF] Philips | The Muonium Antimatter Gravity Experiment[END_REF][START_REF] Cassidy | Atom control and gravity measurements using Rydberg positronium[END_REF][START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF][START_REF] Gai | Astrometric detection feasibility of gravitational effects of quantum vacuum[END_REF] and theoretical [START_REF] Benoit-Levy | Introducing the Dirac-Milne Universe[END_REF][START_REF] Benoit-Levy | The Dirac-Milne cosmology[END_REF][START_REF] Manfredi | Cosmological structure formation with negative mass[END_REF][START_REF] Villata | Dark energy" in the Local Void[END_REF][START_REF] Villata | On the nature of dark energy: the lattice Universe[END_REF][START_REF] Villata | The matter-antimatter interpretation of Kerr spacetime[END_REF][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[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?[END_REF] , "rebellion" of the hypothesis of gravitational repulsion between matter and antimatter. This rebellion will end within the next 3-4 years; the outcome of this rebellion will be either an experimental disproval of the hypothesis or a scientific revolution.

We live in a time of unprecedented experimental efforts to measure the gravitational acceleration of antimatter. After more than one decade of complex development, three competing experiments at CERN (ALPHA-g [START_REF] Amole | Description and first application of a new technique to measure the gravitational charge of antihydrogen[END_REF][START_REF] Bertsche | Prospects for comparison of matter and antimatter gravitation with ALPHA-g[END_REF] , AEGIS [START_REF] Brusa | The AEGIS experiment at CERN: measuring antihydrogen free-fall in earth's gravitational field[END_REF] and GBAR [START_REF] Perez | The GBAR antimatter gravity experiment[END_REF] ) are close to the measurement of the gravitational acceleration of antihydrogen. However, because of the shut-down of CERN that will last for the next 2 years, the first measurements must be postponed until the end of 2021 or the beginning of 2022. In addition to experiments with antihydrogen at CERN, different experimental groups in other laboratories are preparing experiments with muonium 12 and positronium [START_REF] Cassidy | Atom control and gravity measurements using Rydberg positronium[END_REF] . Of course, complete empirical evidence must be the result of two complementary efforts: experiments in laboratories and astronomical observations. It is encouraging that there are already different ideas for astronomical observations; for instance, a study of orbits of tiny satellites in trans-Neptunian binaries has the potential to reveal the eventual gravitational impact of the quantum vacuum [START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF][START_REF] Gai | Astrometric detection feasibility of gravitational effects of quantum vacuum[END_REF] . While a huge majority of theoretical physicists (perhaps too huge to be right) believe that the result of experiments is known in advance, i.e. that antimatter falls exactly in the same way as matter, it may be a good idea to wait and see.

In parallel with the inevitably long preparation of extremely difficult and sophisticated experiments, different authors have tried to reveal the eventual astrophysical and cosmological consequences of repulsive gravity between matter and antimatter. So far, three radically different and completely incompatible theoretical paradigms have been proposed.

Two of these 3 models (Dirac-Milne Cosmology [START_REF] Benoit-Levy | Introducing the Dirac-Milne Universe[END_REF][START_REF] Benoit-Levy | The Dirac-Milne cosmology[END_REF][START_REF] Manfredi | Cosmological structure formation with negative mass[END_REF] and the Lattice Universe [START_REF] Villata | Dark energy" in the Local Void[END_REF][START_REF] Villata | On the nature of dark energy: the lattice Universe[END_REF][START_REF] Villata | The matter-antimatter interpretation of Kerr spacetime[END_REF] ) assume a symmetric Universe composed of equal amounts of matter and antimatter, with antimatter somehow "hidden" in cosmic voids. However, as we will see below, after this common assumption these two models diverge. Dirac-Milne Cosmology introduces CPT violation while within the Lattice Universe CPT Symmetry is respected; in Dirac-Milne Cosmology the Universe expands with constant speed 𝑐 while in the Lattice Universe, expansion of the Universe is accelerated. In both models' preliminary results are intriguing and encouraging.

At the heart of the third model [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[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?[END_REF] is the working hypothesis that quantum vacuum fluctuations are virtual gravitational dipoles; for the first time, this hypothesis makes possible and inevitable to include the quantum vacuum as a source of gravity. The Standard Model matter (i.e. matter made of quark and leptons interacting through the exchange of gauge bosons) is considered as the only content of the Universe, while phenomena usually attributed to dark matter and dark energy are explained as the local [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?[END_REF] and global [START_REF] Hajdukovic | Quantum vacuum and virtual gravitational dipoles: the solution to the dark energy problem?[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe?[END_REF] effects of the gravitational polarization of the quantum vacuum by the immersed baryonic matter. An additional feature is that we might live in a cyclic Universe [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?[END_REF] with cycles alternatively dominated by matter and antimatter; we live in a Universe dominated by matter because the previous cycle was dominated by antimatter.

This variety of models that exclude each other is welcome in a period of great crisis in physics when we are trying to guess how Nature works.

While this Review is strongly limited to antimatter gravity it is important to mention a remarkable series of papers [START_REF] Blanchet | Gravitational polarization and the phenomenology of MOND[END_REF][START_REF] Blanchet | Model of dark matter and dark energy based on gravitational polarization[END_REF][START_REF] Blanchet | Phenomenology of MOND and gravitational polarization[END_REF][START_REF] Bernard | Phenomenology of dark matter via a bimetric extension of general relativity[END_REF][START_REF] Blanchet | Dipolar dark matter as an effective field theory[END_REF] , which are completely outside of the field of antimatter gravity research. The key point is that, in order to reconcile MOND and Dark Matter paradigms, a negative gravitational charge (let us underscore again, completely unrelated to antimatter) was introduced. The essence of this significant work is well described in the abstract of the initial paper [START_REF] Blanchet | Gravitational polarization and the phenomenology of MOND[END_REF] : "The modified Newtonian dynamics (MOND) has been proposed as an alternative to the dark matter paradigm; the philosophy behind is that there is no dark matter and we witness a violation of the Newtonian law of dynamics. In this paper, we interpret the phenomenology sustaining MOND differently, as resulting from an effect of 'gravitational polarization', of some cosmic fluid made of dipole moments, aligned in the gravitational field, and representing a new form of dark matter. We invoke an internal force, of nongravitational origin, in order to hold together the microscopic constituents of the dipole. The dipolar particles are weakly influenced by the distribution of ordinary matter; they are accelerated not by the gravitational field, but by its gradient or tidal gravitational field." Hence, the hypothetical "dipolar dark matter" is composed of permanent gravitational dipoles (i.e. gravitational charges of the opposite sign); it is obvious that the existence of dipoles of any kind (electric, magnetic, gravitational…) assures the existence of the corresponding polarization (i.e. to some extent, dipoles are aligned in an external field). For completeness, let us note the useful calculations [START_REF] Penner | Gravitational anti-screening as an alternative to dark matter[END_REF][START_REF] Penner | Gravitational anti-screening as an alternative to the ΛCDM model[END_REF] that are similar to the "dipolar dark matter" paradigm, but without a specified nature of gravitational dipoles.

It is intriguing and encouraging that very different theoretical motivations and approaches [reconciliation of MOND and Dark Matter paradigms, compatibility [START_REF] Villata | CPT symmetry and antimatter gravity in General Relativity[END_REF] of General Relativity and CPT symmetry; the simplest possible solution to the cosmological constant problem (See Section 3.2)] lead in the same direction: the existence of negative gravitational charge.

Antimatter Gravity Experiments

The beginning of the 21 st century is marked by

• Three, that have been on-going for more than a decade, active experiments at CERN (ALPHAg [START_REF] Amole | Description and first application of a new technique to measure the gravitational charge of antihydrogen[END_REF][START_REF] Bertsche | Prospects for comparison of matter and antimatter gravitation with ALPHA-g[END_REF] , AEGIS 10 and GBAR [START_REF] Perez | The GBAR antimatter gravity experiment[END_REF] ) which are competing to be the first to measure the gravitational acceleration of antihydrogen.

• Preparation of analogous experiments with muonium [START_REF] Philips | The Muonium Antimatter Gravity Experiment[END_REF] (an exotic atom made of an antimuon and an electron) and positronium [START_REF] Cassidy | Atom control and gravity measurements using Rydberg positronium[END_REF] (a system composed of an electron and an antielectron).

An amusing (while non-scientific) question is why so many great experimentalists waste significant time on experiments whose outcomes are known in advance (according to the nearly unanimous prediction of theorists' antimatter must fall in the same way as matter).

All experiments described below will measure the impact of the gravitational field of the Earth on antimatter. Hence, we will get the first empirical evidence about gravitational interaction between matter and antimatter, while the gravitational interaction between antimatter and antimatter will garner no empirical evidence whatsoever.

ALPHA-g Experiment

ALPHA is an extremely successful if not the best antimatter experiment of all time. In 2010, the ALPHA collaboration achieved the first-ever trapping of cold antihydrogen atoms; a seminal success, opening a new era in the study of antimatter. From that time on, for the ALPHA team, production and trapping of antiatoms has become routine, making possible a long-awaited spectroscopy of antihydrogen as a fundamental tool to look for the eventual differences between matter and antimatter.

After two general purpose traps (ALPHA-1 and ALPHA-2) the ALPHA Collaboration has recently constructed a new ALPHA-g apparatus [START_REF] Bertsche | Prospects for comparison of matter and antimatter gravitation with ALPHA-g[END_REF] devoted to the measurement of the gravitational acceleration of antihydrogen. The experiment is pragmatically divided in two stages; in the first stage the goal is limited to the much easier task of determining the sign of the acceleration, while the precise measurement of acceleration is left for the second stage. It is important and encouraging that a proofof-principle measurement 8 has already been completed.

In brief (See Fig. 1), ALPHA-g consists of a vertically oriented apparatus consisting of two symmetric atom and Penning trap arrangements with a high precision region in the centre. Surrounding the cryostat is a radial time projection chamber tracking detector used for locating antiproton annihilations within the trapping volume. The symmetry of the design is aimed towards conducting equivalent experiments on either end of the trap to set limits on or cancel systematic construction and detection errors.

An important feature is that the magnetic potential in the vertical direction (y-direction in Fig. 1) can be tuned via independent control of the trap mirror currents.

Figure 1. Schematic of the ALPHA-g magnet system, with its cylindrical axis of symmetry oriented in the vertical (y) direction. An external solenoid (purple) generates the uniform solenoidal field required for internal Penning traps and operation of the radial time projection chamber detector (gold). Inset shows details on the upper and precision trap. Two independent atom traps surrounding Penning traps are generated by a set of seven mirror coils (red) and a short octupole (green). A precise analysis trap is formed between the two dark orange coils and a long octupole (blue). Adiabatic transport of antihydrogen atoms between trapping can be accomplished through sequencing four transfer mirror coils (grey). External magnetic error fields can be corrected through rectangular correction coil panels (dark grey). Possible trapping regions range in length from approximately 280 mm (single end atom trap) up to 1.3 m (between the extremes of the two end traps). From Reference 9.

The ALPHA-g trap depth is approximately 540𝑚𝐾 for antihydrogen atoms born at the centre of the magnetic volume (i.e. only atoms with energy smaller than 540𝑚𝐾 are trapped). For sub-540mK antihydrogen in a trap roughly 280mm tall, atoms will bounce over the height approximately 1000 times in 10 seconds. The gravitational potential difference for hydrogen over a distance of approximately 280 mm (i.e. during one bounce) and the corresponding magnetic potential change are roughly equal. After the opening of the trap one fraction of bouncing atoms will be detected at the bottom and the other at the top. Based on the measured values of these fractions, the sign of the acceleration can be determined; for it, a few hundred antihydrogen annihilation events are needed-a data rate which is presently achievable during a single 8 h shift on ALPHA.

AEGIS Experiment

The competing AEGIS experiment 10 plans to measure the vertical deviation of a pulsed horizontal beam of cold antihydrogen atoms; the vertical deviation, which is expected to be a few microns, would be measured using a Moire deflectometer. This is visualised in Fig. 2.

A horizontal beam of antiprotons enters the "moiré" setup consisting of three equally spaced elements: two gratings and a spatially resolving emulsion detector. The two gratings with periodicity 𝑑 define the classical trajectories leading to a fringe pattern with the same periodicity at the position of the detector.

In the transit time of the particles through the device is known, absolute force (and the corresponding acceleration) measurements are possible by employing Newton's second law of mechanics.

To infer the force, the shifted position of the "moiré" pattern must be compared with the expected pattern without force. This is achieved using light and near-field interference, the shift of which is negligible.

Figure 2. Illustration of the Moiré deflectometer technique used in AEGIS experiment.

A divergent antihydrogen beam propagates through two identical gratings. Antihydrogen atoms passing the gratings follow a parabolic path and annihilate on a position sensitive detector; the annihilation points form a fringe pattern which is shifted in the presence of a force (it would be shifted down in the case of gravitational attraction and shifted up in the case of the gravitational repulsion). From Reference 10.

GBAR Experiment

The third competing experiment GBAR [START_REF] Perez | The GBAR antimatter gravity experiment[END_REF] is the closest one to our classical vision of a free-fall experiment: the measurement of the time of flight corresponding to a known change of the height.

The fixed change of the height in the GBAR experiment would be small (roughly about 20𝑐𝑚) and for the measurement to be successful the initial speed of 𝐻 ̄atoms mustn't be bigger than a few metres per second; a speed about 3-4 orders of magnitude smaller than the speed of antihydogen in the ALPHA and AEGIS experiments. Hence, while the final measurement in GBAR is more direct and simpler than in the competing experiments, the preparation of the needed ultra-cold antihydrogen is more difficult task and requires to this point the unprecedented cooling of antihydrogen. Just to get an idea about complexity common to all antimatter gravity experiments let us give a few more details.

In the first step GBAR would not produce atoms of antihydrogen (antiproton and positron) but rather antihydrogen ions (antiproton with two positrons). This is motivated by the fact that ion-cooling techniques are more efficient than techniques of cooling neutral atoms.

In the second step antihydrogen ions would be sympathetically cooled with laser cooled matter ions such as 𝐵𝑒 + to temperatures of less than 10𝜇𝐾 (i.e. with velocities of the order of 0.5 𝑚 𝑠 ⁄ ). After that, the extra positron may be photo detached by a laser pulse, with energy of only a few 𝜇𝑒𝑉 above the threshold, in order to obtain an ultracold antiatom. The time of flight of the resulting free fall should be about 200𝑚𝑠, which can be easily measured to extract the acceleration due to Earth's gravity.

Experiments with positronium

Positronium (Ps) is a hydrogen-like atom composed of an electron and a positron. The ground state lifetime of triplet Ps is 1.4 × 10 -7 𝑠, while for the measurement of gravitational acceleration, lifetimes of a few milliseconds or greater are required. Fortunately, the lifetime of positronium is an increasing function of the principal quantum number 𝑛; intuitively it can be understood as the decrease of annihilation rate because larger 𝑛 means a larger distance between the electron and the positron. For a given 𝑛, the lifetime is longer for the higher values of the angular momentum; the lifetime increases [START_REF] Cassidy | Experimental progress in positronium laser physics[END_REF] as 𝑛 [START_REF] Cassidy | Experimental progress in positronium laser physics[END_REF] for non-circular states and 𝑛 5 for circular states (i.e. states with maximal angular momentum 𝑙 = |𝑚| = 𝑛 -1). For instance, lifetime corresponding to 𝑛 = 30 is respectively a few milliseconds and a few seconds for non-circular and circular states.

Hence, the gravitational experiments are possible only with positronium atoms optically excited to long-lived Rydberg states (i.e. states with large 𝑛). The good news is that there is an encouraging initial success in the creation of excited states of positronium by laser.

An experimental programme [START_REF] Cassidy | Atom control and gravity measurements using Rydberg positronium[END_REF] currently underway at University College London (UCL) has as its long-term goal a gravitational free-fall measurement of positronium atoms. On their long way to success they must overcome many obstacles; among these are the production of positronium atoms in a cryogenic environment, efficient excitation of these atoms to suitably long-lived Rydberg states, and their subsequent control via the interaction of their large electric dipole moments with inhomogeneous electric fields.

Let us underscore that the experiment with the antihydrogen is an experiment in the quark sector, while the experiment with positronium would be in the lepton sector of the Standard Model.

Experiments with muonium

Muonium (Mu) is a hydrogen-like system composed of an antimuon 𝜇 + (which is unstable with a lifetime equal to 2.2𝜇𝑠) and an electron 𝑒 -. Measuring muonium gravityif feasiblewould be the first gravitational measurement of a 2nd-generation particle of the Standard Model.

It is obvious that the lifetime of muonium is limited to the lifetime of antimuon and cannot be made longer. Despite the obstacle of a very short lifetime (and some other obstacles) "The Muonium Antimatter Gravity Experiment (MAGE)" is in development [START_REF] Philips | The Muonium Antimatter Gravity Experiment[END_REF] . The MAGE, which is in fact a difficult application of well-established atom interferometry, is illustrated in Fig. 3. A horizontal muonium beam is directed into a three-grating interferometer. The first two gratings set up an interference pattern that has the same period as the gratings. Gravity causes a phase shift in the interference pattern which is equivalent to the deflection of an individual muonium atom; this phase shift (and hence the gravitational acceleration) is determined by scanning the third grating vertically by several grating periods and measuring the resulting sinusoidal variation in detected Mu intensity. The interferometer is aligned using a soft X-ray beam with wavelength comparable to that of the Mu. This X-ray beam will also be used to determine the phase of an undeflected beam.

Just to get an idea of the complexity of this (and all other) antimatter gravity experiments please note that the Schema of the MAGE experiment contains a "Superfluid He film". Let's explain why.

Antimuons 𝜇 + do not annihilate on contact with ordinary matter (since there are no 𝜇 -in ordinary matter). Once stopped, 𝜇 + can combine with a free electron to form cold muonium. Such production of muonium was achieved in many different materials. The trouble is that muonium atoms are emitted with a thermal velocity distribution and uniformly in 𝑐𝑜𝑠 𝛼 (𝛼 being the angle relative to the surface). With such a source of muonium the gravity experiment is impossible. In order to perform this gravity experiment we must have a parallel monochromatic beam of muonium; otherwise the interference patterns of different atoms will have different phases. At this point the superfluid 𝐻𝑒 film is a solution. In fact, muonium has a negative chemical affinity in superfluid helium, so when it diffuses out of the liquid, it is ejected with a nearly constant velocity (6300 𝑚 𝑠 ⁄ ) normal to the surface. Hence, a superfluid He film is an ideal source of muonium, a parallel nearly monochromatic beam(𝛥𝐸 𝐸 ≈ 0.2% ⁄ ).

Theoretical speculations

What if experiments establish that antimatter falls up?

We must respect two experimental facts. First, particles and antiparticles have the same inertial mass (𝑚 𝑖 = 𝑚̄𝑖 , bar denotes an antiparticle). Second, according to the Weak Equivalence Principle (WEP), which is based on indisputable empirical evidence, for each body made of matter, we can use the equality 𝑚 𝑖 = 𝑚 𝑔 , where 𝑚 𝑔 denotes the gravitational mass (it may be better to say the gravitational charge) of matter that is used in Newton's law of gravity. In principle, we must distinguish between active gravitational mass 𝑚 𝑔𝑎 (as a source of the gravitational field) and passive gravitational mass 𝑚 𝑔𝑝 (as a measure of the gravitational force acting on a body in a given gravitational field); fortunately for matter 𝑚 𝑔 = 𝑚 𝑔𝑎 = 𝑚 𝑔𝑝 and we are free of the complication of two different gravitational charges.

Strictly speaking, if, in the gravitational field of the Earth, antimatter falls up, within the framework of Newtonian gravity it means that antiparticles have a negative passive gravitational charge. We cannot measure the active gravitational charge of antimatter, but the most plausible assumption (and the only assumption that respects CPT symmetry) is that, as in the case of matter, active and passive gravitational charges are equal (𝑚̄𝑔 = 𝑚̄𝑔 𝑎 = 𝑚̄𝑔 𝑝 ), while the mass and gravitational charge of an antiparticle have the opposite sign, i.e. 𝑚̄𝑔 = -𝑚̄𝑖or more impressively 𝑚 𝑔 + 𝑚 ̅ 𝑔 = 0. Of course, we must stay openminded; Nature may surprise us with matter-matter, matter-antimatter and antimatter-antimatter gravitational interactions that cannot be described by any combination of signs of passive and active gravitational mass [START_REF] Manfredi | Cosmological structure formation with negative mass[END_REF] . Now, let us remember the cornerstones of contemporary Cosmology. First, the cosmological principle, i.e. the statement that at any particular time the Universe is isotropic about every point (note that this includes homogeneity), leads to the Friedman-Lemaitre-Robertson-Walker (FLRW) metric:

𝑑𝑠 2 = 𝑐 2 𝑑𝑡 2 -𝑅 2 (𝑡) [ 𝑑𝑟 2 1-𝑘𝑟 2 + 𝑟 2 (𝑑𝜃 2 + 𝑠𝑖𝑛 2 𝜃 𝑑𝜗 2 )], (1)
where k=+1, k=-1 and k=0 correspond respectively to a closed, open and flat Universe. The dynamics of the above space-time geometry is completely characterized by the scale factor R(t), which can be determined only within the framework of a specific theory of gravity. Within the framework of General Relativity, the scale factor R(t) is the solution of Einstein's equation 𝐺 𝜇𝜈 = -(8𝜋𝐺/𝑐 4 )𝑇 𝜇𝜈 . Einstein tensor 𝐺 𝜇𝜈 is determined by the FLRW metric, while the Energy-momentum tensor 𝑇 𝜇𝜈 is approximated by the energy-momentum tensor of a perfect fluid; characterised at each point by its proper density 𝜌 and pressure 𝑝.

If a cosmological fluid consists of several distinct components denoted by 𝑛, the result are the cosmological field equations:

𝑅 ̈= - 4𝜋𝐺 3 𝑅 ∑ (𝜌 𝑛 + 3𝑝 𝑛 𝑐 2 ) 𝑛 , (2)
𝑅 ̇2 = 8𝜋𝐺 3 𝑅 2 ∑ 𝜌 𝑛 𝑛 -𝑘𝑐 2 . (3)
The cosmological field equations can be solved only if we know the content of the Universe: the number of different cosmological fluids and the corresponding functions 𝜌 𝑛 and 𝑝 𝑛 . At this point, the relationship between physics and cosmology can be summarized in a single sentence addressed to physicists by cosmologists: Please, tell us the content of the Universe and we will tell you how the Universe evolves in time. The trouble is that the content of the Universe suggested by our best physics (i.e. the Standard Model of Particles and Fields) is apparently wrong. In order to explain observations, our best model of the Universe (Inflationary ΛCDM invokes mysterious content of the Universe (inflation field, dark matter and dark energy) and even with all this additional stuff of unknown nature we have the problem of the initial singularity (in Big-Bang theory), and we do not know the root of the cosmological constant problem and why matter dominates antimatter in the Universe.

A symmetric matter-antimatter Universe

According to our best knowledge we live in a Universe dominated by matter; this matter-antimatter asymmetry is considered one of the biggest mysteries in physics and cosmology. It is obvious that the gravitational repulsion between matter and antimatter cannot have any impact on the Universe if antimatter is not a significant part of the content of the Universe. The only, easily visible way, to "introduce" needed antimatter in the Universe is to abandon the current paradigm of matter-antimatter asymmetry and to assume a symmetric Universe with equal amounts of matter and antimatter, with antimatter somehow hidden in the cosmic voids. This common hypothesis is used in two different models named Dirac-Milne Cosmology [START_REF] Benoit-Levy | Introducing the Dirac-Milne Universe[END_REF][START_REF] Benoit-Levy | The Dirac-Milne cosmology[END_REF][START_REF] Manfredi | Cosmological structure formation with negative mass[END_REF] (a little bit misleading name because Dirac and Milne are not originators of this theory) and the Lattice Universe [START_REF] Villata | Dark energy" in the Local Void[END_REF][START_REF] Villata | On the nature of dark energy: the lattice Universe[END_REF][START_REF] Villata | The matter-antimatter interpretation of Kerr spacetime[END_REF] .

Before we continue the overview of these apparently "wild" models, let us underscore a crucial fact. Preliminary studies show a surprising and astonishing agreement with observations; more precisely models successfully pass the classical cosmological tests such as primordial nucleosynthesis, Type Ia supernovae and the Cosmic Microwave Background.

Of course, a Universe with huge and equal amounts of matter and antimatter is not empty, but it is gravitationally empty if 𝑚 𝑔 + 𝑚̄𝑔 = 0 i.e. if CPT symmetry is valid for gravity; total gravitational charge and average gravitational charge density are equal to zero. Consequently, Eq. (3) reduces to 𝑅 ̇2 = -𝑘𝑐 2 ; real solutions exist only for 𝑘 = -1 (open Universe) and the scale factor of the Universe is a linear function of time, i.e. 𝑅(𝑡) = 𝑐𝑡 . Hence, the FLRW metric (given by Eq. ( 1)) leads to the following metric for a symmetric matter-antimatter Universe:

𝑑𝑠 2 = 𝑐 2 𝑑𝑡 2 -𝑐 2 𝑡 2 [ 𝑑𝑟 2 1+𝑟 2 + 𝑟 2 (𝑑𝜃 2 + 𝑠𝑖𝑛 2 𝜃 𝑑𝜙 2 )]. (4)
By the way, metric (4) was used by Milne 8 decades ago, but of course without any involvement of antimatter; the point of Milne was that expansion exists without gravitation, i.e. in an empty Universe in the limit when General Relativity reduces to Special Relativity.

The hypothesis that antimatter is hidden in the cosmic voids (and there is apparently no other place to hide) is immediately in big trouble. Why antimatter in voids is invisible; why we see voids instead of stars, galaxies and clusters of galaxies made of antimatter?

As a possible explication of the invisibility of antimatter in voids, Dirac-Milne Cosmology [START_REF] Benoit-Levy | Introducing the Dirac-Milne Universe[END_REF][START_REF] Benoit-Levy | The Dirac-Milne cosmology[END_REF][START_REF] Manfredi | Cosmological structure formation with negative mass[END_REF] proposes what can be pictorially called "double antigravity", there is a first gravitational repulsion between matter and antimatter and a second gravitational repulsion between antimatter and antimatter. As a consequence of the second gravitational repulsion (i.e. repulsion between antimatter and antimatter) antimatter stars cannot exist (this is of course a brutal violation of CPT symmetry, but, as already noticed, not necessarily an argument against theory); hence, antimatter in voids is not in the form of stars and galaxies but just in the form of an invisible antimatter cloud that tends to expand (but expansion is limited by the gravitational repulsion of surrounding matter). However, in my opinion, this solution introduces a new problem: while the Universe remains gravitationally empty for matter it is not more gravitationally empty for antimatter (which is now repelled by both matter and antimatter) and apparently the scale factor of the Universe must be different for matter and antimatter.

As explained in a recent Physical Review article [START_REF] Manfredi | Cosmological structure formation with negative mass[END_REF] , a rigorous formulation of "double antigravity" is possible only within the framework of a bi-metric theory of gravity. In simple terms, the postulated negative active gravitational charge of antimatter would repel other antimatter only if the passive gravitational charge of antimatter is positive. On the other side the positive active gravitational charge of matter repels antimatter only if the passive gravitational charge of antimatter is negative. Hence, in one case the passive gravitational charge of antimatter must be positive and in the other case negative; in other words, gravitational properties of matter and antimatter cannot be described by simply assigning a combination of signs to the three types of Newtonian masses (i.e., the inertial mass, active gravitational mass and passive gravitational mass). Instead, Dirac-Milne Cosmology is formulated as a bi-metric theory. In brief, we witness a major reformulation of Dirac-Milne Cosmology, from a simple and elegant but insufficient metric (Eq. ( 4)) to a coherent bi-metric theory. This is the right place for two comments. First, several authors have noted that our universe is very similar to a gravitationally empty or coasting universe [START_REF] Nielsen | Marginal evidence for cosmic acceleration from Type Ia supernovae[END_REF]38 (neither accelerating nor decelerating); if it is true, Dirac-Milne Cosmology is a possible fundamental explanation. Second, there are two different 𝑅(𝑡) = 𝑐𝑡 cosmologies, Dirac-Milne Cosmology (a symmetric matter-antimatter Universe" and Melia-Shevchuk [START_REF] Melia | The RH = ct universe[END_REF] "𝑅 𝐻 = 𝑐𝑡" Universe (a Universe without antimatter); apparently [START_REF] Chardin | Gravity, antimatter and the Dirac-Milne universe[END_REF][START_REF] Manfredi | Cosmological structure formation with negative mass[END_REF] a symmetric matterantimatter Universe (and it is encouraging for Dirac-Milne Cosmology) is in better agreement with observations. Note that in a 𝑅(𝑡) = 𝑐𝑡 cosmology, 𝑅 ̇(𝑡) = 𝑐 and consequently the Hubble parameter H is exactly equal to 1 𝑡 ⁄ (𝐻 = 𝑅 ̇𝑅 ⁄ = 1 𝑡 ⁄ ). Contrary to Dirac-Milne Cosmology, the Lattice Universe is based on strict respect of CPT symmetry. In fact, as a "prelude" to the Lattice Universe, Villata has argued [START_REF] Villata | CPT symmetry and antimatter gravity in General Relativity[END_REF] that CPT and General Relativity are compatible only if matter and antimatter repel each other. In the Lattice model, the Universe is considered to be like a gravitational lattice (analogous to an electrostatic lattice structure i.e. a crystal). The key result is that the alternation of the unlike (positive and negative) gravitational charges in the Universe produces a net accelerated expansion despite the equal amounts of the two components [START_REF] Villata | On the nature of dark energy: the lattice Universe[END_REF] .

So far, scientists working on the development of Dirac-Milne Cosmology and the Lattice Universe haven't published any paper devoted to the study of a single galaxy. The gravitational field in a galaxy is much stronger that it can be according to the existing amount of Standard Model matter and our law of gravity; this anomalous gravitational field is usually attributed to dark matter, or a MOND-type modification of gravity.

It seems obvious (at least to the author of this Brief Review) that antimatter hidden in the voids cannot explain phenomenon of these mysterious central fields in galaxies. If so, Dirac-Milne Cosmology and the Lattice Universe are not a complete alternative to standard Λ𝐶𝐷𝑀 Cosmology. One possible solution is to include dark matter in both Dirac-Milne Cosmology and the Lattice Universe; hence the content of the Universe would be equal amounts of matter and antimatter plus dark matter. However, the introduction of dark matter can violate the crucial assumption of a gravitationally empty Universe. Hence, in Dirac-Milne Cosmology and the Lattice Universe, the only acceptable dark matter must contain equal amounts of positive and negative gravitational mass; dark matter must be dipolar. It is amusing that dipolar dark matter, apparently needed in these two models, was proposed [START_REF] Blanchet | Gravitational polarization and the phenomenology of MOND[END_REF][START_REF] Blanchet | Model of dark matter and dark energy based on gravitational polarization[END_REF][START_REF] Blanchet | Phenomenology of MOND and gravitational polarization[END_REF][START_REF] Bernard | Phenomenology of dark matter via a bimetric extension of general relativity[END_REF][START_REF] Blanchet | Dipolar dark matter as an effective field theory[END_REF] and very successfully developed within the Dark Matter paradigm.

Quantum vacuum and virtual gravitational dipoles

There is a subtle way for a strong gravitational impact of antimatter in a Universe that is dominated by matter. Instead of antimatter hidden in the cosmic voids (which is the cornerstone of Dirac-Milne Cosmology and the Lattice Universe), the impact of antimatter can come from the quantum vacuum 22- 29 which contains the same number of virtual particles and antiparticles. Historically, this is the first proposed paradigm, but we present it as the last one for pedagogical reasons.

It is well established that the quantum vacuum and matter immersed in it interact through electromagnetic, strong and weak interactions [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] . The open question is, if there are also gravitational interactions between the quantum vacuum and the immersed matter?

The heart of the paradigm is the following working hypothesis: (H1) By their nature, quantum vacuum fluctuations are virtual gravitational dipoles. By the way, the motivation for the hypothesis (H1) comes from the question what is the simplest possible solution to the cosmological constant problem. We know that the electric charge of the quantum vacuum is zero because virtual particles and antiparticles (which have the opposite electric charge and make an electric dipole) always appear in pairs. Consequently, it is obvious that the gravitational charge of the quantum vacuum would be zero (i.e. the quantum vacuum would be free of the cosmological constant problem) if particles and antiparticles have gravitational charge of the opposite sign. Of course, we don't know if this logically simplest solution and the real physical solution to the cosmological constant problem are the same.

According to the above hypothesis, a quantum vacuum fluctuation is a system of two gravitational charges (See a very schematic Figure 4) of the opposite sign; consequently, the total gravitational charge of a vacuum fluctuation is zero, but it has a non-zero gravitational dipole moment 𝒑 𝑔 :

𝒑 𝑔 = 𝑚 𝑔 𝒅, |𝒑 𝑔 | < ℏ 𝒄 .
(

) 5
Here, mg 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 ( 5) follows from the fact that the size d of a quantum fluctuation is smaller than the reduced Compton wavelength (i.e. 𝑑 < ƛ 𝑔 = ℏ 𝑚 𝑔 ⁄ 𝑐).

Figure 4. A virtual gravitational dipole is defined in analogy with an electric dipole: two gravitational charges of the opposite sign (𝑚 𝑔 > 0, 𝑚 𝑔 + 𝑚 ̅ 𝑔 = 0 ) at a distance 𝑑 smaller than the corresponding reduced Compton wavelength ƛ 𝑔 . Note that the existence and impact of virtual electric dipoles is well established [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] .

If gravitational dipoles exist, the gravitational polarization density 𝑷 𝑔 , i.e. the gravitational dipole moment per unit volume, can be attributed to the quantum vacuum. It is obvious that the magnitude of the gravitational polarization density 𝑷 𝑔 satisfies the inequality 0 ≤ |𝑷 𝒈 | ≤ 𝑃 𝑔𝑚𝑎𝑥 where 0 corresponds to the random orientations of dipoles, while the maximal magnitude 𝑃 𝑔𝑚𝑎𝑥 corresponds to the case of saturation (when all dipoles are aligned with the external field). The value 𝑃 𝑔𝑚𝑎𝑥 must be a universal constant related to the gravitational properties of the quantum vacuum. Later we will discuss the possibility of the experimental determination of the eventual universal constant 𝑃 𝑔𝑚𝑎𝑥 .

If the external gravitational field is zero, the quantum vacuum may be considered like a fluid of randomly oriented gravitational dipoles (Figure 5a). In this case everything is equal to zero: the total gravitational charge, the gravitational charge density and the gravitational polarization density 𝑷 𝑔 . Of course, such a vacuum is not a source of gravitation (note again that this is the simplest possible solution to the cosmological constant problem). However, the random orientation of virtual dipoles can be broken by the gravitational field of the immersed Standard Model matter. Massive bodies (particles, stars, planets, black holes…) but also many-body systems such 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 5b).

While the behaviour sketched in Figure 5.b is obvious for permanent gravitational dipoles, if you are not familiar with the quantum vacuum, you may wonder if it is also correct for extremely shortliving virtual dipoles. Fortunately, the phenomenon of the electric polarization of the quantum vacuum [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] is well-established in Quantum Electrodynamics and this makes very plausible an analogous gravitational polarization of the quantum vacuum. More precisely, Quantum Electrodynamics (QED) is our first quantum field theory and the quantum vacuum (as an inherent part of QED) is one of the greatest discoveries in the history of science. One of important phenomena is the electric polarization of the quantum vacuum; in particular, the screening of an electric charge by the surrounding virtual electric dipoles. It is immediately obvious (to everyone familiar with the electric polarization of the quantum vacuum) that other kinds of polarization can exist as well. The gravitational polarization of the quantum vacuum is obvious if quantum vacuum fluctuations are virtual gravitational dipoles (defined in full analogy with electric dipoles).

The spatial variation of the gravitational polarization density generates [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?[END_REF] a gravitational bound charge density of the quantum vacuum

𝜌 𝑞𝑣 = -𝜵 ⋅ 𝑷 𝒈 . (6)
You can consider this gravitational bound charge density to be an effective gravitational charge density, which acts as if there is a real non-zero gravitational charge. That is how the magic of polarization works; the quantum vacuum is a source of gravity thanks to the immersed Standard Model matter. The hypothesis (H1) can be combined with a second hypothesis: (H2) Standard Model matter (i.e. matter made of quarks and leptons interacting through the exchange of gauge bosons) is the only content of the Universe.

The hypothesis (H2) excludes dark matter and dark energy as the content of the Universe, while hypothesis (H1) postulates the quantum vacuum as a cosmological fluid free of the cosmological constant problem. Together, these two hypotheses have the following series of intriguing consequences.

The phenomena usually attributed to hypothetical dark matter and dark energy can be considered [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 | Virtual gravitational dipoles: The key for the understanding of the Universe?[END_REF] as a result of the gravitational polarization of the quantum vacuum by the immersed Standard Model matter. Locally, each halo of dark matter can be replaced by the halo of the polarized 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 | Virtual gravitational dipoles: The key for the understanding of the Universe?[END_REF] . Globally, all halos of the polarized quantum vacuum are a cosmological fluid, which, during the expansion of the Universe converts from a fluid with negative pressure -allowing an accelerated expansion of the Universe -to a fluid with zero pressure, which physically means the end of the accelerated expansion [START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe?[END_REF] .

Additionally, the two hypotheses open the possibility that we live in a cyclic universe with cycles alternatively dominated by matter and antimatter [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?[END_REF] . The conversion of matter (or antimatter) of one cycle to antimatter (or matter) of the next cycle, happens in a cataclysmic event similar to the Big Bang, but at a significant macroscopic size (more precisely at a macroscopic size of the scale factor 𝑅 of the Universe); the cause of conversion is an extremely fast and tremendous creation of particle-antiparticle pairs from the quantum vacuum in an extremely strong gravitational field at a relatively small scale factor 𝑅 of the Universe. Consequently, at least mathematically, there is no initial singularity, there is no need for cosmic inflation and there is an elegant explanation of the matter-antimatter asymmetry in the universe: our universe is dominated by matter because the previous cycle was dominated by antimatter and the next one will be dominated by antimatter again.

Dipolar dark matter versus quantum vacuum

We have seen that two different paradigms, "dipolar dark matter" [START_REF] Blanchet | Gravitational polarization and the phenomenology of MOND[END_REF][START_REF] Blanchet | Model of dark matter and dark energy based on gravitational polarization[END_REF][START_REF] Blanchet | Phenomenology of MOND and gravitational polarization[END_REF][START_REF] Bernard | Phenomenology of dark matter via a bimetric extension of general relativity[END_REF][START_REF] Blanchet | Dipolar dark matter as an effective field theory[END_REF] and "virtual gravitational dipoles" [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[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?[END_REF] have a common concept: there is a cosmological fluid composed of gravitational dipoles and there is gravitational polarization [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?[END_REF] of this fluid (that is caused by immersed Standard Model Matter). Of course, this common concept leads to significant mathematical similarities; the basic Eq. ( 6) appears in both theories. However, despite common points, these two theories are fundamentally different and incompatible.

The first major difference is that in one theory dark matter exists while in the other theory dark matter doesn't exist. Dipolar Dark Matter is a successful unification of two apparently incompatible theories, MOND and Dark Matter. According to the Dipolar Dark Matter paradigm, dark matter exists but is composed of permanent gravitational dipoles; as a result of the gravitational polarization, such a kind of dark matter leads to MOND's equations. On the other hand, according to the paradigm that "quantum vacuum fluctuations are virtual gravitational dipoles" dark matter doesn't exist; dark matter is just an illusion caused by 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 | Virtual gravitational dipoles: The key for the understanding of the Universe?[END_REF] .

It is illuminating to consider "dipolar dark matter" and "virtual gravitational dipoles" paradigms from the point of view of analogy between electric and gravitational polarization. Two well known cases of electric polarization are the polarization inside dielectric materials and the polarization inside the quantum vacuum [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] . Dipolar Dark Matter is the gravitational analogue of a dielectric; a still unknown (that remains to be discovered) kind of matter beyond the Standard Model of Particles and Fields. Virtual gravitational dipoles are the analogue of virtual electric dipoles in the quantum vacuum; there is no need to invoke new content of the Universe, Standard Model Matter and the quantum vacuum "enriched" with virtual gravitational dipoles can thus be the only content of the Universe.

Let us underscore the second fundamental difference. The amount of dipolar dark matter in the Universe is a constant; consequently, the ratio of dipolar dark matter and the baryonic matter in the Universe is a constant. Dipolar Dark Matter is a pressureless cosmological fluid. By contrast, the effective gravitational charge of the quantum vacuum is not a constant but depends on the scale radius of the Universe. Hence, the ratio of the effective gravitational charge of the quantum vacuum and baryonic matter is a variable; the polarized quantum vacuum is a cosmological fluid that in some periods of expansion can have negative pressure and consequently has the potential to explain phenomena usually attributed to dark energy.

The third crucial difference is the description of the gravitational field caused by an isolated pointlike body. Within the Dipolar Dark Matter paradigm (and it is also valid in Dirac-Milne Cosmology and the Lattice Universe), the gravitational field of a point-like body (that is far from any other matter or dark matter) is classically described by Newton's inverse square law (or the Schwarzschild metric if General Relativistic effects cannot be neglected). However, within the quantum vacuum paradigm, a point-like body is not a point-like source of gravity, because it is inseparable from the halo of the polarized quantum vacuum around it; a halo that can extend to very large distances (for instance the halo of the Sun is much larger than the Solar System). Consequently, while there is no violation of Newton's law, the point-like body as one source of gravity, together with the inevitable effective gravitational charge of the polarized quantum vacuum as the second source of gravity, produce a composite gravitational field that cannot be described with the simple inverse square law.

In brief, the above differences tell us how crucial the nature of gravitational dipoles is; different kinds of gravitational dipoles can produce radically different effects.

Outlook and Astronomical observations

If, in experiments in our laboratories, antimatter falls upwards, it would be a scientific revolution, but not confirmation of any of the proposed astrophysical and cosmological consequences. It is obvious that eventual discovery of gravitational repulsion between matter and antimatter would not be confirmation of other hypothesis in the proposed theories; experiments with antihydrogen, muonium and positronium cannot tell us if there is antimatter hidden in voids, if there is gravitational repulsion between antimatter and antimatter, if quantum vacuum fluctuations behave as gravitational dipoles… Astrophysical and cosmological phenomena can be revealed only by astronomical observations, and we are lucky that new generation telescopes (from the James Webb Space Telescope to the Extra-Large Telescope) will be operative nearly immediately after a surprising discovery that can come from CERN.

In general, significantly higher precision of astronomical measurements will increase our capacity to distinguish between predictions of different theories.

In addition to cosmological considerations it is illuminating to consider a black whole from the point of view of gravitational repulsion between matter and antimatter and three competing theories described in this review.

Black-white holes

Let us consider a black hole made of matter. If there is gravitational repulsion between matter and antimatter, such an object is a black hole for matter, but not a black hole for antimatter; any antimatter inside the horizon would be violently ejected by repulsion. We propose the name "black-white hole" for such entity, but please note that words black and white do not have the same meaning as in General Relativity.

Everyone thinks that experiments at CERN are just a measurement of the gravitational acceleration of antihydrogen. It is amusing that no one noticed that in fact, experiments at CERN are a test if blackwhite holes exist in the Universe. If antihydrogen falls upwards, black holes must be renamed to blackwhite holes; a black hole made from matter is a black hole for matter but a white hole for antimatter. If antihydrogen falls upwards it is an inevitable phenomenon independent of any theory.

Let us imagine matter falling into a black-white hole. A tiny fraction of falling matter will be ejected back in the form of high-energy antiparticles. Namely, as the result of the collisions of the infalling material (analogous to collisions in our accelerators), different kinds of antiparticles can be created inside the horizon and long-living antiparticles would be violently ejected outside the horizon. If black-white holes exist, they are an inevitable source of high energy positrons and antiprotons in cosmic rays.

An intriguing question is if two different signatures of these black-white holes have already been seen. The first signature may be an unexplained excess of high-energy positrons and antiprotons in cosmic rays [START_REF] Accardo | High Statistics Measurement of the Positron Fraction in Primary Cosmic Rays of 0.5-500 GeV with the Alpha Magnetic Spectrometer on the International Space Station[END_REF] revealed by measurements with the Alpha Magnetic Spectrometer on the International Space Station. The second signature may be a recent detection, at the IceCube neutrino telescope at the South pole, of very high-energy (anti)neutrinos coming from the galactic centre [START_REF] Bai | Neutrino lighthouse at Sagittarius A*[END_REF] ; apparently the Milky Way's supermassive black hole acts as mysterious "factory" of high-energy (anti)neutrinos.

There is a second, more subtle mechanism for creation of particle-antiparticle pairs deep inside the matter horizon. Let us remember that quantum vacuum is an inherent part [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] of the Standard Model of Particles and Fields and that under certain conditions virtual particle-antiparticle pairs from the quantum vacuum can be converted into real particles; we can create something from apparently nothing. For instance, an electron and a positron in a virtual pair can be converted to real ones in a sufficiently strong electric field accelerating them in the opposite direction. The same (i.e. creation of particle-antiparticle pairs from the quantum vacuum) can be done by the gravitational field if particles and antiparticles have gravitational charge of the opposite sign; the only difference is that the needed opposite acceleration is caused by a gravitational field.

Hence, black-white holes might radiate because of particle-antiparticle creation from the quantum vacuum [START_REF] Hajdukovic | Can the New Neutrino Telescopes Reveal the Gravitational Properties of Antimatter[END_REF] . It is obvious that this is a model dependant mechanism (for instance not valid in Dirac-Milne Cosmology and the Lattice Universe). A major question is if Hawking radiation can coexist with quantum vacuum radiation? The answer is: No. Hawking radiation depends on the heretofore assumed model of the gravitational properties of the quantum vacuum. Hawking calculations correspond to the case of gravitational monopoles and cannot be valid if the quantum vacuum is composed of gravitational dipoles.

Trans-Neptunian binaries and quantum vacuum

As a model-dependent phenomenon (not existing in Dirac-Milne Cosmology and the Lattice Universe) let us mention that the quantum vacuum might have a tiny impact on orbits of celestial bodies in the Solar System; it is the gravitational analogue of the Lamb shift [START_REF] Aitchison | Nothing's plenty -the vacuum in modern quantum field theory[END_REF] (i.e. of the impact of the quantum vacuum on energy levels of electrons in atoms) in Quantum Electrodynamics. Apparently, the most promising way to reveal such an impact of the quantum vacuum is the study of orbits of tiny satellites in some trans-Neptunian binaries [START_REF] Hajdukovic | Can observations inside the Solar System reveal the gravitational properties of the quantum vacuum?[END_REF][START_REF] Gai | Astrometric detection feasibility of gravitational effects of quantum vacuum[END_REF] . The key point is that in an isolated binary, the quantum vacuum causes a perihelion shift per orbit ∆𝜔 𝑞𝑣 which is directly proportional to the maximal magnitude 𝑃 𝑔𝑚𝑎𝑥 of the gravitational polarization density 𝑷 𝑔 (See section 3.2 after Eq. ( 5)).

Brief summary

Our current understanding of the Universe (Λ𝐶𝐷𝑀 Cosmology) is both, a fascinating intellectual achievement and the source of the greatest crisis in the history of physics. We do not know the nature of what we call an inflation field, dark matter and dark energy; we do not know why matter dominates antimatter in the Universe and what the root of the cosmological constant problem is.

In about three years from now experiments will end the already 6 decades old theoretical dispute of whether antimatter falls down or falls up. Experiments will tell us if antimatter gravity is crucial or not in the understanding of the greatest mysteries of contemporary physics, astrophysics and cosmology.

So far there are three pioneering theories (Dirac-Milne Cosmology, the Lattice Universe and Cosmology with quantum vacuum fluctuations as virtual gravitational dipoles) that anticipate repulsive gravity as the outcome of the forthcoming experiments. As explained in this Review (and summarized in Table 1) these theories are radically different and mutually excluding which is good for the development of science. For the first time we have a theory that (free of the cosmological constant problem) considers the quantum vacuum as an inevitable source of gravity and provides a common explanation of phenomena usually attributed to dark matter and dark energy; without the need for cosmic inflation and replete with the striking explanation of matterantimatter asymmetry in the Universe (a cyclic universe alternatively dominated by matter and antimatter). The other two models (Dirac-Milne Cosmology and the Lattice Universe) do not use the quantum vacuum as a source of gravity but they achieved intriguing initial success assuming a symmetric matter-antimatter Universe.

Let us wait and see if antimatter gravity experiments will be the birth of a new scientific revolution.

Figure 3 .

 3 Figure 3. Schema of the MAGE experiment. From Reference 12.

Figure 5 .

 5 Figure 5. Schematic presentation of virtual gravitational dipoles in the quantum vacuum: (a) Randomly oriented gravitational dipoles (in the absence of an external field); (b) Halo of non-random oriented gravitational dipoles around a body with baryonic mass 𝑀 𝑏 .

Table 1 .

 1 Cosmological models based on the gravitational repulsion between matter and antimattersimilarities, differences and comparison with the inflation-based ΛCDM CosmologyIn current versions of models, the only content of the Universe are equal amounts of matter and antimatter. Apparently, in the forthcoming improved versions of these models a kind of dark matter (or something replacing dark matter) must be included.

		Dirac-Milne Cosmology	The Lattice Universe	Hajdukovic's Cosmology
	Description of		Matter attracts matter
	gravitational	Matter and antimatter repel each other
	interactions	Antimatter repels	Antimatter attracts antimatter
		antimatter
		(CPT is violated)	(CPT Symmetry is not violated)
	Matter-antimatter	Matter-antimatter symmetric Universe	Our cycle of the
	content of the Universe	(i.e. Universe contains the same amounts of matter	Universe is dominated
		and antimatter with antimatter "hidden" in the voids)	by matter
	Quantum vacuum	So far, these models do not include quantum vacuum	Quantum vacuum is a
		as a source of gravity	crucial source of gravity
	Matter-energy content			The Standard Model
	of the Universe			Matter and quantum
				vacuum composed of
				virtual gravitational
				dipoles
	Cornerstones of ΛCDM
	Dark matter	Phenomena usually attributed to Dark matter remain	What we call DM and
		to be explained in both models	DE are local and global
	Dark energy	There is no need for Dark energy	effects of gravitational
				polarization of the
				quantum vacuum
	Inflation field
	Cosmological Constant
	Problem

Apparently, in all 3 models there is no need for the inflation field So far, this problem is not considered in these models A solution is proposed