MOND-like behavior in the Dirac-Milne universe -- Flat rotation curves and mass/velocity relations in galaxies and clusters
Gabriel Chardin, Yohan Dubois, Giovanni Manfredi, Bruce Miller, Clément Stahl
AAstronomy & Astrophysics manuscript no. 210216_Dirac_Milne_rotation_curves_AA © ESO 2021February 18, 2021
MOND-like behavior in the Dirac-Milne universe
Flat rotation curves and mass/velocity relations in galaxies and clusters
Gabriel Chardin , Yohan Dubois , Giovanni Manfredi , Bruce Miller , and Clément Stahl Université de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, Francee-mail: [email protected] Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis Boulevard Arago, F-75014 Paris, France Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg,Francee-mail: [email protected] Department of Physics and Astronomy, Texas Christian University, Fort Worth, TX 76129, USAFebruary 18, 2021
ABSTRACT
Aims.
Observational data suggest that the observed luminous matter is not su ffi cient to explain several features of the present universe,from gravitational structure formation to the rotational velocities in galaxies and clusters. The mainstream explanation is that themissing mass, although gravitationally active, interacts very weakly with ordinary matter. Competing explanations involve changingthe laws of gravity at low accelerations, as in MOND (Modified Newtonian Dynamics). Here, we suggest that the Dirac-Milnecosmology, a matter-antimatter symmetric cosmology where the two components repel each other, is capable of accounting for suchapparent modification of the Newtonian law, without invoking dark matter. Methods.
Using a simple analytical approximation and 1D and 3D simulations, we study rotation curves and virial velocities, andcompare the mass observed in the simulations to the mass derived assuming Newtonian gravity. Using a modified version of the
RAMSES code, we study the Faber-Jackson scaling relation and the intensity of the additional gravitational field created by antimatterclouds.
Results.
We show that, in the Dirac-Milne universe, rotation curves are generically flat beyond the characteristic distance of ≈ . ≈ g am due to the presence of clouds of antimatter is of theorder of a few 10 − m / s , similar to the characteristic acceleration of MOND. We study the evolution of this additional acceleration g am and show that it depends on the redshift, and is therefore not a fundamental constant. Conclusions.
Combined with its known concordance properties on SNIa luminosity distance, age, nucleosynthesis and structureformation, the Dirac-Milne cosmology may then represent an interesting alternative to the Λ CDM , MOND, and other scenarios forexplaining the Dark Matter and Dark Energy conundrum.
Key words. dark matter – dark energy – cosmology – gravitation
1. Introduction
The Dark Matter enigma has found its first expression in the1930s after the observation by Fritz Zwicky (Zwicky 1933) thatpeculiar velocities in the Coma cluster were far too large (bymore than two orders of magnitude according to Zwicky’s anal-ysis) to account for the bound behavior of the cluster compo-nents if only the visible mass was taken into account. Followingthis initial and remarkably prescient observations and analysis,a long dormant period followed where the dark matter questionwas mostly forgotten. In the 1970s, Vera Rubin and collaborators(Rubin et al. 1980), and Albert Bosma (Bosma 1981), measuringgalactic rotation curves, noticed that they had quite generally aflat behavior at large distances from the core, and their analysiscontributed very significantly to the revival of the Dark Matterenigma. The accumulation of galactic rotation curves then led tothe gradual realization (Bertone & Hooper 2018) that, quite gen-erally, observed galaxy rotation curves are flat at large distances from the galaxy core, which is at odds with the theoretical pre-dictions based on the assumption of dominating mass related toluminous matter.Two main lines of hypotheses were proposed as tentative so-lutions to this enigma: – The first was to conjecture that there really exists a large partof the matter component of our Universe that is dark and in-teracts very weakly, apart from its gravitational interactions.This conjecture was for a long time the dominant hypothe-sis, under the implementation of WIMPs (Weakly InteractingMassive Particles), and later CDM (cold dark matter), afterthe findings that both massive neutrinos (Drukier & Stodol-sky 1984) and supersymmetric particles (Goodman & Witten1985) provided candidates that naturally suited the existingconstraints, in terms of mass and interaction cross-sections,to solve the Dark Matter problem – the so-called “WIMPmiracle". This hypothesis seemed indeed almost necessary
Article number, page 1 of 16 a r X i v : . [ a s t r o - ph . GA ] F e b & A proofs: manuscript no. 210216_Dirac_Milne_rotation_curves_AA since the then mainstream Einstein-de Sitter (EdS) universefeatured a critical density: Dark Matter suggested an elegantway to fill the gap between the small baryonic component,with a density derived and constrained by nucleosynthesis toless than 5% of the critical density, and the critical density ofthe EdS universe.But despite extensive experimental searches, both in directdetection experiments and at the Large Hadron Collider atCERN, no such Dark Matter candidates have been found.Meanwhile, the tensions between the age of an Einstein-de Sitter and the age of the oldest structures in the Uni-verse grew to a point in the mid-1990s where a cosmologicalconstant or some other repulsive component was consideredcompulsory, a few years before the discovery in 1998 of whatis now called Dark Energy, through the SNIa flux measure-ments (Perlmutter et al. 1999; Riess et al. 1998).However, despite its utility, the Λ CDM scenario is not with-out presenting some tensions: the highest tension originatesfrom the local measurements of the Hubble constant H andthe determinations of this same parameter deduced from thecosmic microwave background (CMB) and baryonic acous-tic oscillations (BAO) (Riess 2020). But even on the DarkMatter side, additional tensions exist: there is for exampleno evidence for the cusped density profiles predicted by theDark Matter simulations, and many galaxies apparently haveno need for Dark Matter in their central regions. For somegalaxies, there is even no apparent need for Dark Matter al-together, requiring to explain how the dominant Dark Mattercomponent may have been almost entirely ejected from thesegalaxies (see e.g. (Guo et al. 2020) and references therein). Inaddition, recently, Meneghetti et al. (Meneghetti et al. 2020)have shown that there is an important discrepancy betweenthe lensing properties expected from the Λ CDM simulationsand actual observations. – The second line of explanation rests on a modification ofthe laws of gravitation, MOND (MOdified Newtonian Dy-namics) (Milgrom 1983) being its most popular expression.Although some attempts have been made to find a moti-vated field theory for MOND (e.g. TeVeS (Bekenstein 2004),although ruled out by the multi-messenger observations ofGW170817), the MOND hypothesis is mostly phenomeno-logical, and based on the observation that the Dark Matterproblem seemed mostly confined to regions with values ofthe gravitational field of the order or less than ≈ − m / s .Indeed, MOND proposes that the law of gravitation devi-ates from its Newtonian expression in the following way: a = √ a a N where a is the acceleration, a is the crossoveracceleration of MOND, and a N is the acceleration predictedby Newton’s law. Remarkably, a is of the order of cH , sug-gesting a possible link between cosmology and this crossoveracceleration. We will see in the following that Dirac-Milnemay provide an explanation for this coincidence.Problems with MOND include, apart from its lack of theoret-ical justification, the fact that there seems to exist significantdi ffi culties to represent galactic rotation curves with a singleacceleration parameter a (Marra et al. 2020), although it issupposedly a fundamental parameter. In addition, this accel-eration seems to di ff er when the analysis is done at galacticor at cluster scales. On the positive side, MOND predictedthe Tully-Fisher relation (Tully & Fisher 1977) linking themass to the rotation velocity for structures in a wide range ofmass (at least four orders of magnitude), although again witha shift in slope at the largest scales. A fundamental remarkto which we will come back is the fact that MOND’s law looks akin to the e ff ect of gravitational polarisation (Blanchet2007; Blanchet & Le Tiec 2009).In the present paper, we study the gravitational polarizationpredicted by the Dirac-Milne (D-M) cosmology (Benoit-Lévy &Chardin 2012; Chardin & Manfredi 2018), providing an expla-nation for this apparent modification of the Newtonian law ofgravitation. For this purpose, in Sec. 2, we recall the main char-acteristics of the Dirac-Milne universe. In Sec. 3, we study a sim-ple idealized analytical model, showing that in the Dirac-Milnecosmology rotation curves are indeed expected to be genericallyflat after a characteristic distance, for which we provide an ap-proximate relation. We discuss the law obtained for the rota-tion velocity, notably in relation to the shell and Birkho ff the-orems (Newton 1760; Birkho ff & Langer 1923), with a more de-tailed analysis in Appendix A. In Sec.4, we present preliminaryresults of velocity distributions as a function of mass using amodified version of the RAMSES
3D simulation code incorporat-ing the gravitational behavior of the Dirac-Milne universe, asdescribed in Refs. (Manfredi et al. 2018, 2020). In Sec. 5, weshow that Dirac-Milne follows a Faber-Jackson relation with avery small scatter and an exponent ≈ a . In the finalsection, we summarize our findings, and provide some perspec-tives and possible lines of development for future work.
2. Dirac-Milne universe
The standard cosmological model, although an impressive fitto diverse sets of data, su ff ers from the fact that the two mainingredients of Λ CDM , dark energy and dark matter, supposedto represent 95% of its content, have not been identified de-spite extensive investigations. On the gravitational structure side,as mentioned previously, Λ CDM must face a high tension be-tween the low-redshift and high-redshift constraints on the Hub-ble parameter (Riess 2020), potential di ffi culties with the cusp-core (Flores & Primack 1994)] and the too-big-to-fail problem(Boylan-Kolchin et al. 2011), possibly addressed by baryonicback-reaction (Governato et al. 2012; Teyssier et al. 2013; Wet-zel et al. 2016), and an extremely rare coplanarity of Milky Waysatellites (Ibata et al. 2013; Müller et al. 2018).The Dirac-Milne (D-M) universe, proposed recently byBenoit-Lévy and Chardin (Benoit-Lévy & Chardin 2012;Chardin & Manfredi 2018), suggests a radically di ff erentparadigm for the cosmology of our universe. It features a sym-metric matter-antimatter universe, where matter and antimattere ff ectively repel each other, but where antimatter also repels it-self. As is well-known, when the usual expression of the Equiva-lence Principle is respected for matter and antimatter, such sym-metric matter-antimatter cosmologies are excluded by the non-observation of a di ff use gamma-ray flux (Omnès 1972; Cohenet al. 1998). On the other hand, the Dirac-Milne cosmology isa gravitational implementation of the Dirac particle-hole sea,analog to the electron-hole system in a semiconductor. Giventhe fact that a repulsive and enigmatic Dark Energy componentrepresents ≈
70% of the energy density in the Λ CDM model, itseems interesting to test more extensively this hypothesis of re-pulsion between matter and antimatter. The Dirac-Milne cosmol-ogy is further motivated by the remark by Price (Price 1993) thatthe usual expression of the Equivalence Principle, stating that allparticles must follow the same trajectories given the same initialconditions in a gravitational field, must necessarily be modified
Article number, page 2 of 16abriel Chardin et al.: MOND-like behavior in the Dirac-Milne universe in the case where particles with negative mass are consideredalong with particles of positive mass. Indeed, as shown by Price,two new elements appear in such bound systems: (i) Gravita-tional polarization appears between particles of positive and neg-ative mass whenever they are bound by non-gravitational forces,and (ii) levitation is predicted for a symmetric ( + m , − m ) sys-tem, a gross violation of the usual formulation of the EquivalencePrinciple.A fundamental feature of the D-M universe is that its expan-sion factor varies linearly with time: a ( t ) ∝ t (1)while there is neither Dark Matter, nor Dark Energy in D-M beyond its matter and antimatter components. Being a uni-verse that appears gravitationally empty at large scales, theinitial phases of the D-M universe have timescales di ff eringrather dramatically from the Λ CDM universe: for example, theQuark-Gluon-Plasma transition lasts for about one day, insteadof about 10 microseconds in the Standard Model, while nu-cleosynthesis lasts about 35 years, compared to three minutesin the Standard Model, and recombination occurs at an age ofabout 14 million years, compared to the 380 000 years of the Λ CDM model (Benoit-Lévy & Chardin 2012; Chardin & Man-fredi 2018).Despite these tremendous di ff erences in the initialtimescales, the D-M universe, with only one adjustableparameter H , presents several elements of concordance: its age,equal to 1 / H , is almost equal to the age of the Λ CDM universefor H ≈
70 km s − Mpc − , while the H ( z ) dependence ofcosmic chronometers is nicely reproduced in coasting universes(Melia & Maier 2013), along with primordial nucleosynthesisand the SNIa luminosity distance (Sethi et al. 1999; Chodor-owski 2005; Benoit-Lévy & Chardin 2012). Also, its non-linearstructure formation mechanism appears to reproduce the mainfeatures of the matter power spectrum starting from a singlescale of matter-antimatter domains at decoupling (Manfrediet al. 2018, 2020). In addition, the Dirac-Milne cosmology doesnot su ff er from the horizon problem (Benoit-Lévy & Chardin2012) and therefore does not require inflation.
3. Gravitational setup in the Dirac-Milne universe
It is fundamental to note that although existing in Nature, theDirac particle-hole system has no Newtonian expression, evenwhen the three Newtonian mass parameters (inertial, gravita-tional active, gravitational passive) are used (Manfredi et al.2018). On the other hand, as studied in (Manfredi et al. 2018,2020), the gravitational sector of matter and antimatter in theDirac-Milne universe can be expressed with two separate grav-itational potentials, using the following coupled Poisson equa-tions: ∆ φ + = π G ( ρ + − ρ − ) , (2) ∆ φ − = π G ( − ρ + − ρ − ) . (3)It should be noted that in these equations, although two po-tentials φ + and φ − are invoked, there is a single gravitational con-stant G , and not two independent constants for matter and anti-matter. In particular, following Price (Price 1993), we can derivethe gravitational field for antimatter once the gravitational fieldfor matter is known (and vice-versa). For particles with equal butopposite mass, the gravitational fields: g + ≡ − ∇ φ + (4) and g − ≡ − ∇ φ − (5)exerted on a particle and its antiparticle are opposite in the“Newtonian" regime ( i.e. when the gravitational field created bymatter is much larger than the contribution of antimatter, whichcan then be neglected), as the total force on the bound ( + m , − m )system is zero (Manfredi et al. 2018, 2020).For further use, we also introduce the gravitational force act-ing on a particle sourced only by matter as: g m ≡ − ∇ ( φ + − φ − ) / g am ≡ − ∇ ( φ + + φ − ) / In order to introduce the properties of rotation curves and virialvelocity distributions in the D-M cosmology, let us first considerthe following example of cluster configuration, represented onFig. 1. This configuration was obtained in a 3D simulation us-ing a modified version of the
RAMSES code (Teyssier 2002), thatwe discuss more precisely in the next section. This figure repre-sents a small cluster configuration in the D-M universe at red-shift z ≈
20 for self-gravitating particles, at this stage withoutdissipation. The full tomography of the matter, antimatter, andmatter + antimatter configurations can be found as supplementarymaterial at the following link: https://youtu.be/rdIOCoy8QPM An animation of the formation of structures centered on amassive cluster of the simulation can be found at the followinglink:
In Fig. 1, it can be seen that while matter has the usual clus-tering properties in planes, filaments and nodes (where globularclusters, galaxies and clusters of galaxies accumulate), and isconcentrated in relatively small regions, antimatter has a com-pletely di ff erent distribution, occupying nearly exactly half ofthe total volume, with a much more homogeneous density thanmatter. In addition, a new element is apparent: empty or verylow-density depletion zones surround matter, isolating it fromantimatter and occupying also about half of the total volume.We note that while it will be in most occasions possible to de-fine rather well-defined isolated matter galaxies and clusters, thiswill not be the case for antimatter clouds, which are percolat-ing. Strictly speaking, it is therefore incorrect to speak of “do-mains" for antimatter, since there is path continuity at all dis-tances within the antimatter clouds. The same property of per-colation is respected by the depletion zones, which are thereforeorganized more in tubes surrounding matter structures than inspheres.Let us now show that both these antimatter clouds and thedepletion zones between matter and antimatter lead to an addi-tional surface gravity compared to the Newtonian expectation.For this, we first use a simple analytical model, represented onFig. 2, where the Dirac-Milne universe is represented as the pe-riodic repetition of elementary cubic cells, in which half the vol-ume is occupied by a galaxy (located at the center of the box)and the surrounding almost spherical depletion zone, while theantimatter cloud occupies the other half of the volume in the Article number, page 3 of 16 & A proofs: manuscript no. 210216_Dirac_Milne_rotation_curves_AA
Fig. 1.
Slice of a matter-antimatter simulation of a small volume in theDirac-Milne universe using a modified version of the
RAMSES simu-lation code. The thickness of the slice is 20% of the simulation box,of comoving size 1 h − Mpc. The condensed structures (represented inlight blue) are all made of matter, while antimatter is spread in nearlyhomogeneous extended halos (represented in red) over half of the vol-ume. The second half of the volume is occupied by depletion zones,surrounding the matter galaxies and clusters, isolating them from an-timatter. These depletion zones are a consequence of the gravitationalpolarization expected between positive mass and “negative mass" ob-jects. We note that antimatter clouds percolate, i.e. it is possible to roamat infinity without leaving the antimatter cloud, and the same propertyis respected for the depletion zones surrounding matter.
Fig. 2.
Schematic representation of a galaxy and its surroundings in theDirac-Milne universe. The galaxy core is represented by a condensedobject at the center of the cell, while a depletion zone, occupying about50 % of the cell volume, surrounds the galaxy. Antimatter is spread outbeyond this depletion zone, on the outskirts of the cell, occupying alsonearly 50 % of the cell volume, although its volume may appear smallerin the 2D projection. outer part of the box (see Fig. 2). Although this is clearly anoversimplification, since in this representation all structures aresupposed to have the same mass and size, it will enable us toevidence the salient features of galactic rotation curves in theDirac-Milne universe.More precisely, we postulate that each galactic cell is consti-tuted by a cubic box of volume equal to the average volume per galaxy. We represent the content of this box as the sum of threecontributions:a) A galaxy concentrated in a very small volume at the centerof the box, the density of the galaxy being much higher thanthe average matter density (typically a factor 200 within thevirial radius);b) A depletion zone, mostly empty of matter and antimatter,around the galaxy. In the galactic box, the volume of thedepletion zone is about half of the total volume of the boxwhen antimatter and matter in the box have equal and oppo-site “masses", as supposed in D-M;c) Surrounding this depletion zone, a region filled with antimat-ter with nearly constant density and a volume equal to halfof the total volume of the box.The whole space is then represented as a collection of such cubesin a periodic geometry.We note that although the antimatter clouds occupy half ofthe volume of the box, the radius of the depletion zone extendsalmost to the confines of the box. In order to show this, let us de-rive, in our approximation of spherical symmetry, the extent ofthe depletion zone delimiting the zones of matter (mostly con-densed) and antimatter (spread out almost uniformly). The con-straints are the following: – The total mass of the “galactic cell" is zero, i.e. m + = m − ≡ m – The depletion zone is spherical and of radius r d – The linear size of the individual galactic cell is L , and its totalvolume is therefore L . – The antimatter, due to its internal repulsion, has a constantdensity ρ − .The edge of the depletion zone is defined by the condition thatthe total gravitational force on an antimatter particle becomeszero. In order to express the repulsive force exerted by the anti-matter cloud on an antiparticle at the edge of the depletion zone,we use the fact that the force created by the antimatter cloud ofuniform density would be zero were it not for the spherical de-pletion zone. On the other hand, creating the depletion zone byremoving an homogeneous sphere of antimatter creates a forcedirected towards the center of the box, and its intensity can becalculated by Newton’s shell theorem stating that the situation isequivalent to that where the antimatter mass of the depletion vol-ume is concentrated at the center. The spherical depletion zonewill therefore create an attractive force as it if were concentratedat the origin and with a mass πρ − r d . Since we want the totalgravitational force on an antimatter particle at the edge of thedepletion zone to be zero, it means that this mass must be equalto m + .On the other hand, the remaining mass of antimatter, outsidethe depletion zone, is: m − = ρ − ( L − π r d ) (8)By our first condition, it must be equal to the positive mass m + of the galaxy.Overall, we have therefore the conditions relating ρ − and r d : ρ − ( L − π r d ) = πρ − r d = m + = m − (9)Therefore, in our approximation of spherical symmetry, the anti-matter cloud occupies half the volume of a simulation cube, andthe depletion zone extends until: r d = (cid:114) π L ≈ . L (10) Article number, page 4 of 16abriel Chardin et al.: MOND-like behavior in the Dirac-Milne universe
Fig. 3.
Schematic representation of a periodic structure of identicalgalaxies, each individual galaxy being composed as described in Fig.2. The analytic calculation presented in the text is a crude approxima-tion in the sense that periodic images of the peripheral galaxies are nottaken into account in the derivation of the rotation velocity of the galaxyat the center of the figure. Note in particular the very asymmetric dis-tribution of the antimatter cloud along the vertical and horizontal axis,where the column density is nearly zero, compared to the diagonals ofthe cube, where this column density is maximal.
This means that, counterintuitively, the depletion zone al-most reaches the confines of the average individual box arounda galaxy. Note also that the situation that we have approximatedis in fact the configuration represented on Fig. 3, reproduced atinfinity in a 3D periodic torus, and therefore without sphericalsymmetry, notably concerning the antimatter cloud. Note alsothat in the actual 3D equilibrium configuration, the depletionzone of a cell connects with that of the adjacent cells on its threeaxis.
Let us now estimate the rotation velocity for a matter test parti-cle as a function of the distance to the galaxy in the precedingconfiguration. The galaxy of mass m obviously contributes, for atest body located at the distance r from the galaxy center, to anacceleration equal to Gm / r , as in the Newtonian case.For a matter particle, the gravitational force due to the con-figuration of the antimatter cloud and the depletion zone sur-rounding the galactic core can be estimated as follows. We firstconsider the configuration where antimatter of “negative mass" (see (Manfredi et al. 2018) for a precise definition), is uniformlydistributed in the box. The gravitational force exerted by thisuniform antimatter cloud is then zero everywhere, by symmetry.Then, we represent the depletion zone as the sum of a homo-geneous sphere of positive density (equal and opposite to thatof antimatter) covering the volume of the depletion zone, whichwill compensate the uniform background of the negative massfluid.The situation shown in Fig. 2 can therefore be expressed asthe superposition of three cubes (Fig. 4): Fig. 4.
Our approximation of a galaxy represented on Fig. 2 can be rep-resented as the sum of three cubes: - a cube (a) with uniform “negativemass" density of total mass − m , - a cube (b) with only the galaxy ofmass + m at the center, - a cube (c) with a sphere occupying half the vol-ume of the cube and of homogeneous positive density of matter, equaland opposite to that of the first cube, that together with (a) will createthe empty depletion zone. a) A cube with “negative mass" equal to − m , i.e. twice themass of the galaxy, and uniform density (zero gravitationalfields g + and g − everywhere);b) A cube with the mass m of the galaxy at the centre, i.e. half ofthe total mass of the antimatter in the first box. This galacticpoint-like mass will produce the classical Newton force field g + = − Gm r / r (and g − = − g + );c) A cube with the same positive mass + m as the galaxy, butuniformly distributed over the spherical depletion zone. Thispositive mass will compensate the negative mass − m of thefirst cube contained in the same sphere, thus creating theempty depletion zone, with a volume 50% of the box.As is well known, the homogeneous matter sphere in thethird cube will generate a harmonic restoring force, which is su-perimposed on the Newtonian force of the galaxy, supposed tobe a point galaxy. We note that the second and third cubes havea total mass of 2 m , compensating the “negative mass" − m ofantimatter in the first cube. Space is supposed to be covered byadjacent such cubes (see Fig. 3).At a distance r from the center of the box (and the center ofthe galaxy), the gravitational field created by this homogeneoussphere of positive density will then be: g + = − Gm r r d (11)where m is the positive mass added to create the depletionzone and r is the radial distance vector to the galactic center. Thetotal gravitational field g + exerted on a matter particle orbitingat a distance r from the galaxy center is therefore: g + = − Gm r r − Gm r r d (12)while the relation between orbital velocity and this force is: v r = Gmr + Gmrr d (13)The orbital velocity as a function of the distance r to the centerof the galaxy can be simply deduced from the above expression,and reads as: v ( r ) = (cid:115) Gmr + Gmr r d (14) Article number, page 5 of 16 & A proofs: manuscript no. 210216_Dirac_Milne_rotation_curves_AA
Remarkably, this function has a derivative that vanishes at r = − / r d ≈ . r d , and is almost constant for all r (cid:38) r d / r > r d , the spherical approx-imation breaks down as we have seen in Sec. 3.3. Additionally,beyond the depletion zone, the antimatter cloud is present.Since the depletion zones occupy about 50% of the total vol-ume and since the baryonic density in a D-M universe is de-fined by the baryon / photon ratio η ≈ × − (Benoit-Lévy &Chardin 2012; Sethi et al. 1999), an observer assuming that the Λ CDM cosmology is valid will interpret the depletion zone as ahalo with a density equal to the matter density of the Dirac-Milneuniverse, i.e. : ρ m ρ c = η D − M η Λ CDM × Ω Λ CDM baryon ≈ × − × − × . ≈ .
65 (15)where η = n B / n γ is the number of baryons per photon in thecorresponding cosmology, and ρ c is the critical density for the Λ CDM universe at the present epoch.Now, the precise characterization of the virial radius dependson the cosmological model used and notably of the inflows be-tween the redshift of initial collapse of a structure and the presentepoch. The overdensity ∆ c at the virial radius for Λ CDM is usu-ally approximated as ∆ c ≈ ρ c (Shull 2014), so that a nearlyconstant rotation velocity will be observed for distances: r (cid:38) r d ≈ (cid:115) ∆ c ρ m r v ≈ √ r v ≈ . r v (16)where r v is the virial radius. Therefore, the flat rotation curves,instead of being due to an invisible halo of slowly decreasingdensity, are in fact due in D-M to the asymmetric configurationof matter and antimatter that we have described above.Indeed, the Dirac-Milne universe tells us that in order to de-rive the galactic rotation curves and the virial velocities in a clus-ter of galaxies, it is necessary to take into account not only thematter present between us and the center of the galaxy or thecluster, i.e. matter “below our feet", but also (anti)matter “aboveour heads", which may be interpreted as an External Field E ff ect(EFE) (Chae et al. 2020). This is indeed very counterintuitive,and may even be considered in contradiction with the Newto-nian shell theorem (Newton 1760), stating that inside a sphericalshell, spacetime must be Minkowskian, i.e. the gravitational fieldmust be zero. We further discuss this apparent contradiction inAppendix A.The approximation of spherical symmetry allowed us toshow that flat rotation curves are expected within a large fractionof the volume of the depletion zone. Using a numerical simula-tion, we now explore in more precision the edge e ff ects createdby the antimatter clouds, which are in fact asymmetric. In this subsection, we use our modified version of
RAMSES (seesection 4 for details) to study numerically the spherical approx-imation of the previous subsection. We consider a galaxy shellconfiguration following section 3.1, as described in Fig. 6.More precisely, we considered an initial configuration wherea central galaxy core, confined in a radius equal to ≈
5% of the r v = 1)012345 O r b i t a l v e l o c i t y ( a r b . un i t s ) Dirac-MilneNewton
Fig. 5.
Rotation velocity predicted in the Dirac-Milne universe for apoint mass located at r = ≈ Fig. 6.
Configuration studied in this simulation. A matter galaxy, locatedat the center of the box, is surrounded by a depletion zone, extendingover half the volume of the box, and by an antimatter cloud (representedin magenta), extending on the outskirts of the box over the other half ofthe volume simulation box length, presents an isothermal density and veloc-ity profile. The same mass of antimatter is uniformly distributedat r > r d ≈ . L , see Eq. (10), but its initial velocity is zero.Using RAMSES on a 128 grid, we checked that this con-figuration does not evolve notably. The gravitational field g am created by antimatter is represented in Fig. 7. Due to the non-spherical distribution, the antimatter field is seen to be strongeron the diagonals of the cube for a plane with normal parallelto one of the axes of the cube, e.g. with a normal vector (1, 0, Article number, page 6 of 16abriel Chardin et al.: MOND-like behavior in the Dirac-Milne universe
Fig. 7.
Gravitational field g am created by antimatter on two planes de-fined by their normal vector (1,0,0), for the top panel and ( √ (1 , , e.g. with a normal vector 1 / √ , , cartesian grid (about 2 million cells), and thequantities estimated on the matter region, defined by the distanceto the galactic center, over half the volume, which is the only cutapplied for the selection. A Newtonian regime can be observedin the high field region of the figure, with a gradual transitionfrom the Newtonian regime to a MOND-like regime at an accel-eration of ≈ − m / s . In the middle panel, we have plotted thedistribution of the total gravitational field acting on a matter par-ticle as a function of the gravitational field created by antimatteralone. It can be seen that the antimatter field is rather peaked,with an average value of ≈ × − m / s in this configuration.In order to evidence more clearly the transition to the MOND-like regime, we have represented in the bottom panel the ratiobetween the total gravitational field acting on a matter particleand the gravitational field created by matter only, as a functionof the intensity of the gravitational field created by matter only.While the ratio is almost exactly unity at high values of the field,in the Newtonian regime, this ratio gradually increases, with aMOND-like transition starting at an acceleration of ≈ − m / s , and reaching a factor of ≈ Fig. 8.
Top panel: Total gravitational field acting on a matter particle asa function of the gravitational field created by matter alone. A Newto-nian regime can be observed for the high field region of the figure, witha gradual transition from the Newtonian regime to a MOND-like regimeat an acceleration of ≈ − m / s . Middle panel: Total gravitational fieldacting on a matter particle as a function of the gravitational field createdby antimatter alone. The antimatter field is rather peaked, with an aver-age value of ≈ × − m / s . Bottom panel: Ratio between the totalgravitational field acting on a matter particle and the Newtonian field(created by matter only), as a function of the gravitational field createdby matter only. Almost exactly unity in the Newtonian regime, at highvalues of the field, this ratio gradually increases and reaches a factor of ≈ In the present section, in order to confirm the qualitative andnumerical analysis of the depletion zone presented in the twopreceding subsections, we extend the analysis of Ref. (Manfrediet al. 2020) to incorporate the notion of depletion zone and in-
Article number, page 7 of 16 & A proofs: manuscript no. 210216_Dirac_Milne_rotation_curves_AA homogeneity of the antimatter cloud. Using the Poisson equa-tions (2) and (3), we consider a spherically symmetric geometry,where all quantities depend only on the radius r and the Lapla-cian operator is defined as ∆ φ = r ∂ r ( r ∂ r φ ). We want to solvethese Poisson equations for a typical situation where a positivehigh-density mass (“galaxy") is located in a small spatial regionlocated near r = ρ + ( r ). Incontrast, negative masses are supposed to be thermalized (at lowtemperature T ) and described by a Boltzmann distribution: ρ − ( r ) = ρ exp (cid:32) − m φ − + µ k B T (cid:33) , (17)where k B is Boltzmann’s constant, ρ is a reference density, and µ is a chemical potential that will be chosen so that positiveand negative masses are present in equal amounts in the com-putational box, i.e. : 4 π (cid:82) ρ + r dr = π (cid:82) ρ − r dr . We note thatthe mass m appearing in Eq. (17) is the passive gravitationalmass, which in the D-M model is always positive (Manfredi et al.2018). Then, Eq. (3) becomes a nonlinear Poisson equation for φ − ( r ), which can be solved self-consistently, and the result sub-stituted into Eq.(2) in order to obtain φ + ( r ).However, as discussed at the end of section 3, the externalfield e ff ects require a careful treatment of the boundary condi-tions. Indeed, as they are written, the Poisson equations (2)-(3)describe an isolated system in otherwise empty space. But thisis not the case for a cosmological setting for which the positivemasses are localized in small regions, and the negative massesspread out almost uniformly across about 50% of the availablespace.We therefore solve Eqs. (2)-(3) in a spherical region between r = r = R , using the following boundary condition: φ (cid:48) + ( R ) = π G (cid:82) R ρ + r drR , (18)where the apex stands for di ff erentiation with respect to r .For comparison, the Newtonian case is computed by solving: ∆ φ Newt = π G ρ + , with boundary condition: φ Newt ( R ) = π G =
1, and take R =
20 and temperature T = . r d ≈ .
8. Thus, the ratio of the total volume to the volume of thedepletion zone is approximately R / r d = . ≈
2, as expected.On the outer border of the depletion zone, the gravitational field(defined as: g = − ∂ r φ ) for matter is twice the Newtonian value,while the antimatter field vanishes within the entire depletionzone (see Fig. 10), justifying our hypothesis that the antimattercloud is both very cold and homogeneous. The rotation speedsare defined as: v ( r ) = (cid:112) r | φ (cid:48) ( r ) | . For the D-M model, the rota-tion curve flattens between approximately r d / r d , also inaccordance with our model (see Fig. 11).Finally, we point out that the Poisson equations (2)-(3), withthe boundary condition (18), are equivalent to adding a constantdensity 2 ¯ ρ on the right-hand side of both equations (where ¯ ρ ≡(cid:104) ρ + (cid:105) = (cid:104) ρ − (cid:105) is the average matter or antimatter density) and usingDirichlet boundary conditions φ ± ( R ) = i.e. fixing the valueof the potentials on the sphere ( r = R ). The conditions on thegradients (the fields), i.e. Eq. (18), will be automatically satisfiedbecause of Gauss’s theorem. Hence, the potential φ + acting onmatter results from two sources: the central “galaxy" ρ + and theadditional distribution ρ halo ≡ ρ − ρ − . The latter is analog to thedark matter halo postulated in the standard CDM theory. Thishalo distribution is also represented in Fig. 9. r (r) r / r d r - r + r h a l o Fig. 9.
Matter density ρ + ( r ) (black line), antimatter density ρ − ( r ) (blueline), and pseudo-dark-matter halo density: ρ halo = ρ − ρ − (red line). Gravitational fields r / r d - g N e w t - g + g - Fig. 10.
Gravitational fields: g − (blue line), − g + (black line), and − g Newt (red line) .
V(r) r / r d D - M N e w t o n
Fig. 11.
Rotation speeds for the D-M model (black line) and for stan-dard Newtonian gravity (red line). The decrease of the rotation velocitybeyond r / r d =
4. Simulation results
In order to validate the analytical approximations studied in Sec-tion 2, we modified the Adaptive Mesh Refinement code
RAMSES (Teyssier 2002) in order to implement the gravitational behaviorof matter and antimatter present in D-M, following Eqs. (2)-(3).We note that in most cosmological simulations, the average den-sity is first calculated and subtracted from the local density inorder to calculate the evolution of the scale factor a ( t ), so thatthe Poisson equation is usually written ∆ φ = π Ga δρ . In D-M, this is directly the case, since ¯ ρ = / antimattersymmetric universe, so that δρ = ρ ( t , x ) − ¯ ρ ( t ) = ρ ( t , x ). The newelement introduced in this D-M cosmological version of RAMSES is an extra set of particles with negative mass, which in turn in-troduces the new aspect of a depletion zone.The acceleration of these negative mass particles is given bythe gravitational potential of equation (3) (respectively, equa-tion (2) for positive mass particles). The mass density of bothparticle species are projected separately onto the mesh with acloud-in-cell interpolation, so that the mesh contains both thepositive and negative mass density of the corresponding positiveand negative mass particles. Once these two separate mass den-sities on the mesh are obtained, the positive and negative gravi-tational potentials are derived using the standard conjugate gra-dient algorithm of
RAMSES . The plus and minus accelerationsare computed with a simple finite di ff erence and attributed to thecorresponding set of positive and negative mass particles withthe cloud-in-cell interpolation of the accelerations computed onthe mesh. All particles share a common level-based timestep ob-tained from the smallest Courant condition from particle veloci-ties and free fall time. The cosmological time is a linear functionof the scale factor a ( t ), as in the Milne geometry (Milne 1933). Numerical Setup
The first simulation that we present involves asmall simulation volume, of dimensions 1 h − Mpc, discretizedon a grid of 256 cells. We allowed for up to two refinement lev-els, therefore leading to an e ff ective resolution of ∆ x ≈ h − kpc.According to (Benoit-Lévy & Chardin 2012; Manfredi et al.2020), the initial size of matter and antimatter domains is of theorder of 100 pc at z ≈ z = z =
0. At a redshift of z =
20, 7769halos were identified using the AdaptaHOP algorithm (Aubertet al. 2004), the development of structures in D-M occurringmostly before z =
3. This allows to sample structure charac-teristics such as size and quadratic velocity over a range of aboutthree orders of magnitude in mass (10 − × M (cid:12) ). The totalmass of matter present in this simulation is ≈ . × M (cid:12) (andthe same mass of antimatter), which gives an individual equalmass for each particle of ≈ . × M (cid:12) . For the initial condi-tions, we used a Gaussian velocity distribution with dispersion: v I = (cid:113) Gmr ≈ / s = O (1) km / s.We checked that, even when the particles are initially dis-tributed on a uniform grid with v =
0, the qualitative behaviorof our results is not significantly modified. This is due to the factthat the initial contrast in density in D-M is already of order unityimmediately after the CMB transition, leading to a very e ffi cientvirialization of the first structures within typically one Hubbletime at that epoch, i.e. a few million years.In the simulation, we use the value H =
70 km / s / Mpc for theHubble parameter at the present time. We note that, unlike stan-dard large-scale structure simulations, we did not impose a giveninitial power spectrum but rather a specific matter / antimatter pat- tern. In this first simulation, the matter-antimatter pattern for theinitial condition was generated using an Ising code (with twostates per spin, used here to represent matter and antimatter), ona 256 cartesian grid evolved for a few time steps starting froma random distribution, and using a temperature well below thesecond order transition. This procedure was used in order to cre-ate a fine-grained distribution where matter and antimatter bothpercolate in analogy with an emulsion. The characteristic size ofthe emulsion ( ≈
100 pc at z = Results
Analogous to observations of electrons and holes insemiconductors (Tsidil’kovskii 1975), this small-scale simula-tion exhibits a nearly empty depletion zone between the con-densed clumps of matter and the extended clouds of antimatter.Figure 1 illustrates this geometric distribution, asymmetric be-tween matter and antimatter. We also refer the reader to the sup-plementary movie available at the following weblink: showing that unlike in Λ CDM , structure formation in D-M starts very shortly after the CMB transition, as was alreadypredicted by our earlier 1D simulations (Manfredi et al. 2018,2020). The configuration represented in Fig. 1 is due to the factthat matter and antimatter repel each other, leading to a zonewhere almost no particles, and no antiparticles are present. Asantimatter particles repel each other, antimatter spreads as muchas it can without going close to matter, by which it is also re-pelled. Matter clusters and forms halos as in the conventionalgravitational scenario. Indeed, by looking only at the matterzones, it is di ffi cult at first glance to distinguish, with its planes,filaments and nodes, the matter configuration in D-M comparedto Λ CDM . Obviously, this qualitative statement deserves a moredetailed study.On the other hand, we can look at the common features ofmatter halos present in the 3D simulation. In Fig. 12, we showthe logarithmic density profile of matter and antimatter in 25 ha-los, with masses of the order of 10 solar masses in our simu-lation. These density profiles share common features such as abulge region where the matter density is steeply decreasing withthe distance from the center of the halo, while the density of an-timatter in the same region is zero. After a few virial radii, theantimatter density becomes non zero and the mass of antimatterrapidly increases and becomes of the same order as the mass ofmatter present in the halo, with nearly equal average densities.These extended antimatter clouds, present beyond the depletionzone, create an approximately harmonic restoring gravitationalfield, which adds its contribution to that of matter present inthe halo, and lead to nearly flat rotation curves. This behavior isgeneric to most halos present in our simulation, although someisolated halos can present a somewhat di ff erent behavior whentheir vicinity includes more massive structures. Depending ontheir antimatter environment, they might then appear as beingcompletely devoid of Dark Matter, with accelerations describedby the expected Newtonian behavior or, on the contrary, as be-ing submitted to strongly confining fields, as in the right panel ofFig. 7, and appear as almost entirely constituted of Dark Matter.As discussed in (Manfredi et al. 2020), the highly non-linearstructure formation in D-M appears to give at the present epochthe same order of magnitude for the matter power spectrum at Article number, page 9 of 16 & A proofs: manuscript no. 210216_Dirac_Milne_rotation_curves_AA M a tt e r a n d a n t i m a tt e r L o g d e n s i t y p r o f il e
505 0 25505 0 25 0 25
Distance to halo center (virial radius)
Fig. 12.
In the present figure, we represent the logarithmic spherical density as a function of radius for both matter and antimatter for a set of25 halos with mass ≈ M (cid:12) in our RAMSES simulation. While the density of matter, represented in blue, is steeply decreasing from the halocenter, the density of antimatter, represented in orange, is zero in this region. The logarithmic density is scaled to the average matter and antimatterdensity, equal in D-M at large scales, and converge (equal average densities for matter and antimatter) for almost all halos. It can be seen that theantimatter density becomes non zero after a few virial radii, and that the mass of antimatter becomes of the order of the halo matter mass aftertypically twice the radius at which antimatter density becomes non-zero. The distribution for 25 consecutive halos has been represented in thefigure in order to show that this behavior is quite generic in D-M, although some variations can be seen in the distribution of antimatter around thematter halos. the peak scale as its Λ CDM counterpart. However, at higher red-shifts, the situation is quite di ff erent and bound structures appearmuch earlier in D-M than in Λ CDM . It should be noted thatin the evolution for redshifts between z =
100 and z =
20, theformation of structures in the upper range in mass may be un-derestimated as the size of the simulation box is very limited.Note also that in the present simulations, particles are dissipa-tionless (behaving as Dark Matter particles), which may lead toa modification of the power spectrum at small scales.A few additional remarks are in order: the average velocityof antimatter clouds is much smaller (typically by one order ofmagnitude, depending on redshift) than that of matter. In addi-tion, these peculiar velocities for antimatter correspond to nearlyglobal flows of cold antihydrogen (and antihelium) clouds, i.e. the actual temperature of the antimatter clouds is much colderthan that of matter. Also, as mentioned previously, antimatterclouds percolate with one another, which means that it is possibleto travel continuously at infinity without leaving the antimatterregion. Similarly, the depletion zones also exhibit percolation,meaning that matter structures are confined in tubes rather thanin depletion spheres.In order to better apprehend the gravitational influence ofthese extended antimatter clouds, we now turn to the study ofthe Tully-Fisher and the Faber-Jackson relations.
5. Tully-Fisher and Faber-Jackson relations in theDirac-Milne Universe
The Tully-Fisher relation (TFR) (Tully & Fisher 1977) andFaber-Jackson relation (FJR) (Faber & Jackson 1976) are phe-nomenological scaling relations between the velocities insidegalaxies and clusters and the baryonic mass of the structure con-sidered. The scatter around a relation of the type: m = m v α (19)is surprisingly low, and seems to indicate that either there is noDark Matter, or that there is a strict correlation between the DarkMatter and the baryonic matter component, which seems di ffi -cult to justify in the Λ CDM model. MOND predicts an exponentequal to 4 (Famaey & McGaugh 2012), while most analyses fa-vor somewhat lower values, typically between 3.0 and 3.5 (seee.g. (Bell & de Jong 2001; Lelli et al. 2019)).Let us now show that such power-law relations are a naturalconsequence of the Dirac-Milne cosmology. We have seen that,due to the presence of the antimatter component on the outskirtsof galaxies and clusters of galaxies, rotation curves are almostflat beyond a characteristic distance, which, to a good approxi-mation, is about half the size of the depletion zone (see Fig. 2),and about 2 . r v , where r v is the virial radius as usually defined Article number, page 10 of 16abriel Chardin et al.: MOND-like behavior in the Dirac-Milne universe in the Λ CDM cosmology. At distances r > r d / ≈ . r v froma condensed object, the Dirac-Milne rotational velocity is pre-dicted to be almost constant and equal to v d = √ Gm / r d .The e ff ective radius of the depletion zone – since, as men-tioned before, the depletion zone is in in fact more a set of tubesrather than an isolated sphere – is in turn related to the mass m ofthe structure considered, at least on average, and will be approx-imated here by the following simple equation between m and r d : m ≈ (4 / π r d ρ , where ρ is the average density of matter. Replac-ing this in the above expression for v d , one finds that the asymp-totic and nearly constant velocity is approximately proportionalto r d , and the following approximate relation holds: m ∝ v d (20)In this very crude approximation, where we have assumed spher-ical symmetry, the exponent of the TFR preferred in Dirac-Milneis ≈
3, smaller than the index 4 advocated by the MOND pro-ponents (see e.g. (McGaugh et al. 2000; Lelli et al. 2019)).On the other hand, we note that the MOND exponent of 4 isin any case disfavored by several analyses. For example, Bell(Bell & de Jong 2001) favors an exponent of 3.5 (with randomand systematic 1-sigma slope errors of ≈ . α in the TFR relation varies rather widely depending on thephotometric band used to infer the mass.In order to take into account more realistic situations thanthe idealized “spherical" galaxy studied above, we have first an-alyzed the Faber-Jackson relation (FJR) (Faber & Jackson 1976),using the RAMSES simulation described in the preceding section.Using the set of 7769 halos identified by the AdaptaHOP algo-rithm (Aubert et al. 2004), we calculated the average quadraticvelocity within the virial radius. We plot this quantity as a func-tion of the mass of the halo within the same radius. The result-ing scatter plot is shown in Fig. 13, where the scatter beyond thepower-law relation appears to be very small.We consider this first analysis to be very encouraging, butwe note that the Faber-Jackson relation can be obtained also inthe context of Λ CDM , with a similar exponent. An importantdi ff erence lies in the fact that in Dirac-Milne, there is no DarkMatter, and that therefore the scaling relation between baryonicmass and virial velocity is tighter. On the other hand, we clearlyneed more realistic simulations, in particular regarding hydrody-namics and feedback, which will allow us to test the Tully-Fisherrelation for spiral galaxies, in addition to the Faber-Jackson re-lation.
6. MOND-like behavior in the Dirac-Milne universe
As mentioned previously, the MOND phenomenology can bejustified if there exists gravitational polarization (Blanchet 2007;Blanchet & Le Tiec 2009), while Dirac-Milne predicts such apolarization. We have indeed seen in Section 4 that the D-M uni-verse exhibits a MOND-like behavior for idealized, “spherical"galaxies. On the other hand, the question remains open whetherthe value of the additional field created by antimatter and pre-dicted by Dirac-Milne can justify the characteristic acceleration a ≈ . × − m / s that seems to best fit the MOND behavior.In the present section, we study the value of this additional grav-itational acceleration and show that this value cannot be reducedin D-M to a constant and that, in particular, it depends on theredshift at which it is measured. Velocity dispersion [km/s] m a ss [ M ] = 2.56 ± 0.01, M = (1.35 ± 0.02) × 10 M Fig. 13.
Faber-Jackson relation for halos with mass between ≈ and ≈ × solar masses in the Dirac-Milne universe, using simulations ofa modified version of the RAMSES code. The mass versus virial velocityrelation exhibits a power-law behavior with a very small scatter. Theexponent in the power-law relation m = v α is ≈ . α ≈ .
45 and ≈ .
85 in the lower andthe upper part of the plot, respectively. This behavior is similar to thepower-law exponent predicted by Λ CDM , but somewhat smaller thanthe exponent 4 favored by the MOND proponents.
In order to study the intensity of the additional gravitationalfield created by the antimatter clouds on the outskirts of galaxiesand clusters, we will compare the modulus | g m | of the gravita-tional field produced by matter alone given in equation (6) to | g + | , the total gravitational field acting on matter, and producedboth by matter and antimatter as in equation (4). Similarly, wewill use the gravitational field produced by antimatter alone | g am | to quantify the additional gravitational field present in D-M, dueto the gravitational polarization between matter and antimatter,mimicking a MOND acceleration transition.In the top panel of Fig. 14, the total gravitational | g + | fieldacting on a matter particle, created by both matter and antimatter,is plotted as a function of the gravitational field | g m | that wouldbe created by matter if it were alone. These gravitational fieldsare calculated on a cartesian grid 256 (about 16 million cells),and the quantities estimated on the matter region, excluding theantimatter region, which is the only cut applied for the selec-tion. The figure clearly shows two regimes: the high field part onthe right part of the figure shows the Newtonian regime, wherethe gravitational field | g + | matches almost exactly the expectedNewtonian gravitational field | g m | created by matter. It can beseen that for accelerations smaller than ≈ × − m / s , a non-Newtonian behavior appears, with a flattening of the observedacceleration, similar to the MOND behavior (but, on this figure,for a di ff erent value of the acceleration parameter a ). Althoughthe dispersion increases at low accelerations, it can be seen that,in this situation where the halo mass range exceeds three ordersof a magnitude in mass range, the vast majority of the points lieabove the diagonal, indicating that the antimatter field | g am | rein-forces quite generally the matter field | g m | , mimicking a MONDbehavior. Article number, page 11 of 16 & A proofs: manuscript no. 210216_Dirac_Milne_rotation_curves_AA
In the bottom panel of Fig. 14, we represent the modulus ofthe total gravitational field | g + | acting on a matter particle, cre-ated by both matter and antimatter, as a function of the gravita-tional field | g am | that would be created by antimatter if it werealone. We can see that the additional acceleration created by an-timatter in the matter and depletion zones is significantly smalleron average than the matter gravitational field observed in theNewtonian regime, and has a relatively peaked distribution, at ≈ × − m / s in the cluster simulation presented.Clearly, the MOND-like transition observed in Fig. 14 at anacceleration of ≈ × − m / s is here significantly smaller thanthe value of ≈ . × − m / s of the fundamental acceleration a favored by MOND. Therefore, one may ask whether the D-Mmatter-antimatter scenario has any relevance as a possible expla-nation of MOND. In order to answer this question, we note thatthe average modulus of the gravitational field created by anti-matter depends on the total mass of the simulation box and alsoon the redshift.In Fig. 15, we have plotted the evolution as a function of red-shift of the distribution of the gravitational field | g am | created byantimatter for two simulations, the first one of “galactic" sizeand total matter mass of about 2 . × solar masses, the sec-ond one of “cluster" size and total matter mass of about 10 solar masses. These two figures present the following features:the distribution of the antimatter additional field is rather peakedat all redshifts. But although its overall shape is almost indepen-dent of redshift, reflecting the hierarchical build-up of gravita-tional structures, it is clear that the value of the peak dependson the redshift and on the total mass present in the simulation.Using the non-linear structure formation of D-M as function ofredshift shown in Fig. 4 of (Manfredi et al. 2020), we predictthe evolution of the average modulus | g am | of the gravitationalfield created by antimatter as a function of redshift for a Dirac-Milne universe of cosmological dimensions, i.e. extending muchbeyond the homogeneity scale of ≈
200 Mpc.The quantity: | g am | ≈ π G ρ ( z )3 π k peak ( z ) (21)where ρ ( z ) is the average density at redshift z , and π/ k peak ( z ) isthe scale of the largest structures at redshift z , provides an esti-mate of the gravitational field created by antimatter at this red-shift z . The evolution of this quantity is represented in Fig. 16.This figure shows two striking features: the modulus of the ad-ditional gravitational field created by antimatter is of the orderof a few 10 − m / s at the present epoch ( z ≈
7. Conclusions and perspectives
The present study has evidenced new elements of concordancebetween our universe and the Dirac-Milne universe. In particu-lar, it proposes an explanation for the observation of flat rotationcurves in galaxies, which is usually attributed to the presence ofDark Matter or to a modification of the laws of gravitation akinto MOND. We have seen in particular that, due to the combinedinfluence of the depletion zone and the antimatter clouds, flatrotation curves are generic in the Dirac-Milne universe, leadingto a systematic overestimate of the mass present in galaxies andclusters beyond a few ( (cid:39) .
5) virial radii.
Fig. 14.
Top panel: scatter diagram showing the relation between thegravitational field | g m | that would be created by matter if it were alone,on the x-axis, as a function of the total gravitational field | g + | acting ona matter particle, created by both matter and antimatter, on the y-axis.While the high field part on the right part of the figure shows the ex-pected Newtonian behavior, it can be seen that for accelerations smallerthan ≈ × − m / s , a non-Newtonian behavior appears, with a flatten-ing of the acceleration observed, in a behavior analogous to the MONDbehavior.Bottom panel: Scatter diagram showing the relation between the gravi-tational field | g am | that would be created by antimatter if it were alone,on the x-axis, and the total gravitational field | g + | acting on a matter par-ticle, created by both matter and antimatter, on the y-axis. It can be seenon this figure that the gravitational field created by antimatter is muchmore uniform, with values smaller by typically one order of magnitude,than the Newtonian regime of the previous figure, created for the mostpart by matter. The additional force experienced in D-M di ff ers both fromthe interpretation of MOND, with its modified expression forthe gravitational field using a fundamental acceleration constant a , or from the local expression of the Λ cosmological termconjectured by Gurzadyan and collaborators (Gurzadyan 1985;Gurzadyan & Stepanian 2019). On the other hand, in the samespirit as Gurzadyan, we propose a common explanation to thetentative Dark Energy and Dark Matter components of Λ CDM ,using in D-M a single constant G , instead of two constants G and Λ . This provides an explanation for the otherwise rather extraor-dinary and fine-tuned coincidence of the Dark Energy and DarkMatter densities by linking them to the dynamical evolution ofthe matter and antimatter components. Article number, page 12 of 16abriel Chardin et al.: MOND-like behavior in the Dirac-Milne universe
Fig. 15.
Evolution as a function of redshift of the distribution of themodulus of the gravitational field | g am | created by antimatter for twosimulations. In the first simulation, on the top panel, the total mass of thesimulation is of galactic size, ≈ . × M (cid:12) . The second simulation,on the bottom panel, is of cluster size and has a total mass of ≈ M (cid:12) .Both distributions are rather peaked at all redshifts, and with an overallshape and width largely independent of redshift. On the other hand, thenumerical value of the peak of the distribution clearly depends on theredshift. For both simulations, of limited size and mass, the peak valueof the antimatter field di ff ers from the fundamental constant postulatedby MOND, a ≈ . × − m / s . We have also noted that the gravitational polarization be-tween the positive and negative mass components (Price 1993)is at the origin of a MOND-like behavior. Blanchet and collab-orators (Blanchet 2007; Blanchet & Le Tiec 2009) have indeedshown that gravitational polarisation could explain the MONDphenomenology, although these authors did not have in mindthat this gravitational polarization could be due to antimatter.On the other hand, our analysis di ff ers significantly from that Fig. 16.
Evolution as a function of the scale factor of the average modu-lus of the gravitational field created by antimatter in the D-M universe.The scale factor is normalized such that at the present epoch the scalefactor a =
1. At our epoch, the value of the average modulus of the an-timatter gravitational field is of a few 10 − m / s , i.e. leading to a valueof the a acceleration parameter similar to that postulated by MOND. of Hajdukovic (Hajdukovic 2011, 2014) and Penner (Penner2016), who conjectured that MOND could be justified by thegravitational polarization of the vacuum, i.e. without taking intoaccount the gravitational polarization of matter and antimatterstructures.Additionally, using both a simple analytical model and RAMSES simulations, we showed that the Dirac-Milne cosmol-ogy predicts a rather well-defined power law between mass onthe one hand, and rotation or virialized velocity, on the other, ina large mass range of gravitational structures. This may providean explanation for the impressive correlation evidenced in theTully-Fisher and Faber-Jackson relations in galaxies and clus-ters, although the exponent α in the relation m ∝ v α was shownto be closer to 3, rather than the value of 4 advocated by theMOND proponents.In future studies, we intend to realize a more realistic treat-ment of structure formation by including hydrodynamics andfeedback in our RAMSES simulations, and also to study largersimulation volumes, extending beyond the homogeneity scale (cid:38)
200 Mpc) predicted by Dirac-Milne, and observed in our uni-verse. This is a challenge for the Dirac-Milne cosmology, as the“domains" of matter and antimatter assembled at decoupling,which initiate the formation of larger structures, are of limitedgeometrical extension (of the order of 100 parsec at z = H , a question ofparamount interest in the present context, where the tensions onthis parameter are at the ≈ . σ level between “local" and cos-mological measurements (Riess 2020).Concerning the average value of the additional gravitationalfield created by the antimatter clouds, we have shown that thetransition acceleration a at our epoch ( z =
0) is of the order of10 − m / s , a striking similarity with the MOND formalism. On Article number, page 13 of 16 & A proofs: manuscript no. 210216_Dirac_Milne_rotation_curves_AA the other hand, we noted that the coherence length of this addi-tional “MOND" gravitational field is very large, of the order of100 Mpc. At galactic scales, although the corrective harmonicgravitational field created by the antimatter clouds exists andis at the origin of flat rotation curves beyond a few virial radii,its value is usually significantly smaller than 10 − m / s in suchstructures. Also, we have shown that the modulus of this param-eter depends on the redshift and is therefore not a fundamentalconstant, di ff ering fundamentally from the MOND formalism.Finally, we note that both nucleosynthesis and the almostpurely non-linear structure formation in the Dirac-Milne uni-verse (Benoit-Lévy & Chardin 2012; Manfredi et al. 2018, 2020)set strong constraints on the size of initial “domains" of matterand on the later hierarchical (largely bottom-up) developmentof structures. A complementary way to test our hypothesis willbe to predict the mass distribution of stars and black holes re-sulting from the very early collapse from such matter domains.Compared to the black hole mass distribution derived by theLIGO and Virgo collaborations (The LIGO Scientific Collabora-tion et al. 2020), now in possession of about 70 candidate events,mostly binary black holes, this could represent an important ad-ditional test of the Dirac-Milne scenario. Acknowledgements.
We are indebted to Benoit Famaey, James Rich and YvesSacquin for their thorough reading of the manuscript and their insightful com-ments. Needless to say, they are not responsible for the errors and approximationsremaining in this paper. This work has made use of the Horizon Cluster hostedby the Institut d’Astrophysique de Paris. The work of the YT (Turk et al. 2010),IPython (Perez & Granger 2007), Matplotlib (Hunter 2007), NumPy (van derWalt et al. 2011) and SciPy (Virtanen et al. 2020) development teams is alsogratefully acknowledged. The halo catalogs have been computed using the Adap-taHOP algorithm (Aubert et al. 2004).
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Appendix A: Is the shell theorem violated in theDirac-Milne cosmology ?
The previous discussion on the rotation curves and the virial ve-locity distributions in the Dirac-Milne universe has evidencedthe surprising property that in this cosmology, to a good approx-imation, around a central massive object, a harmonic restoringforce is felt in addition to the usual Gm / r Newtonian force,leading to nearly flat rotation curves. At distances comparableto the radius of the depletion zone, although mostly empty, thisadditional harmonic restoring force is far from negligible sinceit is on average equal to the force of the central galaxy (or, moregenerally, any massive structure surrounded by its depletion zoneand antimatter cloud). It may seem that this violates blatantly theshell theorem of Newton (Newton 1760) and, in General Rel-ativity, the Birkho ff theorem (Birkho ff & Langer 1923). In thepresent appendix, we discuss this question, which presents inter-esting features.The first element of answer comes from noticing that, in or-der to describe the gravitational behavior of matter and antimat-ter in the Dirac-Milne universe, two coupled Poisson equationsare required: ∆ φ + = π G ( ρ + − ρ − ) , (A.1) ∆ φ − = π G ( − ρ + − ρ − ) . (A.2)The possibility to express the Dirac-Milne gravitational be-havior with coupled equations using Laplacian operators im-plies, through the Gauss theorem, that in a situation of spheri-cal symmetry, the gravitational field in an empty region must bezero, since the mass enclosed is zero and spacetime in this re-gion should be Minkowskian. However, there is a caveat to thisassertion, related to the boundary conditions of the mass con-figuration. The previous statement on the shell theorem beingrespected may indeed seem at odds with our decomposition interms of three spheres (see Fig. 4) of the average galaxy environ-ment in the Dirac-Milne universe, where a harmonic restoringforce is observed in addition to the usual Newtonian force withinthe depletion zone. The short answer to this apparent contradic-tion is that the situation lacks spherical symmetry and that theharmonic restoring force that we have derived just approximatesthis asymmetric configuration of the antimatter cloud surround-ing a spherical depletion zone (see in particular Fig. 3). Note inparticular that in actual configurations, the depletion zones arenot spherical, but percolate with surrounding depletion zones,with a similar percolation property for the antimatter clouds.Also, the approximation used is valid only at distances smallerthan r d , where r d is the approximate size of the depletion zonesurrounding the massive structure, and becomes increasingly in-accurate when we exceed radial distances larger than ≈ r d / i.e. the cube with uniform repulsive background andthe cube containing a sphere with uniform positive mass den-sity compensating, within the volume of the sphere, the negativemass background of the first cube. As soon as we have acceptedthe property that the gravitational field created by cube (a) withuniform density is necessarily zero everywhere, which seems un-avoidable by symmetry, it is also clear that the contribution ofthe second cube will create a harmonic restoring force, althoughthe inner sphere, in the superposition of the two cubes, is nowempty. The situation is even stranger when we consider the configu-ration with complete spherical symmetry, where the whole spaceis filled with a uniform negative background, to which we super-impose a sphere centered on the origin with a positive uniformdensity, compensating the negative mass fluid inside the sphere(and only there). This time we cannot invoke the asymmetry ofthe situation and it seems that we have a gross violation of theshell and Birkho ff theorems (Newton 1760; Birkho ff & Langer1923) since the gravitational field appears to be nonzero in anempty region with exact spherically symmetry.However, the expression of Birkho ff ’s theorem (Birkho ff &Langer 1923) only states that any spherically symmetric solutionof the vacuum field equations must be static and asymptoticallyflat, and represented by a Schwarzschild metric. We must thennote that we have in fact filled out the entire space with a negativemass fluid of constant density, and therefore with infinite nega-tive global mass. This configuration is necessarily not static andthe “cosmological" aspects must now be taken into account inthe dynamical situation. Our decomposition in three cubes withrespective masses − m , + m and + m , on the other hand, restoresa total mass zero, and has not a diverging mass and potentialat infinity (Seeliger 1895), but involves a configuration withoutspherical symmetry.Two important additional comments can be made: – The first comment is based on the remarkable analysis byGurzadyan as early as 1985 (Gurzadyan 1985), in the earlydays of Dark Matter searches, which in several respectsreaches conclusions similar to those of the present analysis .In his first publication on this topic (Gurzadyan 1985),Gurzadyan notes that the Newtonian 1 / r force law couldbe extended by requiring the property, realized in the New-tonian case, that for a configuration with spherical symme-try, the gravitational action of a mass can be reduced to thesituation where all the mass is concentrated at the origin. Re-markably, this requirement leads not only to the Newtonianpotential, but also to an additional harmonic force, attractiveor repulsive depending on the sign of Λ , which appears asthe fluid analog of the cosmological constant, and we repro-duce here, with a slight change of notation, Equation (5) ofRef. (Gurzadyan 1985): F ( r ) = Ar − + Λ r (A.3)In this sense, the introduction of a cosmological constantin Newtonian cosmology finds a natural justification in theMilne cosmology (Milne 1933). Gurzadyan notes that thissecond component has the property that the shell theorem(zero gravitational field inside an empty spherical shell)is not respected. More importantly, in recent publications,(see e.g. (Gurzadyan & Stepanian 2019) and referencestherein), Gurzadyan conjectures that the flat rotation curvesobserved in galaxies can be explained by the same (local)expression as the cosmological repulsive term. However,with his expression of a generalized gravitational potentialusing two gravitational constants G and Λ , Gurzadian does The last sentence of this paper is particularly noticeable: “The small-ness of the cosmological constant evidently excludes the checking ofEquation (5) by means of any experimental methods; however, the con-tribution of the second member in (5) can be evaluated from the analysisof the structure of galaxy clusters, their haloes, etc. The possibility ofthe existence of a long-range force of the above type may a ff ect in acertain way the ideas concerning the future of the open Universe".Article number, page 15 of 16 & A proofs: manuscript no. 210216_Dirac_Milne_rotation_curves_AA not consider the possibility that these two terms could beexpressed with a single constant G but with a sign reversal.For example, the potential term in the GR metric in Equation(1) of Ref. (Gurzadyan & Stepanian 2019) leads directlyin Equation (2) to the two components, Newtonian andharmonic restoring force, of the gravitational field that wehave derived previously for Dirac-Milne. – The second comment is related to the fact that, as mentionedpreviously, cosmological simulations of self-gravitatingstructures routinely use e ff ective negative mass without ex-plicitly stating it. Indeed, as mentioned previously, cosmo-logical simulations first calculate the average density in orderto derive the average cosmological expansion of a ( t ), the cos-mological scale parameter. After subtraction of this averagedensity, a symmetric (in terms of sign of mass) mass distribu-tion is then obtained, usually with symmetrical overdensityand underdensity distributions. Piran indeed remarked (Piran1997), following previous numerical simulations of gravita-tional structures with Dubinski and collaborators (Dubinskiet al. 1993), that while we are used to representing gravita-tional structures in terms of (collapsing) positive mass, it isalso possible and useful to consider the problem in terms of(expanding) voids of negative mass. In their expansion, thesevoids will e ff ectively develop into the largest structures in theuniverse. As studied later by Sheth (Sheth & Van De Wey-gaert 2004), now followed by the work of several authors (fora review, see for example (Pisani et al. 2019) and referencestherein), the study of voids constitutes today a powerful testof the cosmology at play in our universe.We further note, following Piran, that the matter surround-ing an underdense region will create in its expansion a highdensity ridge along the rim of the underdense region, anddensity even diverging at shell crossing (see e.g. Figs. 1 and2 of Ref. (Dubinski et al. 1993)). A region nearly empty, oreven totally empty, will therefore, in a cosmological context, e ff ectively “repel" the surrounding matter, due to its fasterexpansion compared to its surroundings.Piran goes as far as to suppose that there could exist a speciesof matter with negative gravitational mass and positive iner-tial mass, violating the usual expression of the EquivalencePrinciple. But curiously, Piran describes the interactions ofthis new species endowed with negative gravitational massand positive inertial mass as attractive between themselves.We note that this is not the behavior actually observed in“voids" as the negative density (underdensity) flattens out in-stead of becoming more negative. Of course, under the ordi-nary assumption that only positive mass particles exist, themaximum “negative" mass (underdense) region is limited bythe condition of zero matter density. This leads to the funda-mental new element introduced by the introduction of “nega-tive" mass particles: a depletion zone develops, and we haveseen the fundamental role that it plays in terms of mimick-ing the behavior of extended Dark Matter clouds or, alterna-tively, providing a MOND-like behavior. As we have studiedin detail in (Manfredi et al. 2018), if we want to describe thebehavior of the repulsive underdense regions using a secondspecies of “negative" mass, this is not possible in a Newto-nian description, even with the three parameters of inertial,passive gravitational and active gravitational mass. In orderto implement the gravitational behavior of the Dirac particle-hole system, let us stress again that it is necessary to use abimetric description (Manfredi et al. 2018).In conclusion, the harmonic restoring force predicted by theDirac-Milne model, adding its contribution to the usual New-tonian force term, is an unavoidable consequence of the com-bined influence of the antimatter cloud and empty depletion zonedeveloping in the D-M universe around a localized structure ofpositive mass. But unlike the additional Λ term conjectured byGurzadyan, this harmonic restoring force does not violate theshell theorem. This unexpected restoring force is simply ex-plained in terms of the asymmetric distribution of the positiveand negative mass components in the D-M cosmology.term conjectured byGurzadyan, this harmonic restoring force does not violate theshell theorem. This unexpected restoring force is simply ex-plained in terms of the asymmetric distribution of the positiveand negative mass components in the D-M cosmology.