Structure formation in a Dirac-Milne universe: comparison with the standard cosmological model Giovanni Manfredi, Jean-Louis Rouet, Bruce N. Miller, Gabriel Chardin

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Giovanni Manfredi, Jean-Louis Rouet, Bruce N. Miller, Gabriel Chardin. Structure formation in a Dirac-Milne universe: comparison with the standard cosmological model. Physical Review D, Ameri- can Physical Society, 2020, 102 (10), pp.103518. ￿10.1103/PhysRevD.102.103518￿. ￿hal-02999626￿

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Structure formation in a Dirac-Milne universe: Comparison with the standard cosmological model

Giovanni Manfredi * Universit´e de Strasbourg, CNRS, Institut de Physique et Chimie des Mat´eriaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France

Jean-Louis Rouet Universit´ed’Orl´eans, CNRS/INSU, BRGM, ISTO, UMR7327, F-45071 Orl´eans, France

Bruce N. Miller Department of Physics and Astronomy, Texas Christian University, Fort Worth, Texas 76129, USA † Gabriel Chardin Universit´e de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France

(Received 15 July 2020; accepted 12 October 2020; published 13 November 2020)

The presence of complex hierarchical gravitational structures is one of the main features of the observed universe. Here, structure formation is studied both for the standard (ΛCDM) cosmological model and for the Dirac-Milne universe, a matter-antimatter symmetric universe that was recently proposed as an alternative “coasting” cosmological scenario. One-dimensional numerical simulations reveal the analogies and differences between the two models. Although structure formation is faster in the Dirac-Milne universe, both models predict that it ends shortly before the present epoch, at cosmological redshift z ≈ 3 for the Dirac-Milne cosmology, and at z ≈ 0.5 for the ΛCDM universe. The present results suggest that the matter power spectrum observed by the Sloan Digital Sky Survey might be entirely due to the nonlinear evolution of matter and antimatter domains of relatively small initial dimensions, of the order of a few tens of parsecs comoving at cosmological redshift z ¼ 1080.

DOI: 10.1103/PhysRevD.102.103518

I. INTRODUCTION nuclear matter of the standard model of particle physics, which is routinely observed in particle accelerators and The standard cosmological model (ΛCDM), which has makes up the world around us—constitutes today less than emerged in its present form over the last two decades, is 5% of the total mass-energy content in the ΛCDM model, capable of accurately reproducing most cosmological ≈25% — the rest being composed of cold dark matter (CDM, ) observations among others primordial nucleosynthesis, and dark energy (in the form of a cosmological constant Λ, the cosmic microwave background (CMB) radiation, bar- ≈70%). This is a rather unfortunate situation, which has yonic acoustic oscillations (BAOs), or type-1a supernovae stimulated a lot of work, both experimental (searching (SN1a) luminosity distance. Not all such measurements are experimental evidence for dark matter) and theoretical equally compelling, but their combination seems to require (e.g., introduction of new degrees of freedom, modification a universe composed of much more than just ordinary of the gravitational force, and relaxing assumptions of the (baryonic) matter. Some tensions have appeared more standard model such as homogeneity at large scales). recently between the values of the Hubble constant H0 Several authors have noted that our universe is very close derived from the early or late universe observations [1]. to a “coasting” universe, i.e., a universe that neither ≈5σ These discrepancies are statistically significant ( ). No decelerates nor accelerates, akin to the one originally systematic errors have been found yet, although it is too proposed by Edward Arthur Milne [4]. A review of early to ascertain whether these discrepancies are a glimpse coasting cosmologies was published recently by Casado of some “new physics” ahead [2,3]. [5]. For example, Sarkar and coworkers have recently Despite its descriptive success, the ΛCDM model is argued that the present SN1a data are still unable to clearly not a finished theory. Baryonic matter—ordinary demonstrate convincingly the acceleration of the universe expansion rate [6,7]. Further, Blanchard et al. [8] have *[email protected] shown that cosmic acceleration by all “local” cosmological † [email protected] probes (redshift z<3) is not statistically compelling.

2470-0010=2020=102(10)=103518(11) 103518-1 © 2020 American Physical Society MANFREDI, ROUET, MILLER, and CHARDIN PHYS. REV. D 102, 103518 (2020)

Finally, when a mild evolution of the SN1a luminosity is TABLE I. Interactions between matter (þ) and antimatter (−) allowed, the SN1a data of the Pantheon supernovae sample particles in the Dirac-Milne universe, from [17]. [9] show that the Milne universe constitutes a fairly decent fit to the SN1a data (see for example Fig. 4 of [9]). Also, it Type of matter Type of matter Interaction is well known that the age of the ΛCDM universe is nearly þþAttraction identical to the age of a Milne universe, equal to 1=H0. −−Repulsion In 2012, Benoit-Levy and Chardin [10] proposed an − þ Repulsion þ − alternative universe where matter and antimatter are present Repulsion in equal amounts and their gravitational interaction is repulsive, which they named the “Dirac-Milne” (D-M) universe to highlight its two main features, namely the bimetric extension of general relativity or, alternatively, as “ presence of antimatter (hence, Dirac) and its coasting mentioned above, as the description of the Dirac electron- ” expansion law (Milne). This universe, analogous in its hole system in a gravitational field. gravitational behavior to the Dirac electron-hole system, In the same work, we provided a detailed comparison of gravitational structure formation in the D-M and Einstein- avoids annihilation between matter and antimatter domains Ω ¼ 1 Ω ¼ 0 after cosmological recombination. Although this scenario is de Sitter (EdS: M ; Λ ) universes, using a local clearly unconventional, it should be noted that there is to 1D model embedded in a spherically expanding universe. For both cases, we observed gravitational structure for- date no direct experimental evidence on the gravitational mation, with clusters and subclusters developing from an behavior of antimatter, while several experiments are being almost uniform initial condition. Both models display developed at CERN to measure the acceleration of anti- power-law behavior in the wave number spectrum of the hydrogen atoms “free-floating” in the gravity field of the matter density in both the linear and nonlinear regimes. Earth. The first results of the Gbar [11], ALPHA-g [12], However, there is one crucial difference. Whereas for EdS and AEgIS [13] collaborations are expected within a couple the formation of structures continues indefinitely, in the of years and deviations from perfect matter-antimatter following we will see that structure formation freezes out symmetry will have profound consequences on current for the Dirac-Milne universe a few billion years after the big cosmological theories. bang, a feature shared with the ΛCDM model. With its null total mass, the D-M universe is gravita- In the present work, we compare the D-M universe to a tionally empty on large scales and thus displays a coasting universe with finite positive cosmological constant expansion, i.e., without acceleration nor deceleration. This (“ΛCDM”), using the same 1D Newtonian approach. It behavior leads to dramatically different timescales in the must be stressed, however, that our “ΛCDM” universe is early phases of the Universe. For example, the quark- essentially nonlinear, in contrast to the standard cosmologi- gluon-plasma transition lasts for about one day, instead of a cal model, for which the evolution of the power spectrum is few microseconds as in the Standard Model, and nucleo- almost entirely linear. This choice was dictated by our synthesis lasts about 35 years, compared to the three intention to closely compare how the D-M and the nonlinear minutes in the Standard Model, while recombination occurs “ΛCDM” models evolve from similar initial conditions at at an age of about 14 million years, compared to the 380 recombination. Our numerical results reveal striking simi- Λ 000 years of the CDM model. larities, but also significant differences between the two Despite these tremendous differences in the initial time- cosmologies, with the D-M model predictions appearing to scales, the D-M universe is remarkably concordant with be compatible with the hierarchical structures observed in only one adjustable parameter, namely H0 [10]. In par- today’s universe. 1 ticular, its age, equal to =H0, is almost identical to the age A short outline of the present work is as follows: In the next Λ ≈ 70 of the CDM universe for H0 km=s, the SN1a section, we briefly summarize the essential futures of the Λ luminosity distance is very close to that of CDM D-M universe. In Sec. III, we define the comoving coor- [10,14], and so is its primordial nucleosynthesis [10,15]. dinates used in the numerical code. Some features of the Since the distance to its horizon diverges, it also does not comoving equations of motion reveal an interesting relation- suffer from the horizon problem, nor does it need primor- ship between the D-M and ΛCDM cosmologies. The results dial inflation to explain the current homogeneity at large of the computer simulations are presented in Sec. IV, which scales. A more complete description of the properties of the also contains a direct comparison to data from the Sloan D-M universe can be found in Refs. [10,16]. Digital Sky Survey (SDSS). Conclusions are drawn in Sec. V. More recently [17], we developed a theoretical basis for the D-M model, whereby the unconventional matter- II. GRAVITATIONAL PROPERTIES OF THE antimatter gravitational interaction can be accounted for, at DIRAC-MILNE UNIVERSE the Newtonian level, by two gravitational potentials that obey two coupled Poisson’s equations (see Table I). This In the Dirac-Milne universe, whereas matter attracts model may be viewed as the Newtonian limit of some matter, all other gravitational interactions are repulsive,

103518-2 STRUCTURE FORMATION IN A DIRAC-MILNE UNIVERSE: … PHYS. REV. D 102, 103518 (2020) as is summarized in Table I, reproduced from Ref. [17]. described in Sec. IV, that structure formation stops around Hence, matter can form gravitational structures, whereas the same epoch in numerical simulations of both universes. antimatter, being repelled by everything else, tends to spread We further note that the standard cosmological Poisson across the universe. Such spread is rather uniform, but not equation is completely: since matter repels antimatter, the latter is −3 expelled from matter-dominated regions (galaxies) and Δϕ ¼ 4πGðρ − ρ¯Þ¼4πGðρ − ρ0a Þ; ð4Þ forms a low-density almost homogeneous background dis- ’ tributed over the underdense regions in between matter s where ρ¯ is the average matter density, so that the D-M and structures. As in the analog electron-hole system in a semi- ΛCDM models have a similar form for the Poisson conductor, the matter and antimatter regions are separated equation for matter. However, in the D-M case ρ¯ ¼ 0 from each other by a depletion zone that precludes the and the negative mass density ρ− plays the role of ρ¯. occurrence of annihilation events, in accordance with the observations. A simple analytic model predicts that this depletion zone occupies ≈50% of the volume of space, III. COMOVING COORDINATES as confirmed by 3D simulations that will be presented Let us now consider an expanding distribution of matter elsewhere. with spherical symmetry. In this case, the gravitational field It must be stressed that the Dirac-Milne scenario, in the has only one component Erðr; tÞ, which depends on time Newtonian limit, cannot be simply described by a combi- and on a single spatial variable r. This type of system was nation of the signs of the inertial and gravitational (active studied extensively in the past [18–26]. and passive) masses. Instead, as shown in [17], the Dirac- In the presence of a finite cosmological constant Λ, the Milne model can only be accounted for by two gravitational Newtonian equation of motion for matter particles reads as: potentials that obey two distinct Poisson equations: d2r c2Λ Δϕ ¼ 4π ðρ − ρ Þ ð Þ ¼ E ðr; tÞþ r; ð5Þ þ G þ − ; 1 dt2 r 3

Δϕ− ¼ 4πGð−ρþ − ρ−Þ: ð2Þ where Er ¼ −∂rϕ is the gravitational field and c is the speed of light. The factor 3 comes from the 3 space The above model may be seen as the Newtonian limit of a dimensions. bimetric gravity model. We consider an expanding universe with scale factor aðtÞ In the forthcoming numerical simulations, we will make and, following [27], we define the generalized “super- the further simplifying hypothesis that antimatter consti- comoving” coordinates (denoted by an overcaret) as: tutes a low-density background uniformly distributed everywhere in space. This approximation appears to be r ¼ aðtÞr;ˆ ð6Þ justified for the study of gravitational structure formation, as overdense regions are very much dominated by matter dt ¼ b2ðtÞdt:ˆ ð7Þ anyway. Using this approximation, one can neglect the evolution of antimatter and replace it with a homogeneous Note that we also introduced a scaled time tˆ, which defines background that decreases as the inverse of the cube of the a new epoch-dependent “clock.” The velocity transforms as scale factor aðtÞ: ˆ 3 dr a dr ρ−ðr;tÞ¼ρ0=a ; ¼ þ a_ r;ˆ ð8Þ dt b2 dtˆ ρ ’ where 0 denotes today s matter density. In the following, where a dot stands for differentiation with respect to t. “ ” the subscript 0 will be used systematically to refer to By defining v ≡ dr=dt and vˆ ≡ dr=ˆ dtˆ, Eq. (8) can be quantities evaluated at the present time. Then, the Poisson rewritten as equation for matter becomes a −3 vˆ ¼ v − HðtÞr; ð9Þ Δϕ ¼ 4πGðρ − ρ0a Þ; ð3Þ b2 where we have dropped for simplicity the subscript “+” where HðtÞ¼a=a_ is the Hubble parameter. Therefore, ≡ ˆ 2 denoting matter. The dilute repulsive background could be vpec av=b represents the peculiar velocity, i.e., the viewed as a cosmological constant that decreases with time, velocity fluctuations around the Hubble flow HðtÞr. the corresponding vacuum energy decreasing as a−3. This Using the transformations of Eqs. (6) and (7), the similarity with ΛCDM underpins the observation, further comoving equation of motion becomes

103518-3 MANFREDI, ROUET, MILLER, and CHARDIN PHYS. REV. D 102, 103518 (2020) 2 ˆ _ _ ˆ ̈ 4 2Λ Ω 1 d r 2 a b dr 4 a b ˆ c 4 ̈¼ − 2 M − Ω ð Þ þ 2b − þ b rˆ ¼ E þ b r;ˆ ð10Þ a H0 2 Λa ; 15 dtˆ2 a b dtˆ a a3 r 3 2 a ˆ ðˆ ˆÞ where Er r; t is the scaled gravitational field. As the _ Ω 1=2 3 a M density must scale as ρˆðr;ˆ tˆÞ¼a ðtÞρðr; tÞ in order to ¼ H0 þ ΩΛ : ð16Þ a a3 preserve the total mass, we scale the gravitational field ˆ ˆ 2 as Erðr;ˆ tÞ¼a ðtÞErðr; tÞ, so that the Poisson equation where H0 ¼ða=a_ Þ . The quantities Ω and ΩΛ are, remains invariant in the scaled variables. 0 M respectively, the densities of matter (both baryonic and We choose the same time scaling that was used for the 4 ¼ 3 dark) and vacuum normalized to the critical density, with EdS universe [17], i.e., b a , so that the coefficient in Ω ¼ Λ 2 ð3 2Þ Λ front of the gravitational field is time-independent. This Λ c = H0 . Approximate values [28] in the CDM Ω ¼ 0 3 Ω ¼ 0 7 yields: model are M . and Λ . , which will be used in the forthcoming simulations. For a case without radiation, d2rˆ 1 drˆ c2Λ an analytical solution of the Friedmann equation (16) can þ a1=2a_ þ a2ärˆ ¼ Eˆ þ a3r:ˆ ð11Þ dtˆ2 2 dtˆ r 3 also be obtained, see Appendix. Inserting the Friedmann equations (15)–(16) into the The above comoving equation of motion can be used for equation of motion (11), we obtain both the D-M and the ΛCDM cosmologies, by taking the respective scale factors aðtÞ. 2 2 d rˆ H0 drˆ ω þ ðΩ þ Ω 3Þ1=2 ¼ J0 ˆ þ ˆ ð Þ ˆ2 2 M Λa ˆ 3 r Er; 17 A. Dirac-Milne universe dt dt pffiffiffiffiffiffiffiffiffiffiffiffiffi In the Dirac-Milne cosmology, the cosmological con- where ω 0 ¼ 4πGρ0 is the Jeans’ frequency and we used Λ ¼ 0 ð Þ J stant vanishes ( ) and the scale factor a t is linear in the following relation: time (coasting universe): 2 ð Þ¼ ð Þ 2Ω ¼ ω2 ð Þ a t t=t0; 12 H0 M 3 J0: 18 where t0 denotes the present epoch. We note that, for the For a homogeneous density, the gravitational field is D-M universe, the age of the universe is exactly equal to ˆ 2 −1 Er ¼ −ðω 0=3Þrˆ, and the two terms on the right-hand side t0 ¼ H0 , where H0 is the Hubble constant at the present J time [10]. of Eq. (17) exactly cancel each other. As in the D-M case, For positive-mass particles, the equation of motion (11) we assume a locally planar perturbation embedded in this then becomes: expanding universe. In this locally planar system, the factor 1=3 can be dropped from the first term on the right-hand 2 ˆ ˆ side of Eq. (17) and this term can be incorporated into an d r H0 1 2 dr ˆ þ a = ¼ E : ð13Þ ’ dtˆ2 2 dtˆ r approximate 1D Poisson s equation Next, we consider a locally planar perturbation embedded ∂ ˆ Er ¼ −4π ½ρˆðˆ ˆÞ − ρ ð Þ in this expanding universe. In this locally planar system, the ∂ˆ G r; t 0 : 19 comoving version of Poisson’s equation (3) can be approxi- r mated by its one-dimensional (1D) counterpart (we still use The 1D equation of motion then becomes the notation rˆ for the local comoving coordinate):

2 ∂ ˆ d rˆ H0 drˆ Er þ ðΩ þ Ω 3Þ1=2 ¼ ˆ ð Þ ¼ −4πG½ρˆðr;ˆ tˆÞ − ρ0; ð14Þ 2 M Λa Er: 20 ∂rˆ dtˆ 2 dtˆ where we recall that ρ0 represents, in comoving coordi- For ΩΛ ¼ 0 and ΩM ¼ 1, we recover the EdS universe nates, a uniform background of negative-mass particles. [17]. The ΛCDM simulations presented in the forthcoming The numerical results of the D-M universe presented in the section will be performed in this planar reference frame, next sections will be based on N-body simulations of the using Eqs. (19) and (20). ’ scaled equation of motion (13) and Poisson s equation (14). Interestingly, the equations of motion for the D-M universe [Eq. (13)] and for the ΛCDM universe [Eq. (20)] only differ ΛCDM B. universe in the coefficient of the fictitious friction terms (the respective For the ΛCDM universe, the cosmological constant is Poisson equations are also identical). This friction coefficient nonzero and the scale factor is a solution of the Friedmann can be written as a function of the scale factor (leaving aside equations (neglecting radiation): the prefactor H0=2):

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2.0 interacting particles, using a velocity Verlet scheme. 5 DM Typical simulations employed N ≈ 2.5 × 10 particles. CDM To reduce the level of fluctuations, for each case we 1.5 EdS performed ensemble averaging over 5 statistically equiv- alent initial conditions. As mentioned in the preceding sections, we consider a

f(a) 1.0 3D expanding spherically symmetric universe and then study planar perturbations in the comoving coordinates. This reduces the problem to one spatial dimension in the 0.5 local comoving coordinate rˆ, which will be represented in the numerical results. In this 1D approximation, the particles are in fact infinite sheets with uniform surface 0.0 mass density. Boundary conditions are taken to be spatially 0.00.51.01.52.0periodic. More details on the model can be found in a Refs. [17,20–23]. As already stated (see Sec. II), only matter (positive mass) is evolved in the numerical simu- FIG. 1. Coefficient fðaÞ of the friction term in the various lations. Antimatter (negative mass) is approximated by a equations of motion, see Eq. (21). uniform background with constant density ρ0 in the comoving frame. In the present set of simulations, no 1=2 attempt is made to simulate the depletion zone between f − ðaÞ¼a ; D M matter and antimatter. ð Þ¼ðΩ þ Ω 3Þ1=2 fΛCDM a M Λa ; For both models, the initial condition is set at recombi- ð Þ¼1 ð Þ nation, corresponding to redshift z ¼ 1080; note that this fEdS a ; 21 redshift does not correspond to the same cosmological where we also added the EdS case for comparison. Note that, time in the D-M (≈14 million years) and ΛCDM for the D-M and ΛCDM cosmologies, the friction coefficient (≈380 000 years) universes, see [17]. The initial matter grows with a, which will lead to the freezing of the density is the sum of a spatially uniform term ρ0 and a small ρ˜ ð Þ¼ gravitational structures before or around the present epoch perturbation with the 1D power spectrum P1D k jρ˜ j2 ∼ p (roughly, z ≈ 3 for D-M and z ≈ 0.5 for ΛCDM). k k , where k is the wave number. Initial power spectra The three cases are represented in Fig. 1. The friction of this form, with p ∈ ½0; 4, were used in a number of earlier term is weaker at early times (a ≲ 0.3) for the D-M model works on structure formation [21,25]. In the present work, we compared to ΛCDM leading to a faster formation of larger take p ¼ 3, which produces a spectrum that is largest at small structures in the D-M universe, as will be shown in the wavelengths, and then study the clustering of matter at forthcoming simulations. For 0.3 ≲ a<1, the situation is increasingly larger scales. The standard 3D power spectrum ð Þ ð Þ ¼ ð Þ2π 2 reversed. Interestingly, the integrated coefficient between P k can be obtained by setting: P1D k dk P k k dk, ð Þ¼ ð2π 2Þ the big bang and the present time yielding P k P1D= k . Hence, the initial 3D spectrum Z behaves as the standard Harrison-Zeldovich spectrum: 1 PðkÞ ∼ k. The decrease of PðkÞ at z ¼ 0 for large wave f¯ ¼ fðaÞda; 0 numbers originates from the fact that the initial 1D spectrum becomes flat due to statistical noise when 2π=k reaches the is almost identical for the D-M universe (f¯ ¼ 0.666) and average initial interparticle distance. the ΛCDM universe (f¯ ¼ 0.676). This is an interesting For the sake of comparison, the initial spectra are taken indication that, although the two models have very different to be statistically identical (same slope) for the D-M and past histories, they should lead to a similar universe at the ΛCDM models. The ensuing evolution is fully nonlinear in present time. In particular, they should both stop forming both cases. This is the expected behavior in the D-M structures at similar epochs, as will be confirmed by the universe, where the density fluctuations are already large at forthcoming numerical simulations. z ¼ 1080 [10], so that collapse and nonlinear evolution occur almost immediately. In contrast, in the standard cosmological model, the evolution of the power spectrum IV. SIMULATION RESULTS is almost entirely linear, the nonlinearity bringing only a In this section we present the results of numerical relatively minor correction [29]. Here, however, the idea is simulations obtained with the Dirac-Milne and ΛCDM to closely compare the two universes starting from the same models described in the preceding sections. The simula- initial configuration. Therefore, we choose to focus only on tions were performed with an N-body code [17] that solves the nonlinear development of structures even for the the relevant equations of motion (13) or (20) for N ΛCDM case.

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105 CDM their nonlinear evolution almost immediately after the Dirac-Milne CMB transition, which results in rapid virialization of 4 10 the velocities of the initial structures, effectively decoupled

3 from the cosmological expansion. The initial velocity 10 spread for the simulations presented here can be read from

2 the corresponding figure at the end of this section (Fig. 7). 10 The evolution is labeled by either the scale factor a or the cosmological redshift z ¼ð1 − aÞ=a. In the numerical 10 code, densities are measured in units of ρ0 and time in units of ω−1. Lengths are measured in units of an arbitrary 1 J0 10-2 10-1 1 length scale λ and the gravitational field (an acceleration) is λω2 λ expressed in units of J0. In practice, is an adjustable parameter which is chosen here so that the power spectrum ¼ 0 Λ FIG. 2. Power spectra at z for the D-M and CDM at z ¼ 0 issued from the simulations has a peak at the same universes. wavelength as the spectrum obtained from observations, such as the Sloan Digital Sky Survey (SDSS) [30]. Then, The initial velocities are set using the Zeldovitch once λ has been fixed, all other normalizations (e.g., for the approximation, so that only the growing mode is excited particle velocities or the amplitude of the power spectrum) [21]. The influence of the initial condition for the particle follow uniquely without any additional assumptions. velocities has been studied by varying the amplitude of the velocity distributions. For the Dirac-Milne model, the 101 uncertainty and influence of these initial velocities should Dirac-Milne be relatively small, as the initial structures are initiating CDM 1

104 z=-.9 z=0. 102 z=9. 10-1 z=99. 1 z=341. z=1080. 10-2 10-2 10-4 10-3 10-2 10-1 1 101 a 10-6 105 10-8 4 10-2 10-1 1 10 102 103 10

103 105 2 4 z=-.95 10 10 z=0. z=4.4 103 101 z=24. 102 z=116. z=1080. 1 Dirac-Milne 10 CDM -1 1 10 10-3 10-2 10-1 1 101 10-1 a 10-2 FIG. 4. Top panel: wave number kpeak corresponding to the 10-3 Λ 10-3 10-2 10-1 1 10 peak of the power spectra for the Dirac-Milne and CDM universes as a function of the scale factor aðtÞ. Bottom panel: corresponding value of the peak power at kpeak. A striking feature FIG. 3. Evolution of the power spectra for the cases D-M (top of these simulations is that kpeak evolves in time, describing the panel) and ΛCDM (bottom panel), for different cosmological nonlinear evolution in both the D-M and ΛCDM models, whereas redshifts z. The thick lines correspond to the present epoch in the standard ΛCDM analysis the usual assumption is that the (z ¼ 0). A negative value of z ¼ −0.9 for D-M corresponds to nonlinear evolution represents only a small correction, while the −1 a ¼ 10 (t ≈ 140 Gy), while a negative value of z ¼ −0.95 for peak position is fixed at kpeak ∼ 0.018 h Mpc corresponding to ΛCDM corresponds to a ¼ 20 (t ≈ 65 Gy). the mode entering the horizon at matter-radiation equality.

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We show the results of two typical simulations, one each For both cosmologies, it is clear that structure formation for the D-M and ΛCDM models. has stopped at, or slightly earlier than, the present epoch. The power spectra at z ¼ 0 for the two cosmologies are For the D-M case (Fig. 3, top panel), the initial spectrum at shown synoptically in Fig. 2. As mentioned above, the z ¼ 1080 peaks around 100 h−1 kpc, when structure for- horizontal axis has been scaled so that the spectra peak at mation begins. Then, the evolution proceeds by collecting −1 0.018 h Mpc , where h ¼ H0=ð100 km=s=MpcÞ,asin larger and larger clusters in a bottom-up fashion. At the the observed SDSS spectrum [30]. The shape of the present epoch, the spectrum displays a nonlinear power-law contemporary D-M and ΛCDM spectra are virtually behavior extending over four decades in wave number identical, supporting the idea that, although the details of space, with a peak around 50 h−1 Mpc, the formation of the the evolutions are very different (coasting expansion for largest structures being frozen since z ≈ 3 in the D-M D-M as opposed to a sequence of accelerations and universe. We stress again that the transition between the decelerations for ΛCDM ), the end result at z ¼ 0 is rather P ∼ k and the P ∼ k−3 regimes comes about because of similar. This conclusion is in line with our earlier obser- strong nonlinear effects—whereby clusters of matter coa- vation (see Fig. 1) that, while the respective comoving lesce into bigger clusters, then into even bigger clusters, equations of motions are different, “averaging” (in a loose and so on and so forth—until this process stops shortly way) between a ¼ 0 and a ¼ 1 produces effectively the before the present epoch. For D-M, this freezing of same results. Further, the expected slopes for long (P ∼ k) structures is due to the presence of a homogeneous back- and short (P ∼ k−3) wavelengths are correctly recovered by ground of negative mass, which acts as a cosmological the Dirac-Milne simulations. This power law behavior in constant that decreases with time, as was mentioned the nonlinear regime suggests a self-similar matter distri- in Sec. II. bution in each model, pointing to the existence of a robust A similar nonlinear build-up of gravitational structures is fractal dimension [21,31]. also seen in our ΛCDM simulations (Fig. 3, bottom panel), The evolution of the power spectra from z ¼ 1080 to in contrast with the standard ΛCDM model which is z ≈−0.9 (corresponding to a ≈ 10) is displayed in Fig. 3. essentially linear [29]. This is due to our choice of initial

z=341. 3 1.0 103

v 0 density 0 -1.0 103 z=99. 8 3.6 103

v 0 density 0 -3.6 103 z=9. 14 3.6 103

v 0 density 0 -3.6 103 z=0. 24 1.5 103

v 0 density 0 -1.5 103 z=-0.90 29 2.3 102

v 0 density 0 -2.3 102 -1920 0 1920 -1920 0 1920 r (h-1 Mpc) r (h-1 Mpc)

FIG. 5. Dirac-Milne universe. Left frames: matter density ρðr;ˆ tÞ in comoving space for the positive-mass particles, at five epochs characterized by different redshift z. Right frames: corresponding distributions in the phase-space. The vertical axis is the peculiar velocity, in km=s.

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z=116. 7 5.1 103

v 0 density 0 -5.1 103 z= 24. 11 6.1 103

v 0 density 0 -6.1 103 z= 4. 16 5.1 103

v 0 density 0 -5.1 103 z= 0. 18 3.3 103

v 0 density 0 -3.3 103 z= -0.95 24 2.7 102

v 0 density 0 -2.7 102 -1920 0 1920 -1920 0 1920 r (h-1 Mpc) r (h-1 Mpc)

FIG. 6. ΛCDM universe. Left frames: matter density ρðr;ˆ tÞ in comoving space for the positive-mass particles, at five epochs characterized by different redshift z. Right frames: corresponding distributions in the phase-space. The vertical axis is the peculiar velocity, in km=s. condition, dictated by our wish to highlight the differences structures is around 50 Mpc, in accordance with the power between D-M and ΛCDM starting from a similar spectra of Fig. 2. At the same epoch, the peculiar velocities configuration. start decreasing, signalling a local cooling. As was already noticed in our earlier work [17], structure Figure 7 shows the “thermal” dispersion of the peculiar ≡ hð − Þ2i1=2 formation is initially faster for the D-M universe. This fact velocities, see Eq. (9), defined as vp;th v Hr , is even more apparent from the evolution of the peak wave where the average is taken over all the particles. At the number of the power spectrum, as shown in Fig. 4, where present epoch (a ¼ 1), D-M predicts peculiar velocities of we clearly see that structure formation has stopped at a similar epoch for both universes. The peak power also grows faster in the D-M universe, and is larger than the 104 corresponding ΛCDM power during the epochs between z ≈ 100 and z ¼ 0. This discrepancy might yield a differ- 103 ence in the predicted abundance of clusters or quasars at high redshift, which could be tested against observation. Again, it is important to note that the evolution in time of 102 the position of the peak for our ΛCDM simulation reflects the development of structures in the nonlinear regime, Λ 10 while in the standard CDM analysis, the peak position is ΛCDM time-independent and fixed by the mode entering the Dirac-Milne horizon at matter-radiation equality [29]. 1 -3 -2 -1 2 Structure formation in comoving space is shown in 10 10 10 1 10 10 a Figs. 5 and 6 for the D-M and ΛCDM universes, respec- tively. Again, in both cases the gravitational structures FIG. 7. Dispersions of the peculiar velocities for the D-M and freeze, around z ≈ 0.5 (a ≈ 0.67) for ΛCDM and z ≈ 3 nonlinear ΛCDM cosmologies, as a function of the scale (a ≈ 0.25) for D-M. The typical size of the contemporary parameter aðtÞ.

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100000 construction of larger structures at about the same time—a few billion years after the big bang—and in the case of ΛCDM approximately at the epoch where the dark energy component is supposed to become predominant. The ) 3 gravitational structures observed around the present epoch 10000 in our simulations are surprisingly similar for the two Mpc

-3 models. This is all the more remarkable, as the two universes have undergone a very different earlier history— coasting expansion for D-M vs. a sequence of accelerations P(k) (h SDSS data and decelerations in ΛCDM. 1000 CDM linear Further, we have compared the D-M power spectrum Dirac-Milne Dirac-Milne smooth with observational data from the SDSS survey [30]. Although the D-M spectrum agrees relatively well with 0.01 0.1 the data points around the main peak, it displays signifi- cantly less power at shorter wavelengths. This discrepancy k(hMpc-1) may be attributed to the 1D geometry and lack of baryons FIG. 8. Comparison of power spectra at z ¼ 0. Black squares: physics and feedback effects in our approach. More SDSS data points; red line: best fit linear ΛCDM model (from complete 3D simulations, currently underway, should be Ref. [30]). Solid black line: Dirac-Milne spectrum (this work); able to settle this important issue. black dashed line: Dirac-Milne spectrum averaged with a moving It is important to note that the mechanisms of con- triangular window over 10 points. struction of large-scale structures, although similar in their contemporary stages, are remarkably different in their early the order of 400 km=s, which is comparable with the values stages. The Dirac-Milne cosmology begins its structure usually reported in the literature [32,33]. In future epochs, formation very early (a few tens of millions of years after significant cooling can be observed. the CMB transition) and from relatively small matter pools 105 ≈10 Finally, in Fig. 8, we compare the power spectrum (typically solar masses, with domain size pc ¼ 1080 obtained from our own Dirac-Milne simulation (solid comoving at z ) [10], directly in a nonlinear regime and dashed black curves) with data points from the with a matter-antimatter density contrast of order unity. SDSS survey (black squares), as well as a best-fit linear Then, structure formation proceeds as a bottom-up non- ΛCDM model to these data (red line), both taken from linear construction, with small clusters coalescing into Ref. [30]. Although the D-M simulated spectrum agrees larger and larger ones, until this process stops near the relatively well with the data points around the main peak, at present epoch. shorter wavelengths it displays significantly less power In contrast, ΛCDM starts its structure formation from a (about one order of magnitude) than the SDSS data (note nearly homogeneous universe with a linear growing mode that, as is apparent from Fig. 2, our nonlinear ΛCDM that requires an elusive dark matter component to trigger spectrum at z ¼ 0 is very similar to the corresponding D-M the onset of nonlinear collapse and reionization at a much one). This is an important issue that will have to be later stage. In this respect, Dirac-Milne appears to provide a addressed in future, more sophisticated, simulations. The much more unified approach, starting from a single scale most obvious limitations of the present approach are the 1D and building up in a nearly pure bottom-up construction, nature of the modeling and the absence of baryonic physics with three orders of magnitude of comoving scale growth, and related feedback, which are fundamental at smaller the entire matter power spectrum. scales but probably less important for the position of the Further, the Dirac-Milne cosmology may provide an main peak at smaller k (large scales). explanation for what appears in ΛCDM as a remarkable coincidence, since the small oscillation of the BAOs (a few V. CONCLUSIONS AND PERSPECTIVES percent) occurs at a scale where strong clustering and inhomogeneity is clearly visible at the same 100–150 Mpc In this paper, we compared nonlinear structure formation scale. In Dirac-Milne, there is a single scale, resulting from in the ΛCDM and the Dirac-Milne cosmologies. This is a the inhomogeneities of the matter-antimatter emulsion at follow-up to our previous study comparing the Dirac-Milne z ≈ 1080, followed by roughly three orders of magnitude of and Einstein-de Sitter universes, where it was shown that bottom-up nonlinear growth of this initial scale. the structure formation in the Einstein-de Sitter universe Encouraged by the present study, we will present in a never fully stops, this universe being critical and right at the forthcoming publication the results of a set of 3D simu- border of recollapse. The present study highlights the lations using a modified version of the RAMSES simulation similarities in the recent (z ≲ 5) and contemporary stages code [34]. There, we shall investigate the impact of the of the D-M and ΛCDM universes: both universes stop their asymmetry between the geometrical distributions of the

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105 matter component, collapsed in planes, filaments and -2 4 10 cluster node structures, and the antimatter clouds, unable 10 3 to collapse due to their internal repulsion, and separated 10 2 from matter by a depletion zone. These further 3D 10 10

simulations may allow us to investigate the origin of the a 10-3 1 baryonic acoustic oscillations (BAOs), an issue not 106 107 addressed here, which in the Dirac-Milne model have a 10-1 ≈20 10-2 EdS much larger value ( Gpc) and play a completely ΛCDM (ΩR=0) -3 10 -5 negligible role in the formation of structures. Indeed, in ΛCDM (ΩR=5.10 ) the Dirac-Milne universe, all the structures develop from a 10-4 106 107 108 109 1010 1011 single, nonlinear, self-similar process. t (y)

ACKNOWLEDGMENTS FIG. 9. Time evolution of the scale factor aðtÞ for the EdS Λ Ω ¼ 0 We are indebted to Jim Rich for his thorough reading of universe and for the CDM universe without ( R ) and with Ω ¼ 5 10−5 the manuscript and many insightful comments. We also ( R × ) radiation. The inset is a zoom at early epochs. thank Cl´ement Stahl for several useful suggestions. Needless to say, the authors are solely responsible for which gives the errors or imprecisions that may still remain in this paper.  pffiffiffi  The numerical simulations were performed on the com- 1=3 4Ω g expð 6ω 0tˆÞ puter cluster at the Centre de Calcul Scientifique en r´egion aðtˆÞ¼ M i pffiffiffiJ ; ðA4Þ Ω ð1 − ð 6ω ˆÞÞ2 Centre-Val de Loire (CCSC). Λ gi exp J0t

APPENDIX: INTEGRATION OF THE with FRIEDMANN EQUATION pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 þ ΩΛa =Ω − 1 For the D-M universe, the behavior of the scale factor g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii M : ðA5Þ i 1 þ Ω 3 Ω þ 1 aðtÞ is known analytically, see Eq. (12). For the ΛCDM Λai = M case, one has to solve the Friedmann equation (neglecting radiation, as was done throughout this work) in order to Note that Eq. (A3) has a vertical asymptote: when a → ∞, obtain aðtÞ. As our equation of motion (20) is written in then t → t∞, where comoving coordinates, it is convenient to rewrite the Friedmann equation (16) using comoving variables. With 1 ω 0tˆ∞ ¼ − pffiffiffi logðg Þ the help of Eq. (7), we get: J 6 i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi da ˆ 3 With the parameters of the present work, ωJ0t∞ ≈ 8.775. ¼ H0a ΩM þ ΩΛa : ðA1Þ dtˆ Once the expression of aðtˆÞ is known, it is possible to integrate Eq. (7) to obtain the relation between the real time Integrating Eq. (A1) yields t and the scaled time tˆ: Z rffiffiffi Z a tˆ pffiffiffiffiffiffiffiffi 1 0 2 0 pffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi da ¼ ω 0 dtˆ ; ðA2Þ 2 1 þ g expð 3=2ω 0tˆÞ Ω J i J a 0 Λ 03 3 0 H0t ¼ pffiffiffiffiffiffiffi log pffiffiffiffi pffiffiffiffiffiffiffiffi : ðA6Þ i a 1 þ Ω a 3 Ω ˆ M Λ 1 − gi expð 3=2ωJ0tÞ where a ¼ 1=ðz þ 1Þ, with z ¼ 1080, is the initial value ˆ i i i For t ¼ 0, Eq. (A6) gives the initial time ti, corresponding of the scale factor and we have introduced the Jeans to the recombination epoch. frequency ω 0 using Eq. (18). We obtain: J If one includes radiation in the model (i.e., ΩR > 0), no pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 analytical solution can be found, and the Friedmann 1 þ ΩΛaðtˆÞ =Ω − 1 log pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM equation should be integrated numerically. However, the ˆ 3 1 þ ΩΛaðtÞ =ΩM þ 1 difference is minimal, as is shown in Fig. 9, which displays pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ω ¼ 0 1 þ Ω 3 Ω − 1 pffiffiffi the evolution of the scale factor for R and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛai = M ˆ Ω ¼ 5 10−5 − log ¼ 6ωJ0t; ðA3Þ R × , together with the expression for an EdS 1 þ Ω 3 Ω þ 1 2 3 Λai = M universe (a ∼ t = ).

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