arXiv:1612.04572v3 [gr-qc] 18 Sep 2017 2 1 da scale Briscese large Fabio the to dynamics. applications matter system: N-body Newtonian h Schr The cie sacasclNbd ytmi hc h indi- DM the gravitationally. of which only agglomerates interact in that bounded system particles represent de- N-body particles safely classical be vidual a can as CDM [10]. scales universe scribed the cosmological within the at describe bodies Therefore, to massive sufficient of is evolution Newtonian where equations time the Einstein cases the non-relativistic, realistic struc- of are for limit bound velocities and where typical scales, this small the scales, but At fluid; smaller form. dust tures at suc- pressureless fails is a CDM assumption as scales, described cosmological cessfully large At crucial. comes is particles DM [9]. for currently issue search are open the an there and elusive, Even candidates, [8]. still DM supernovae many are and particles [1], [7], DM BAO LSS though [6], including leansing data, in [5], is cosmological CMB (CDM) all matter with dark at cold agreement fluid fact, cold In a as scales. described cosmological is DM (almost) and interact gravitationally, that only species particle standard unknown the of In stituted DM). of review historical (DM) Λ an and Matter for Dark sheets [3] collisionless (see superclusters, by [2] as shaped is universe, struc- [1], scale the filaments large in of formation (LSS) the tures tat accepted widely is It Introduction 1 wl eisre yteeditor) the by inserted be (will No. manuscript EPJ CMcsooia oe 4,D sasmdt econ- be to assumed is DM [4], model cosmological -CDM oo5 08 oe Italy. Rome, Gr 00185 Severi, Francesco UK 5, Matematica Tyne, Moro Alta upon di Newcastle Nazionale 8ST, Istituto Enginee NE1 Electrical 207, and D Building Physics Mathematics, of Department nti otx,tesuyo h vlto fCMbe- CDM of evolution the of study the context, this In ojcue h mrigeetv lnkcntn scompu is constant as Ca Planck system the effective on emerging and the Nelson, conjecture, by introduced quantization stochastic ecie ntrso h Schr¨ the of terms in described ag cl tutr omto sdsusd nparticula In discussed. is formation structure scale large ftevldt fteSchr¨ the of validity the of Abstract. date version: Revised / date Received: n h aso h ois and bodies, the of mass the and PACS. scales. cosmological large at dynamics o omlg 88.k–0.0- tcatcprocesses Stochastic 05.40.-a – 98.80.-k Cosmology ¨ igrPisneutosa h ag- ii fthe of limit large-N the as equations dinger-Poisson ~ nti ae ti rudhwtednmc ftecasclNe classical the of dynamics the how argued is it paper this In ∼ M 5 / 3 G 1 / 2 ( /<ρ> ρ < N/ o igrmto snmrcldul fteNbd simulations N-body the of double numerical as method dinger o igrPisneutosi h large the in equations dinger-Poisson > ρ < ) 1 / 6 steraeaedniy h eeac fti euti the in result this of relevance The density. average their is hr is where , G poNzoaed iiaMtmtc,Citt`a Universitaria Matematica, Fisica di Nazionale uppo h rvttoa constant, gravitational the ig otubi nvriy ecsl iyCmu,Elli Campus, City Newcastle University, Northumbria ring, 2 lto tandi h -oysmlto,oeincreases one simulation, N-body the in attained olution on of bound mass LSS. lower (average) of practical the formation a the of galactic describe course, order to of Of the aims one formation of if the be halos galactic can study it to but wants halos, mass one the if as galaxies far as e.g. study; [11,12,13,14,15], not case does the M This be results. question final to the and seem arbitrary, affects is choice this simulation whether N-body the in ticles particles. clus- of of millions of and of made particles, galaxies formation of of hundreds the ters of shows made simulation effective halos masses such the galactic of from fact, variety up In wide built suffi- particles. a are is with abundances description different halos this and how mass, explain same to the cient have par- all these do simulation the ticles in dark although and elementary particles, of matter agglomerations huge represent system 1]wscridottaigteeouinof evolution mass the of tracing particles out identical carried [11, simulation simulations was MILLENNIUM numerical [11] the N-body instance, For to resort 12,13,14,15]. to is system atta decreasing that fact ,ti niggvsafrhragmn nsupport in argument further a gives finding this r, × smc mle hntems fteojcsta we that objects the of mass the than smaller much is n ih ru httecoc ftems ftepar- the of mass the of choice the that argue might One h rtadms aua a otetti N-body this treat to way natural most and first The 10 e ntrso h aaeeso h N-body the of parameters the of terms in ted 30 oeocnetr.Acrigt h Calogero the to According conjecture. logero Kg M steslrms.Teprilsi h N-body the in particles The mass. solar the is N hudb uhsalrta h asof mass the than smaller much be should ii.Ti euti ae nthe on based is result This limit. tna -oysse a be can system N-body wtonian M n hrfr nraigteres- the increasing therefore and , N and M M ≃ r h number the are fdr matter dark of 10 9 M otx of context M ⊙ where , oe rmthe from comes N .eA. P.le , ∼ M rk ⊙ 10 son 10 ≃ 2 Fabio Briscese: Title Suppressed Due to Excessive Length the computational effort to solve the N-body dynamics ScM is based on the hypothesis that, in the Newto- numerically. nian limit, it is possible to describe the evolution of DM 9 Thus, although the choice M 10 M⊙ in the N- by means of a wave function ψ, such that the DM density ≃ is given by ρ = M ψ 2, where M represents the effec- body simulations is fit for purpose, it is still arbitrary. DM | | However, an indication on the plausibility of this value tive mass of DM particles. The wave function ψ obeys the comes from scalar field DM models [16]. In fact, a mas- coupled Schr¨odinger-Poisson equations (SPEs) sive and non-interacting scalar field with lagrangian den- ~2 µ 2 i~∂ ψ = ∆ψ + MV ψ , sity L = ∂µΦ∂ Φ mφ Φ forms bound DM halos [17] t − 2a(t)M (the generalization− to self-interacting| | scalar fields with (1) quartic potential has been studied in [18]). In the case ∆V =4πG ρ , of spherically symmetric scalar field configurations Φ = where V is the Newtonian potential, ρ is the energy den- exp [ im t]σ(r)/√2, the size of the halo is given by l φ sity of the universe, a(t) is a scale factor introduced to − ~ ∼ MP /σ(0) ( /mφc). From such relation it is quite evi- take into account the expansion of the universe, and ~ is a dentp that, even if M /σ(0) 1 so that l is much bigger P ≫ parameter representing an effective Planck constant. The than the Compton wavelength of DM particles, halos of Newtonian potential V is determined through the Pois- size above & 10kpc are formed only for ultralight DM par- son equation in (1). The form of the DM density used ticles. The typical orbiting velocity in the halo is v /c o ∼ to run cosmological simulations in [24,25,26,27,28,29,30, σ(0)MP , and using l 10 kpc and vo 100 km/s for 31,32,33,34,35,36,37,38,39] is ρ = (M ψ 2 ρ )/a(t), p ∼ ∼ −6 crit low luminosity spiral galaxies, one has σ(0)/MP 10 , | | − −23 ∼ where ρcrit is a parameter representing a comoving criti- while m 10 eV [17]. We stress that this estimate of cal density of the universe, although some authors assume m coincides∼ (in order of magnitude) with that obtained in φ ρcrit = 0, so that the SPEs reduce to the Schr¨odinger- [36,37,38,39]. Furthermore, the mass M of the halo in this Newton equations [40]. However, since we are not inter- 3 2 9 simple model is given by M σ(0)M /m 10 M⊙ ested in discussing the explicit form of the DM density, we ∼ q P φ ∼ − will use the generic expression ρ. Furthermore, here we are [17], indeed m 10 23 is an upper bound for the DM only focused on the relation between the ScM and N-body particle mass yielding∼ a lower bound for the masses of ha- simulations, and this is not related to the expansion of the los that can be realized. We mention that scalar field dark universe; thus hereafter we set a(t) = 1. matter might be useful to resolve potential small scale The SPEs can be viewed as the non-relativistic limit of problems of CDM, see [19] for an exhaustive discussion of the Klein-Gordon and Dirac equations, and their theoret- this issue. ical justification follows from the correspondence princi- Due to the huge numerical effort to solve the N-body ple that relates classical and quantum mechanical phase- problem in realistic situations when N 1010, it would be ∼ space-distribution functions in the semiclassical limit [42]. desirable to have a simple analytical model from which it is In this case ~ coincides with the Planck constant, and M possible to extract the most important physical properties is the mass of the elementary DM particles. Numerical so- of this N-body system. An alternative is to describe the N- lutions of the SPEs point towards ultralight DM particles body dynamics statistically, by means of the phase-space of mass M 10−23 eV [36,37,38]. distribution of the bodies f(t,x,p), where the evolution of Alternatively,∼ the ScM has been introduced as a nu- f(t,x,p) is given by the Boltzmann equation. In the case merical double of the N-body description of DM at large of LSS, N is large and collisions are suppressed; moreover, scales; see e.g. [24,25,35] and references therein. In this the dynamics is only affected by the Newtonian potential case, the effective particles in the N-body system are again [20], so that the Boltzmann equation reduces to a Vlasov huge agglomerates of DM elementary particles, and there- (or collisionless Boltzmann) equation [21]. Although this fore M is huge in comparison with the mass of elementary model is simple from a conceptual point of view, there is DM particles. Moreover, the constant ~ in (1) does not no general solution of the Vlasov equation. However, the coincide with the Planck constant, nor is fixed by the N- relevant physical information can be extracted from the body problem, but it is merely a free parameter that can (n) momenta M of the distribution function. To do so, one be chosen at will. Furthermore, due to the correspondence should solve the infinitely coupled hierarchy of equations principle [42], ~ determines the phase-space resolution in (n) for the momenta M , and it turns out that the only the ScM. To ensure the match with N-body simulations coherent way to neglect higher cumulants is to neglect one must require ~/M 10−4Mpc c, so that ~ is huge in them entirely [22]; but in this case one reduces to the dust comparison with the true∼ Planck constant.· model, which we know to be inappropriate to describe It is worth to emphasize the difficulty in reconciliat- halo formation, while giving a good description of LSS at ing such huge values of M and ~ with the derivation of larger scales. the SPEs from the fundamental quantum mechanical evo- We can assume that, for our purposes, it is sufficient lution of DM particles. In fact, even though one might to study the evolution of smoothed density and velocity assume that M represents the mass of huge agglomerates fields [23]. A possibility is to use the so called Schr¨odinger of DM particles, one encounters the insurmountable prob- method (ScM), which has been proposed as numerical lem of explaining the extremely large value of ~. In fact, technique to describe the dynamics of CDM [24,25,26, since the Planck constant is fundamental, its value should 27,28,29,30,31,32,33,34,35,36,37,38,39]. not be affected by the Newtonian and semiclassical limits. Fabio Briscese: Title Suppressed Due to Excessive Length 3

In this paper we discuss the relation between the ScM references therein); but we want to exploit the result of and the N-body description of DM used in numerical simu- Nelson in the context of the N-body dynamics. lations [11,12,13,14,15], and we argue how the SPEs can At this point, one can ask the question of the na- be obtained as the large N limit of the Newtonian N- ture of the random field B(t) responsible for the emergent body system. This argument is valid beyond the context quantization. One of the most studied possibilities is that of LSS formation, and it implies that any Newtonian N- B(t) is the random zero-point radiation field of the elec- body system of identical bodies can be described by means tromagnetic field [44]. Another possibility, conjectured by of the SPEs, and that makes this finding of wide interest. Calogero [45], is that the random noise B(t) is the resul- For completeness, we mention that the correspondence be- tant of the gravitational interaction of the particle with tween the ScM method and the Vlasov equation has been all the other particles of the universe. In fact, apart from extensively studied, and we remand the interested reader the interaction with neighboring bodies which is not small to the existing literature; see [35] and references therein. and must be included in the potential φ, the gravitational Our staring point is the Newtonian N-body system interaction with far bodies behaves as a small background of DM agglomerates considered in [11,12,13,14,15]. We noise, and its random behavior comes from the fact that show that the dynamics of this system satisfies the hy- the classical dynamics of a N-body system is chaotic (see pothesis of the so called Nelson stochastic quantization for instance [46] for a review of classical chaos). In the con- [43] in the large N limit. That implies that the evolution text of cosmological simulations, it has been shown that of the system can be described statistically, by means of chaos appears at scales smaller than a critical transition the Schr¨odinger equation. The stochastic background re- scale 3.5Mpc/h, where h is the dimensionless Hubble sponsible for the Nelson quantization is given, as in the parameter,∼ while the dynamics appears to be nonsensitive Calogero conjecture, by the gravitational interaction be- to initial conditions (thus non-chaotic) at larger scales; see tween the N bodies, and its stochastic character is due [47] for more details. to the chaotic behavior of the N-body dynamics. What is Therefore, any particle of the N-body system experi- more, the Calogero conjecture also allows to estimate the ences a stochastic gravitational acceleration due to the rest order of magnitude of the effective Planck constant. of the system. Exploiting this idea, Calogero has shown To begin, let us discuss briefly the hypothesis of the that the order of magnitude of the induced Planck con- Nelson stochastic quantization [44], and let us consider a stant is [45] particle of mass M which moves according to the New- ton laws of motion. The further assumption is that this ~ M 5/3G1/2(N/<ρ>)1/6 , (4) particle constantly undergoes a Brownian motion with no ∼ friction, and with a diffusion coefficient ~/M inversely pro- where N and M are the number and the mass of the bod- portional to its mass M. Therefore, the trajectory of this ies, <ρ> is the average density of the system, and G the particle will be given by gravitational constant. Let us briefly describe how (4) can be obtained on the basis of semiquantitative arguments. The relevant quan- M x¨ = ∇φ + B(t) (2) − tities in our analysis are the dimensional parameters G, where ∇φ represents all the conservative forces, and M, and <ρ>; plus N, which of course is dimensionless. B(t) is− a random variable with zero mean, representing From these quantities we can define the unit of time T as a small random noise. Nelson has shown [43] that, un- −1/2 der these hypotheses, the motion of the particle can be T (G<ρ>) . (5) ∼ described by means of a stochastic process, and the prob- We want to estimate the characteristic time τ of the stochas- ability distribution f(x) of the particle can be expressed tic acceleration that each particle of the system undergoes as f(x) = ψ 2, in terms of a wave function ψ satisfying | | due to all the other particles. Since the N-body dynamics the Schr¨oedinger equation is chaotic, it is plausible that the characteristic frequency

2 of this motion ν 2π/τ should be a growing function of ~ ~ (3) ∼ i ∂tψ = 2M ∆ψ + φ ψ . N, and since the background gravitational noise is due to − a collective stochastic effect, it is also plausible to assume Therefore, in this picture the quantum behavior of the that ν is proportional to the square root of N, so that dynamics of the particle is not a fundamental property of the nature, but it is induced by the random field B(t). τ N −1/2 T. (6) It is necessary to note that the Nelson quantization only ∼ implies the emergence of the Schr¨oedinger equation (3), The ”quantum” of action (that is, the characteristic ac- which of course, does not encompass all the features of tion) associated with the stochastic gravitational noise is quantum mechanics. For instance, all the properties re- obtained multiplying τ by the gravitational energy per lated to the measurement processes in quantum mechan- particle ǫ, which is estimated as ics, e.g. entanglement, have not been derived in the con- text of Nelson quantization. However, we are not inter- ǫ G(NM)2R−1N −1 GM 5/3N 2/3 <ρ>−1/3 , (7) ested in discussing the validity of the Nelson quantization ∼ ∼ as a real theory of quantum world, and we refer the reader where the length R (NM/ < ρ >)1/3 represents the interested in this problem to the literature (see [44] and average linear size of≡ the N-body system. Therefore, the 4 Fabio Briscese: Title Suppressed Due to Excessive Length effective Planck constant is obtained as ~ ǫ τ, which fi- M 2/3G1/2(N/<ρ>)1/6 obtained from (4). For instance, nally gives Eq. (4). At that point we should≡ emphasize that in the case of the Millennium simulation [11], where the this argument is not a pure dimensional analysis since, N-body problem is solved for N 1010 particles of mass 9 2 ≃ −26 3 even though the exponents of the dimensional quantities M 10 M⊙, using <ρ> 3H0 /8πG 4 10 Kg/m , ≃ −1 ≃ ≃ × in (4) are fixed by their dimensions, the dependence on the where H0 h 100 km/sMpc, with h 0.73, is the dimensionless quantity N is fixed by the assumption made Hubble constant,≃ × one has× ~ 2 1066Kgm2≃/s. This cor- in (6), which plays a fundamental role in the derivation of responds to a value ~/M 10≃−4×Mpc c in the range of val- (4). We stress that (6) can be justified in a more rigorous ues used in numerical simulations,≃ e.g.· ~/M 10−4Mpc c way, and we refer the reader to [45], where this relation in [35,37] or ~/M 10−6Mpc c in [36,38].∼ · has been derived through a more detailed analysis of the In conclusion,∼ in this paper· it has been shown that properties of the Newtonian N-body system. the dynamics of the classical Newtonian N-body system Let us come back to our gravitational N-body system. is well described in terms of the SPEs in the large N limit. Using the Calogero conjecture, we have argued that, due This is due to the stochastic quantization of the N-body to the classical gravitational interaction with all the other system induced by the random gravitational background particles, any particle in the system undergoes a stochastic produced by the N bodies, as in the Calogero conjecture. gravitational noise which plays the role of the stochastic Moreover, the emerging effective Planck constant in the random noise B(t) in (2). Thus, the dynamics of each par- SPEs can be computed by means of (4) in terms of the ticle of the system is given in terms of a wavefunction ψ parameters of the corresponding N-body system. solution of (3), where ~ is given by (4). At that point, we When applied to LSS formation, this finding gives a can express the wavefunction of the entire system using further argument in support of the validity of the Schr¨odinger the Hartree-Fock approximation, so that the number den- method as numerical double of the N-body simulations of sity of the N-body system will be n(x) = N ψ 2. This is DM dynamics at large cosmological scales [11,12,13,14, the analog of the derivation of the famous Gross-Pitaevskii| | 15], and offers a natural justification for the huge value ~ equation [50] for a Bose-Einstein condensate by means of of often used in numerical solutions of SPEs. These re- the Hartree-Fock approximation, see [49] for a detailed sults are particularly remarkable, since this derivation of analysis of the quantum many-body system of bosons. Fi- SPEs in the context of the Schr¨odinger method is the first nally, the potential in (3) is φ = M V , where V is the practical application of the Nelson quantization and of the gravitational potential solution of the Poisson equation Calogero conjecture to a realistic physical problem. ∆V =4πGρ, where ρ = Mn(x). During the proofreading of this manuscript, the author has noticed a paper [51], where it has been presented a Of course, this analysis is accurate only for N large, generalized Schr¨odinger equation derived from the theory and therefore we conclude the dynamics of the N-body of scale relativity, and its application to the problem of system is well described by the SPEs (1) in the large N dark matter halos formation has been discussed. Due to limit. Finally, we stress that the resolution of the SPEs in the links to the results presented in this manuscript, such the phase-space is fixed by ~, which in turn is fixed by the paper has been included in the literature corresponding N-body problem. However, for given values Acknowledgments: the author is very grateful to F. of <ρ> and N, this resolution is improved decreasing M, Calogero for useful discussions on the Calogero conjecture, which is also true for the corresponding N-body system. and to G. Rigopopulos and I. Moss for useful discussions We can now exploit this result in the context of the on the draft version of this paper. LSS formation. In fact, if we come back to the descrip- tion of the CDM dynamics at large scales as a Newtonian 10 system of N 10 bodies (which in our case represent References huge aggregations∼ of DM particles) of mass M, which in- teract only gravitationally, we immediately realize that 1. V. 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