Cold dark matter: Bose-Einstein condensation of gluons in Anti-de Sitter space time

: In the same way as the realization of some of the famous gedanken experiments 1 imagined by the founding fathers of quantum mechanics has recently led to the current renewal 2 of the interpretation of quantum physics, it seems that the most recent progresses of observational 3 astrophysics can be interpreted as the realization of some cosmological gedanken experiments 4 such as the removal from the universe of the whole visible matter or the cosmic time travel leading 5 to a new cosmological standard model. This standard model involves two dark components 6 of the universe, dark energy and dark matter. Whereas dark energy is usually associated with 7 the cosmological constant, we propose to explain dark matter as a pure QCD effect, namely a 8 gluon Bose Einstein condensate, following the transition from the quark gluon plasma phase to 9 the colorless hadronic phase. Our approach not only allows us to assume a ratio Dark/Visible 10 equal to 11/2 but also provides gluons and (anti-)quarks with an extra mass of vibrational 11 nature. Such an interpretation would comfort the idea that, apart from the violation of the 12 matter/antimatter symmetry satisfying the Sakharov’s conditions, the reconciliation of particle 13 physics and cosmology needs not the recourse to any ad hoc ﬁelds, particles or hidden variables. 14


Introduction
The new cosmological standard model model involves two dark components of the 20 universe, dark energy [1] and dark matter [2].Whereas dark energy is commonly 21 associated with the cosmological constant, both of us, Gilles Cohen-Tannoudji [3] and 22 [4], and Jean-Pierre Gazeau [5] have independently tried to address the challenging issue 23 of the dark matter component in the cosmological energy density.

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The approach of GC-T [3] and [4] aimed at interpreting dark matter as a component 25 of the cosmological energy density, which, together with dark energy, would constitute 26 the world matter, namely what, according to de Sitter, must be added to the visible matter 27 in order, for a cosmological theory to obey the principle of the relativity of inertia.On 28 the other hand, the interpretation by J-PG in [5] in terms of a pure vibration energy due 29 to positive curvature was partially based on mass formulae in terms of energy and spin 30 in de Sitter and/or Anti-de Sitter spacetimes, that are established in the quantum context 31 with the reasonable assumption that the proper mass of an elementary system (in the 32 Wigner [6,7] sense) is independent of the space-time metric.

33
In the present paper, we explain how our two approaches are complementary in 34 proposing the value 11/2 = 5.5 for the ratio Dark/Visible (the observed one is currently 35 estimated to be 27/5 = 5.2) and interpreting (cold) dark matter as a gluonic Bose the logical impossibility of supposing matter not to exist.I can call this the "material 86 postulate" of the relativity of inertia.This can only be satisfied by choosing the system 87 A, with its world-matter, i.e. by introducing the constant λ, and assigning to the time 88 a separate position amongst the four coordinates.On the other hand, we have the 89 'mathematical postulate' of the relativity of inertia, i.e. the postulate that the g µν shall 90 be invariant at infinity.This postulate, which, as has already been pointed out above, 91 has no physical meaning, makes no mention of matter.It can be satisfied by choosing 95 By revisiting the Einstein-de Sitter debate about the concept of inertia, one can 96 notice a perplexing irresolution concerning the position of that constant λ or Λ in the left-97 hand side (as a fundamental constant) or in the right-hand side (as a phenomenological 98 world matter term) in the Einstein equation [10].This is also the insistent "little music" 99 pervading the content of the present contribution.In an isotropic and homogeneous cosmology, the solution of the Einstein's equation is the Robertson metric depending on the time-dependent radius of the universe R(t) and a curvature index k.The coordinate r is dimensionless; the dimension is carried by R(t), which is the cosmological scale factor which determines proper distances in terms of the comoving coordinates.These quantities obey the Friedmann-Lemaître equations of a perfect fluid with which is phenomenologically modeled the material content of the universe (c = 1).

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and a third equation expressing energy conservation, ρ = −3H(ρ + P) . ( In these equations, the density ρ and isotropic pressure P express the stress energy momentum of the perfect fluid: The cosmological term is taken to the right-hand side of the Einstein's equation and may 123 be interpreted as a contribution to the stress energy tensor that reduces to minus the 124 pressure multiplying the metric g µν (the density and the pressure sum to zero).This in [3] and [4] as the (anti-de Sitter) world matter identified with dark matter.The results of simulation, that can be considered as an algorithmic performance of 150 the final stage of the cosmological gedanken experiment [11], are particularly spectacular.

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The figure 1 can be interpreted as showing the complex topology of the spacetime of the 172 AdS is represented by the one-sheeted hyperboloid embedded in R 5 equipped with 173 the metric: . The Lie algebras of groups dS and AdS group are generated 175 by the ten Killing vectors 176 There exists a crucial difference between dS and AdS with regard to the question 177 of time.While there is no globally time-like Killing vector in dS, there is one in AdS, 178 namely K 50 .This fact has heavy consequences for attempting to properly define "energy 179 at rest" in dS, as is shown below.In a given unitary irreducible representation (UIR) of dS and AdS groups their respective generators map to self-adjoint operators in Hilbert spaces of spinor-tensor valued fields on dS: • For de Sitter, Q • For Anti-de Sitter, Q While the relation between mass and energy in Minkowski is not ambiguous, these notions in de Sitterian/Anti-de Sitterian geometry have to be devised from a flat-limit viewpoint, i.e. from the study of the contraction limit Λ → 0 of these representations.In this respect, a mass formula for dS has been established by Garidi [13]: This definition should be understood through the contraction limit of representations: More precisely, with we have This result was proved in [14] and discussed in [15].One should notice the possible Concerning AdS a mass formula similar to (10) has been given in [10,16]: One here deals with the AdS group representations U AdS (ς AdS , s) with ς AdS ≥ s + 1 (discrete series and its lowest limit), and their contraction limit holds with no ambiguity Now, the contraction formulae ( 12) and ( 14) give us the freedom to write which agrees with the Einstein position that the proper mass of an elementary system 201 should be independent of the geometry of space-time, or equivalently there should not 202 exist any difference between inertial and gravitational mass.

203
Let us now disclose a property of AdS which is essential for our interpretation of dark matter in our universe.Since the invariant ζ AdS is the lowest value of the discrete spectrum of the AdS time generator we define the positive rest energy as It results from Equation ( 13): with frequency ω AdS := c. Hence, to the order of h, an AdS elementary system 204 in the Wigner sense is a deformation of both a relativistic free particle with rest energy 205 mc 2 and a 3d isotropic quantum harmonic oscillator with ground state energy 3 2 hω AdS .

206
A complete proof of this feature in the 1 + 1 AdS case is given in [17].

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We do not find such a limpid result with dS.Nevertheless let us formally define which can assume any real value.The counterpart of (17) reads: There is a noticeable simplification in both cases for spin s = 1/2 : for dS : for AdS: The choice E rest dS = mc 2 should be privileged for obvious reasons.Moreover, in the massless case and spin s, we have for dS : for AdS: Therefore, while for dS the energy at rest makes sense only for massless fermionic systems and is just zero, on the contrary, for AdS the energy at rest makes sense for any spin, and in particular for spin 1 massless bosons we get Save the proper energy mc 2 common to the three relativities, the second term in Equation to call it a de Sitter horizon; and, for reasons that that will appear clearer below we call 238 the past event horizon, an anti-de Sitter horizon.

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The methodology underlying this phenomenological description is the one of the   The cosmic evolution is schematized on the thick line, on which the cosmic time, with Our interpretation is inspired by the seminal work of Brout, Englert and Gunzig [24] 301 which states

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Cosmology, because it is concerned with the variation of g µν within a distribution of 303 matter and not without, is described -at least in the mean -by only that part of g µν 304 which is its determinant that may be represented by a scalar field φ in Minkowski 305 space .

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This leads us to equate in Eq. ( 25) the critical density ρ c to the energy density ρ φ of the dilaton, the covariant (comoving) quantum field φ, determinant of the Friedman, Robertson, Walker (FRW) metric of the effective comoving dark universe.According to our methodology of effective field theory, φ has the equation of state W φ = P φ /ρ φ = −1/3.Thus ρ c = ρ φ = −3P φ and this insures the vanishing of the total active mass of the vacuum, the zero point of energy: with Eq. ( 25) becomes, with all densities and pressures (including ρ DE ) being rescaled at time τ, ρ This means that the dark matter density is what must be added to the visible matter

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If one wants to match the standard models of cosmology and particle physics, one has to move on the thick line of Figure 2, either "bottom up", ψ to point δ that marks the transition from the QCD quark gluon plasma to the colorless hadronic phase, or, "top down" from point β and through point γ, the electroweak symmetry breaking per the BEH mechanism, to point δ that represents the low energy frontiers of the standard model of particle physics and the high energy frontiers of the one of quantum cosmology.Our idea is thus to interpret ρ ind AdS as a comoving density that, when evaluated at point δ, would plays the role of the anti-desitterian world matter, induced by QCD, to cancel the contribution of the comoving CC at point δ.Now, it turns out that following an idea of Sakharov [29] and the work of Adler [30] such a contribution can be rigorously evaluated (or at least estimated) [31,32].The idea of Sakharov was that the non-renormalizable Einstein-Hilbert action would be an effective theory resulting from the coupling of a renormalizable gauge theory to a renormalizable gravitational theory quadratic in the curvature.The aim of Adler was to use the methodology of effective theories to evaluate the cosmological term induced by integrating out, in the effective action, the quantum fields of the standard model: where T(0), the trace anomaly, can be evaluated in terms of the flat space time vacuum expectations of renormalized products of gauge and matter fields (called condensates).
In QCD, these condensates involve a mass scale parameter where µ is the renormalization scale and b 0 = 11N c − 2N f where N c is the number of colors and N f the number of quark flavors, that plays, in QCD, the same role as the scale parameter a(θ) where θ is the temperature.The mass scale parameter presents an essential singularity at g 2 = 0, so the induced cosmological term cannot be evaluated perturbatively.Anyhow, if one can use some non-perturbative technique such as the lattice gauge quantization, one can expect all the condensates contributing to the trace anomaly to be proportional, with a negative factor, to the constant b 0 .For instance, the contribution of the gluon pairing amplitude to the trace anomaly reads In the following quote from [4] it was argued:

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The minus sign in the right hand side shows that when the constant b 0 = 11N c − 2N f 367 is positive, all the QCD condensates contribute negatively to the energy density, which 368 means that the QCD world-matter is globally an anti-de Sitter world-matter (domi-369 nance of an anti-de Sitter world-matter over a smaller de Sitter world-matter).

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The multiplicative factor b 0 allows reading, thanks to the well-known property that N f , which is not the number of quark flavors, but rather the number of fermions that 392 constitute a nucleon, is also equal to 3. Eq. ( 30) thus allows us to conjecture the value of 393 the ratio Dark/Visible to be equal to 11/2, at point δ but also today since pure numbers 394 need not to be rescaled.This the main outcome of [4] that we make ours and that, to our 395 knowledge has never been made elsewhere.402 Indeed, the baryon mass is of order N, which can be written as respectively.In Reference [5] it was suggested that dark matter originated from this 448 Now, it is tempting to establish a parallel between dark matter and CMB, since the latter is viewed as the emergence of the photon decoupling, precisely when photons started to travel freely through space rather than constantly being scattered by electrons and protons in plasma.Hence, one may assert that a (considerable) part of the gluonic component of the quark epoch freely subsists after hadronization within an effective AdS environment.As an assembly of N G non-interacting entities with individual energies E n = (n + 2)hω AdS and degeneracy g n = (n + 1)(n + 3) [18], those remnant gluons are assumed to form a grand canonical Bose-Einstein ensemble whose the chemical potential µ is fixed by the requirement that the sum over all occupation probabilities at temperature T yields Since this number is very large one expects that this Bose-Einstein gas condensates at temperature 1/3 (32) to become the currently observed dark matter.The above formula involving the value

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the system B without a world-matter, and with complete relativity of the time.But here 93 also we need the constant λ.The introduction of this constant can only be avoided 94 by abandoning the postulate of the relativity of inertia altogether [underlined by us].

100 2 . 2 .•••121 3 . 1 .
The performance of the gedanken experiment with ΛCDM 101 The more and more precise measurements of the cosmic microwave background 102 (CMB) radiation by the COBE, WMAP, and Planck experiments allowed the performance 103 of the above mentioned gedanken experiment leading to the assets of ΛCDM, the new 104 standard model of cosmology, namely: 105 The rediscovery of the cosmological constant that, as mentioned above, is essential 106 for the validity of the foundational principle of the relativity of inertia.107 The replacement of the big bang singularity which prevented any causal description 108 of the early universe by an inflation mechanism that remains conjectural but can 109 explain quantitatively the primordial fluctuations observed in the CMB.110 The discovery of two non-visible components of the cosmological energy density, 111 which together amount to about 95% of the full content of the universe, the dark 112 energy that is commonly associated with the cosmological constant, and the dark 113 matter which raises the theoretical questions some of which are addressed in the 114 present paper.1153. A possible kinematics in quantum cosmology: desitterian/anti-desitterian 116 comoving world-matter densities 117 In this section we first remind the Friedmann-Lemaitre model involving density 118 and pressure of the material content of the universe before describing the consequences 119 on the kinematic symmetry of space time if one decides to view pressure as an effective 120 curvature.A reminder about the cosmological formalism 122 (www.preprints.org)| NOT PEER-REVIEWED | Posted: 12 July 2021 doi:10.20944/preprints202105.0320.v3

152 165 •
dark universe: a web of dark filaments that are tensionless dark strings freely moving 153 in a void space (the white regions in the figure) with negative curvature related to 154 the cosmological constant, whereas the spacetime inside the filaments has a positive 155 curvature.156 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 12 July 2021 doi:10.20944/preprints202105.0320.v33.4.dS/AdS quantum elementary systems in Wigner's sense 157 3.4.1.dS and AdS geometries 158 Here we abandon the cosmological conception of Λ as a part (pressure or density) 159 of the right side of the Einstein equation to instead adopt the fundamental constant point 160 of view according to the place of Λ should lie on the left of the Einstein equation, as it 161 was discussed in [10].Minkowski, de Sitter (dS) and anti-de Sitter (AdS) space-times are 162 maximally symmetric.dS and AdS symmetries are one-parameter deformations [12] of 163 Minkowski symmetry.In terms of the cosmological constant Λ we respectively have 164 • dS negative curvature − √ Λ dS /3 = −H/c (H : Hubble parameter), AdS positive curvature |Λ AdS |/3.166 The corresponding kinematical groups are the proper orthochronous Poincaré group 167 R 1,3 SO 0 (1, 3) (or R 1,3 SL(2, C)) the dS SO 0 (1, 4) (or Sp(2, 2)) and AdS SO 0 (2, 3) 168 (or Sp(4, R)) groups.169 dS space-time is conveniently represented by the one-sheeted hyperboloid embed-170 ded in the 5d Minkowski space

[ 5
and spinorial part S αβ acting on the field 183 components.184 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 12 July 2021 doi:10.20944/preprints202105.0320.v3The physically relevant UIR's of the Poincaré, dS and AdS groups are denoted by 185 P > (m, s) (">" for positive energies), U dS (ς dS , s), and U AdS (ς AdS , s), respectively.These 186 UIR's are specified by the spectral values • of their quadratic and quartic Casimir 187 operators.The latter define two invariants, the most basic ones being predicted by the 188 relativity principle, namely proper mass m for Poincaré and ς dS , ς AdS for dS and AdS 189 respectively, and spin s for the three cases (details on their respective ranges are given in 190

192
breaking of dS irreducibility into a direct sum of two Poincaré UIR's with positive and 193 negative energy respectively.To some extent the choice of the factors c < , c > , is left to 194 a "local tangent" observer.In particular one of these factors can be fixed to 1 whilst 195 the other one is forced to vanish.This crucial dS feature originates from the dS group 196 symmetry mapping any point (x 0 , P) ∈ H dS into its mirror image (x 0 , −P) ∈ H dS with 197 respect to the x 0 -axis.Under such a symmetry the four dS generators L a0 , a = 1, 2, 3, 4, 198 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 12 July 2021 doi:10.20944/preprints202105.0320.v3(and particularly L 40 which contracts to energy operator!)transform into their respective 199 opposite −L a0 , whereas the six L ab 's remain unchanged.200

208( 21 ) 3 . 4 . 3 .
stands for the ground state energy of a 3d isotropic quantum harmonic oscillator 209 whose excited states apart from degeneracy are spaced at equal energy intervals of 210 hω AdS .211 Although the situation is more involved in the AdS massless case, for instance 212 for photons or gluons, where Gupta-Bleuler & gauge structures have to be introduced 213 in the description of quantum states, the spectrum of the time generator L 05 is still of 214 the harmonic type, but with possible different degeneracies.Moreover the concept of 215 helicity in AdS has to be reconsidered in terms of conformal symmetry.Details are 216given in[18] [19][20].One important point to notice is that the quantum states in 217 the massless cases ζ AdS = s + 1, are described in holographic terms of vector-valued 218 functions on the three-dimensional Shilov boundary of the Cartan classical domain R IV 219 of the fourth type [21] whereas the massive cases ζ AdS > s + 1 are described in terms 220 of holomorphic functions in R IV .The latter is diffeomorphic to the left group coset 221 Sp(4, R)/K, where K is the maximal compact subgroup S(U(1) × SU(2)).Its Shilov 222 boundary is diffeomorphic to [0, π] × S 2 ("Lie sphere"), or equivalently to the null cone 223 in R 5 equipped with the (+, −, −, −, +) metric.The latter can be viewed as the future 224 horizon of the AdS space time.225 From the elementary quantum context to the quantum cosmological context 226 All what has been done in the present section in the elementary quantum context can 227 be transposed in the cosmological context by coming back to the standard conception 228 considering the cosmological term as a part of the right hand side of the Einstein's 229 equation.The phenomenological description of the matter content of the universe in 230 terms of a perfect fluid characterized by a density and a pressure involves, in a four 231 dimensional Minkowskian spacetime, a thermodynamical interpretation of the Friedman 232 Lemaître differential equations that assimilates the boundaries in the far future (point ω 233 in Figure 2 below, in which the Hubble radius L(a) = H −1 (a) is plotted versus the scale 234 factor a(t) in logarithmic scale) and the remote past (point α in Fig. 2) implied by the 235 Hubble expansion to event horizons with quantum properties [22].The future event horizon 236 occurs in the region where the dark energy (attributed to CC) dominates, which leads us 237

240 4 . 4 . 1 . 4 . 1 . 1 .
effective theories according to which, if there are parameters very large or very small 241 with respect to the quantities of physical interest (with the same dimensions), one 242 can integrate out the very small and/or very large parameters and obtain a simpler, 243 approximate description (said semi-classical) of the phenomena in terms of a family of 244 effective theories depending only on finite but variable effective parameters (said running 245 or comoving): so, in our interpretation, the de Sitter and anti-de Sitter world matter 246 densities are meant to be effective co-moving world matter densities.247 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 12 July 2021 doi:10.20944/preprints202105.0320.v3Matching the standard models of particle physics and cosmology 248 Our interpretation of the assets of ΛCDM 249 From time dependent densities to effective co-moving densities 250 The way how we interpret the assets of ΛCDM in terms the co-moving de Sitter 251 and anti de Sitter world matter densities is illustrated by Figure 2. 252

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that is proportional to the logarithm of the scale factor, is made implicit, which allows 254 somehow to solve the "problem of time in cosmology" by replacing all dimensioned 255 quantities depending on the local time t, by "effective co-moving densities" that are 256 scaled by the scale factor depending on a global time, said thermal [23], τ = 2π a t because 257 it depends on the temperature θ = U a 2π , related to the acceleration a through the 258 Unruh's constant U = h k B c at a power corresponding to their dimensions.259 In particular, this means that the dS and AdS curvatures discussed above in the 260 quantum elementary context have to be replaced, in the quantum cosmological context 261 by effective co-moving curvatures.For instance, the Hubble radius between β and 262 behaving like a 2 must be rescaled by a factor a −1 because it is a length.The boundary of 263 the Hubble 3-sphere is a co-moving horizon.So, the comoving Hubble radius behaves like the 264 comoving radius of the universe, which means, in terms of densities, that the number of 265 degrees of freedom in the bulk equals the number of degrees of freedom on the boundary.266 It turns out that this "holographic" relation can be extended to the whole region between 267 and ψ in which pressure-less matter dominates over radiation, in such a way that 268 holography is at work in the full expansion region from β to ψ.

269 4 . 1 . 2 .
Our interpretation of the flatness sum rule 270 All quantum fluctuations exit from the co-moving horizon in the primordial infla-271 tion phase, enter it in the expansion phase and re-exit it in the late inflation phase.No 272 information-carrying quantum fluctuation with a wavelength smaller than λ − on the 273 left of the past informational, or anti-de Sitter event horizon α this is the reason why we 274 call it an anti-de Sitter horizon), or with a wavelength larger than λ + on the right of the 275 future informational, or de Sitter event horizon ω enters the co-moving horizon.276 The interpretation of the holographic relation, that is at work in the full expansion 277 phase from point β to point ψ is particularly clear at point ψ that marks the transition 278 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 12 July 2021 doi:10.20944/preprints202105.0320.v3from the expansion phase to the re-inflation phase at which the function R(t) presents an 279 inflexion point ( R = 0) which, through the second Friedman equation (4) leads to equate 280 the total bulk energy, or total active mass, with the contribution of the cosmological 281 constant (CC).It is clear, since the pressure associated with Λ is negative, that this cannot 282 be realized without a contribution with a positive pressure, that is with an anti-de Sitter 283 world matter ρ ind AdS (ψ) which exactly cancels at point ψ the contribution of CC.Such an 284 anti-de Sitter world matter can be interpreted as the constant of integration resulting 285 of integrating out the wave lengths smaller than λ − , namely beyond the anti-de Sitter 286 horizon.The cancelation of CC by this anti-de Sitter world matter amounts to replace 287 in our quantum cosmology, the local time t by the global time τ.One could say that 288 considering these two times amounts to a complexification of the time, and that t and τ 289 are complex conjugate variables if the densities in our quantum cosmology are analytic 290 functions depending on the global time τ they do not depend on its complex conjugate, 291 namely the local time.292 More generally, the flatness sum rule that expresses the vanishing of the spatial 293 curvature [4] equates the sum of the visible energy density ρ vis (the baryonic ρ b and 294 radiative ρ R energy densities), the dark energy density and the dark matter energy 295 density, which amounts to nothing but the total active mass in the effective comoving 296 dark universe with a radius equal to its Hubble radius, to the so called critical density 297 ρ c = 3H 2 8πG N .The latter is the energy density at the boundaries in the far past and in 298 the far future of the Hubble horizon in the absence of any integration constant and any 299 spatial curvature.The flatness sum rule reads as: 300

371 boson and fermion loops contribute in quantum field theory with opposite signs, see 372 Figure 3 ,Figure 3 .
Figure 3, the relative contributions of the components of the QCD vacuum to the full

396 4 . 2 . 1 .
Baryons as "chromo-magnetic" monopoles 397The superconductor analogy was used in[33] by Nielsen and Olesen who pro-398 posed a suggestive model of the QCD vacuum involving unconfined chromo-magnetic 399 monopoles moving freely along magnetic flux lines.The interpretation of baryons as 400 color magnetic monopoles b) had been proposed by Ed.Witten[34] in the following 401 quote (where N has to be replaced by N c ):

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supplementary mass granted to (anti-)quarks by the ADS environment.In the present 439 paper we instead explain the existence of dark matter as holding its origin from the 440 gluonic component of the QGP.441 Equation (24) tells us that the energy at rest of a spin 1 massless boson in an AdS 442 background is purely "dark" and is twice the elementary quantum hω AdS .Hence the 443 QGP gluons in the AdS background at the point δ acquire an effective mass 2hω AdS .444 The latter is qualitatively determined through the equipartition k B T c f ≈ hω AdS .Hence, 445 2hω AdS /c 2 = 144 × m u ≈ 317 MeV/c 2 .One should notice that this gluonic effective 446 mass is about 4/3 times the effective mass acquired by quarks and antiquarks in that 447 QGP-AdS environment.

449 ζ( 3 )
≈ 1.2 of the Riemann function is standard for all isotropic harmonic traps (see for 450 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 12 July 2021 doi:10.20944/preprints202105.0320.v3scales, dark matter is rather distributed in the halo of ordinary matter.But we think 502 that this feature cannot be objected as an argument against our approach, because our 503 interpretation of dark matter as a Bose-Einstein condensate, precisely aimed to correct 504 the defects of the conventional cold dark matter model, has been some how comforted, 505 as shown above, by the fuzzy dark matter paradigm.On the other hand, it seems that 506 a very recent work, based on deep learning, has shown the existence of a local web of 507 dark matter at galactic scales [45].

preprints.org) | NOT PEER-REVIEWED | Posted: 12 July 2021 doi:10.20944/preprints202105.0320.v3 must
[5]se at the points where are located the heavy black holes.And this is precisely This feature led one of us in[5]to infer that at the 434 point δ , i.e., at the hadronization phase transition, "chemical freeze-out" temperature 435 T c f 1.8 × 10 12 K, the ratio "dark/visible" r := 3 2 hω AdS mc 2 for light quarks u (mass 436 m u ≈ 2.2 MeV/c 2 ) and d (m d ≈ 4.7 MeV/c 2 ) are given by r(u) ≈ 108 and r(d) ≈ 49 1/(1/N).But 1/N is 403 the "coupling constant" of the strong interactions, which characterizes the interaction 404 among mesons.1/Nplays in QCD roughly the role that α plays in spontaneously 405 broken gauge theories of the weak and electromagnetic interactions.The fact that the 406 baryon mass is of order 1/(1/N) is analogous to the fact that the Polyakov-'t Hooft 407 monopole mass is of order 1/α.408 .415 monopoles have been integrated out, in giant black holes at the center of galaxies or 416 galaxy clusters, we expect that in simulations of the distribution of galaxies, the filaments 417 b) As a special tribute to Georges Lochak (1930-2021), French physicist known for his work on magnetic monopoles.Preprints (www.