The dark universe as the ground state of our quasi-classical quantum cosmology

: 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 interpret dark matter in terms of a pure vibration 8 energy due to positive curvature and held by quarks and/or by a gluon Bose Einstein condensate 9 accompanying baryonic matter at the hadronization transition from the quark gluon plasma phase 10 to the colorless hadronic phase. Such an interpretation, partially based on mass formulae in terms 11 of energy and spin in de Sitter and Anti de Sitter respectively, would comfort the idea that, apart 12 from the violation of the matter/antimatter symmetry satisfying the Sakharov’s conditions, the 13 reconciliation of particle physics and cosmology does not need the recourse to any ad hoc ﬁelds, 14 particles or hidden variables.


Introduction
The new cosmological standard model model involves two dark components of the 19 universe, dark energy and dark matter. Whereas dark energy is commonly associated 20 with the cosmological constant, both of us, Gilles Cohen-Tannoudji [1] and [2], and 21 Jean-Pierre Gazeau [3] have independently tried to address the challenging issue of the 22 dark matter component in the cosmological energy density. 23 The approach of GC-T [1] and [2] aimed at interpreting dark matter as a component 24 of the cosmological energy density, which, together with dark energy, would constitute 25 the world matter, namely what, according to de Sitter, must be added to the visible matter 26 in order, for a cosmological theory to obey the principle of the relativity of inertia. On 27 the other hand, the interpretation by J-PG in [3] in terms of a pure vibration energy due 28 to positive curvature was partially based on mass formulae in terms of energy and spin 29 in de Sitter and/or Anti de Sitter spacetimes, that are established in the quantum context 30 with the reasonable assumption that the proper mass of an elementary system (in the 31 Wigner [4,5] sense) is independent of the space-time metric. 32 In the present paper, we intend to merge our two approaches in a less conjectural 33 work, each of them possibly filling the gaps of the other's. 34 In Section 2, we review the history of the new cosmological standard model, known 35 as ΛCDM from the Einstein-de Sitter debate at the onset of modern cosmology to its 36 cosmological gedanken experiment and its qualitative results 48 2.1. The Einstein de Sitter debate 49 Our common starting point is the history of the debate that was raised between 50 Einstein [6] and de Sitter [7], at the onset of modern cosmology. This debate was about a 51 critical cosmological gedanken experiment, the one which would consist of "removing 52 all the visible matter from the universe" in order to decide whether or not an isolated attempts, leading to a sort of repulsive force (negative pressure) preventing the universe 69 from collapsing under its own gravitation, and which de Sitter assimilates to a world 70 matter insuring the validity of the postulate of the relativity of inertia; and the third one, 71 the 'system B' according to de Sitter consists of a universe that is empty except for the 72 cosmological term. About these last two models, de Sitter summarizes the debate in a 73 postscript added to Ref. [7] and quoted here: 74 Prof. Einstein, to whom I had communicated the principal contents of this paper, 75 writes 'to my opinion, that it would be possible to think of a universe without matter is 76 unsatisfactory. On the contrary the field g µν must be determined by matter, with- be invariant at infinity. This postulate, which, as has already been pointed out above, 85 a) The proper mass is predicted by special relativity if we adopt Wigner point of view of elementary system [4,5] for the validity of the foundational principle of the relativity of inertia.

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• The replacement of the big bang singularity which prevented any causal description 102 of the early universe by an inflation mechanism that remains conjectural but can 103 explain quantitatively the primordial fluctuations observed in the CMB.

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• The discovery of two non-visible components of the cosmological energy density, 105 which together amount to about 95% of the full content of the universe, the dark 106 energy that is commonly associated with the cosmological constant, and the dark 107 matter which raises the theoretical questions some of which are addressed in the 108 present paper. 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. These quantities obey the Friedmann-Lemaître equations of a perfect fluid with which is phenomenologically modeled the material content of the universe.
and a third equation expressing energy conservation, In these equations, the density ρ and isotropic pressure P express the stress energy momentum of the perfect fluid:
160 dS space-time is conveniently represented by the one-sheeted hyperboloid embed- AdS is represented by the one-sheeted hyperboloid embedded in R 5 equipped with 164 the metric: . The Lie algebras of groups dS and AdS group are generated 166 by the ten Killing vectors There exists a crucial difference between dS and AdS with regard to the question  In a given unitary irreducible representation (UIR) of dS or AdS group their generators map to self-adjoint operators in Hilbert spaces of spinor-tensor valued fields on dS: • For de Sitter, 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 [11]: This definition should be understood through the contraction limit of representations: More precisely, with we have This result was proved in [12] and discussed in [13]. One should notice the possible Concerning AdS a mass formula similar to (10) has been given in [9,14]: 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 192 should be independent of the geometry of space-time, or equivalently there should not 193 exist any difference between inertial and gravitational mass.

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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 := |Λ AdS | 3 195 sense is a deformation of both a relativistic free particle with rest energy mc 2 and a 3d 196 isotropic quantum harmonic oscillator with ground state energy 3 2h ω AdS . A complete 197 proof of this feature in the 1 + 1 AdS case is given in [15]. 198 We do not find such a limpid result with dS. Nevertheless let us formally define

c. Hence an AdS elementary system in the Wigner
which can assume any real value. The counterpart of (17) reads: There is a noticeable simplification for spin s = 1/2 : for AdS: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 14 May 2021 doi:10.20944/preprints202105.0320.v1 The choice E rest dS = mc 2 should be privileged for obvious reasons. Moreover, in the massless case and spin s, we have 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

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(21) stands for the ground state energy of a 3d isotropic quantum harmonic oscillator 200 whose excited states apart from degeneracy are spaced at equal energy intervals of 201h ω AdS . A similar interpretation holds for Equation (24).     It turns out that this "holographic" relation can be extended to the whole region between 246 and ψ in which pressure-less matter dominates over radiation, in such a way that 247 holography is at work in the full expansion region from β to ψ.
Our interpretation of this sum rule involves the dilaton, the covariant (comoving)   non-conservation (leptogenesis) (see for instance [17,18]). Now, through the so-called

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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 [20] and the work of Adler [21] such a contribution can be rigorously evaluated (or at least estimated) [22,23]. 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 M(g, µ) = µ exp(−1/b 0 g 2 ) 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 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 14 May 2021 doi:10.20944/preprints202105.0320.v1 In the following quote from [2] it was argued 328 The minus sign in the right hand side shows that when the constant b 0 = 11N c − 2N f 329 is positive, all the QCD condensates contribute negatively to the energy density, which 330 means that the QCD world-matter is globally an anti-de Sitter world-matter (domi-331 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 333 boson and fermion loops contribute in quantum field theory with opposite signs, see 343 Figure 3. A "tadpole" diagram in which a boson (resp. fermion) exchanges a virtual dilaton with a vacuum loop involving a particle identical to it, is transformed trough the interchange of identical particles, into a positive, i.e. increasing the mass (resp. negative, i.e. increasing the energy) self-energy diagram. Now, since the transition from the Quark Gluon Plasma (QGP) to the colorless 344 hadronic phase occurs in the region of expansion in which we use the methodology of 345 effective theories, we assimilate the full content of the universe at point δ to an "effective 346 dark universe" for which the radius of the universe is equal to the Hubble radius. This 347 means that we have (by thought) sent the baryonic matter at the Hubble horizon namely 348 made of its energy density a de Sitter world matter. This is the key point of [2]: the term 349 in Eq. (28) proportional to N f is a de Sitter world matter that represents, at point δ, the 350 kinetic energy of the quarks, called "valence quarks" that constitute the baryonic matter,  what is seen in Figure 1. instance [26,27] and the recent comprehensive historical account [28]. Measurements

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Returning to our approach to elementary systems in dS or AdS space-times, Equa-392 tion (21) tells us that the energy at rest of a fermion in an AdS background decomposes 393 into a "visible" mass part, like in Minkowski, and a "dark" part which is like the 394 ground state energy of a quantum three-dimensional isotropic harmonic oscillator with 395 frequency equal to |Λ AdS | 3 c. This feature led one of us in [3] to infer that at the 396 point δ , i.e., at the hadronization phase transition, "chemical freeze-out" temperature 397 T c f 1.8 × 10 12 K, the ratio "dark/visible" r := 3 2h As explained in [29], it has been shown, thanks to the physics of ultra-cold c) atoms, 416 that Bose Einstein condensation can occur in non-condensed matter but also in gas, and 417 that this phenomenon is not linked to interactions but rather to the correlations implied by 418 quantum statistics.

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Recently, the interpretation of dark matter in terms of a Bose-Einstein condensate 420 as drown interest in the framework of the so called "fuzzy dark matter" (FDM) model 421 [30]. Originated from the idea that the dark matter particle is an ultralight particle, the For testing the ability of such a model to account for observations, it was necessary to 427 use simulations. Such high precision simulations were done in [32] and they confirm 428 the expected order of magnitude of 10 −22 eV for the mass of the dark matter particle.

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Whereas such small masses seem highly unrealistic for particles to be considered in 430 particle physics, we think that our scalar covariant quantum dilaton field φ can have a 431 temperature dependent mass of this order of magnitude.

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To conclude the present article, we want to come back to the notion of a "realized"