# Cold Dark Matter: A Gluonic Bose–Einstein Condensate in Anti-de Sitter Space
Time

Gilles Cohen-Tannoudjia and Jean-Pierre Gazeaub a Laboratoire de recherche sur
les sciences de la matière, LARSIM CEA, Université Paris-Saclay, F-91190
Saint-Aubin, France
<EMAIL_ADDRESS>b Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France
<EMAIL_ADDRESS>
###### Abstract.

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

Keywords: cosmological constant; dark matter; dark energy; de Sitter; Anti-de
Sitter; quark gluon plasma; gluon Bose-Einstein condensate

###### Contents

  1. 1 Introduction
  2. 2 The new standard model of cosmology, the performance of the critical cosmological gedanken experiment and its qualitative results
    1. 2.1 The Einstein de Sitter debate
    2. 2.2 The performance of the gedanken experiment with $\Lambda$CDM
  3. 3 A possible kinematics in quantum cosmology: desitterian/anti-desitterian comoving world-matter densities
    1. 3.1 A reminder about the cosmological formalism ($c=1$)
    2. 3.2 Dark matter as an anti-de Sitter world matter
    3. 3.3 Simulation, a gedanken cosmological experiment algorithmically performed
    4. 3.4 dS/AdS quantum elementary systems in Wigner’s sense
      1. 3.4.1 dS and AdS geometries
      2. 3.4.2 Compared classifications of Poincaré, dS and AdS UIR’s for quantum elementary systems
      3. 3.4.3 From the elementary quantum context to the quantum cosmological context
  4. 4 Matching the standard models of particle physics and cosmology
    1. 4.1 Our interpretation of the assets of $\Lambda$CDM
      1. 4.1.1 From time dependent densities to effective co-moving densities
      2. 4.1.2 Our interpretation of the flatness sum rule
      3. 4.1.3 The primordial inflation and the minimal “beyond the standard model” (BSM) assumption
    2. 4.2 The dark matter induced by QCD
      1. 4.2.1 Baryons as “chromo-magnetic” monopoles
      2. 4.2.2 Magnetic flux lines as dark matter filaments
    3. 4.3 Bose Einstein condensation in the cosmological context
  5. 5 Discussion

##  1\. Introduction

The _new cosmological standard model_ model involves two dark components of
the universe, dark energy [Peebles&Ratra(2003)] and dark matter
[Arbey&Mahmoudi(2021)]. Whereas dark energy is commonly associated with the
cosmological constant, both of us, Gilles Cohen-Tannoudji [Cohen-
Tannoudji(2018)] and [Cohen-Tannoudji(2019)], and Jean-Pierre Gazeau
[Gazeau(2020)] have independently tried to address the challenging issue of
the dark matter component in the cosmological energy density.

The approach of GC-T [Cohen-Tannoudji(2018)] and [Cohen-Tannoudji(2019)] aimed
at interpreting dark matter as a component of the cosmological energy density,
which, together with dark energy, would constitute the _world matter_ , namely
what, according to de Sitter, must be added to the visible matter in order,
for a cosmological theory to obey the principle of the relativity of inertia.
On the other hand, the interpretation by J-PG in [Gazeau(2020)] in terms of a
pure vibration energy due to positive curvature was partially based on mass
formulae in terms of energy and spin in de Sitter and/or Anti-de Sitter
spacetimes, that are established in the quantum context with the reasonable
assumption that the proper mass of an elementary system (in the Wigner
[Wigner(1939), Bargmann$Wigner(1948)] sense) is independent of the space-time
metric.

In the present paper, we explain how our two approaches are complementary in
proposing the value $11/2=5.5$ for the ratio Dark/Visible (the observed one is
currently estimated to be $27/5=5.2$) and interpreting (cold) dark matter as a
gluonic Bose Einstein Condensate (BEC) that is a relic of the quark period.
These proposals were already present in [Cohen-Tannoudji(2019)] but the BEC
interpretation is here given a firmer basis thanks to the quantum features of
elementary systems in Anti-de Sitter spacetime whose the description was given
in [Gazeau(2020)].

In Section 2, we review the history of the new cosmological standard model,
known as $\Lambda$CDM, from the Einstein-de Sitter debate at the onset of
modern cosmology to its current assets. We recall that $\Lambda$ stands for
the cosmological constant and CDM for Cold Dark Matter.

Section 3 is devoted to the key concept of a _(normal- or anti-) de Sitter
comoving world matter density_ with which we intend to fill the gaps between
our two approaches.

In Section 4 we conjecture, as a merging of our two approaches, that the cold
dark matter, identified with the gravitational potential induced by matter, is
affected by a Bose-Einstein condensation mechanism.

Finally, in Section 5, we compare our approach with other schemes that also
assume a Bose-Einstein mechanism, but involving unknown particles.

##  2\. The new standard model of cosmology, the performance of the critical
cosmological gedanken experiment and its qualitative results

###  2.1. The Einstein de Sitter debate

Our common starting point is the history of the debate that was raised between
Einstein [Einstein(1917)] and de Sitter [de Sitter(1917)], at the onset of
modern cosmology. This debate was about a critical cosmological gedanken
experiment, the one which would consist of “removing all the visible matter
from the universe” in order to decide whether or not an isolated particle,
acting as a test body would have inertia. The answer to this question refers
to what is known as the _Mach’s principle_ or what Einstein and de Sitter
named _the principle of the relativity of inertia_. It is summarized in the
following quoted statement by de Sitter [de Sitter(1917)], in which he
introduces the concept of _world matter_ :

> _To the question: If all matter is supposed not to exist, with the exception
> of one material point which is to be used as a test-body, has then this
> test-body inertia or not? The school of Mach requires the answer No. Our
> experience however decidedly gives the answer Yes, if by ‘all matter’ is
> meant all ordinary physical matter: stars, nebulae, clusters, etc. The
> followers of Mach are thus compelled to assume the existence of still more
> matter: the _world-matter_._ a)a)a)The proper mass is predicted by special
> relativity if we adopt Wigner point of view of elementary system
> [Wigner(1939), Bargmann$Wigner(1948)]. Its existence results from the
> symmetry of empty Minkowski, de Sitter, and Anti-de Sitter space-times as
> being one of the two invariants (the other one being the spin) of the
> representations of their respective kinematical groups. This point is
> developed in Subsection 3.4.

The debate of Einstein and de Sitter concerned three possible cosmological
models: the first one was the first Einstein’s closed, static model consisting
of large masses sent at spatial infinity, a model that Einstein abandoned
following the criticism of de Sitter; in the second model which de Sitter
calls the ‘system A’, Einstein re-introduces the ‘cosmological term’,
involving the cosmological constant that he had ignored in previous attempts,
leading to a sort of repulsive force (negative pressure) preventing the
universe from collapsing under its own gravitation, and which de Sitter
assimilates to a world matter insuring the validity of the postulate of the
relativity of inertia; and the third one, the ‘system B’ according to de
Sitter consists of a universe that is empty except for the cosmological term.
About these last two models, de Sitter summarizes the debate in a postscript
added to Ref. [de Sitter(1917)] and quoted here:

> _Prof. Einstein, to whom I had communicated the principal contents of this
> paper, writes ‘to my opinion, that it would be possible to think of a
> universe without matter is unsatisfactory. On the contrary the field
> $g^{\mu\nu}$ _must be determined by matter, without which it cannot exist_
> [underlined by de Sitter]. This is the core of what I mean by the postulate
> of the relativity of inertia’. He therefore postulates what I called above
> the logical impossibility of supposing matter not to exist. I can call this
> the “material postulate” of the relativity of inertia. This can only be
> satisfied by choosing the system A, with its world-matter, i.e. by
> introducing the constant $\lambda$, and assigning to the time a separate
> position amongst the four coordinates. On the other hand, we have the
> ‘mathematical postulate’ of the relativity of inertia, i.e. the postulate
> that the $g^{\mu\nu}$ shall be invariant at infinity. This postulate, which,
> as has already been pointed out above, has no physical meaning, makes no
> mention of matter. It can be satisfied by choosing the system B without a
> world-matter, and with complete relativity of the time. But here also we
> need the constant $\lambda$. The introduction of this constant can only be
> avoided by abandoning the postulate of the relativity of inertia altogether_
> [underlined by us].

By revisiting the Einstein-de Sitter debate about the concept of inertia, one
can notice a perplexing irresolution concerning the position of that constant
$\lambda$ or $\Lambda$ in the left-hand side (as a fundamental constant) or in
the right-hand side (as a phenomenological world matter term) in the Einstein
equation [Gazeau&Novello(2011)]. This is also the insistent “little music”
pervading the content of the present contribution.

###  2.2. The performance of the gedanken experiment with $\Lambda$CDM

The more and more precise measurements of the cosmic microwave background
(CMB) radiation by the COBE, WMAP, and Planck experiments allowed the
performance of the above mentioned gedanken experiment leading to the assets
of $\Lambda$CDM, the new standard model of cosmology, namely:

  * •

The rediscovery of the cosmological constant that, as mentioned above, is
essential for the validity of the foundational principle of the relativity of
inertia.

  * •

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

  * •

The discovery of two non-visible components of the cosmological energy
density, which together amount to about 95% of the full content of the
universe, the _dark energy_ that is commonly associated with the cosmological
constant, and the _dark matter_ which raises the theoretical questions some of
which are addressed in the present paper.

##  3\. A possible kinematics in quantum cosmology: desitterian/anti-
desitterian comoving world-matter densities

In this section we first remind the Friedmann-Lemaitre model involving density
and pressure of the material content of the universe before describing the
consequences on the kinematic symmetry of space time if one decides to view
pressure as an effective curvature.

###  3.1. A reminder about the cosmological formalism ($c=1$)

In an isotropic and homogeneous cosmology, the solution of the Einstein’s
equation

(1) $\mathsf{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathsf{R}=8\pi
G_{N}T_{\mu\nu}+\Lambda g_{\mu\nu}$

is the Robertson metric

(2)
$\mathrm{d}s^{2}=\mathrm{d}t^{2}-R^{2}(t)\left(\frac{\mathrm{d}r^{2}}{1-kr^{2}}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\phi^{2}\right)$

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.

(3)
${{H}^{2}}\equiv{{\left(\frac{{\dot{R}}}{R}\right)}^{2}}=\frac{8\pi{{G}_{N}}\rho}{3}-\frac{k}{{{R}^{2}}}+\frac{\Lambda}{3}$
(4)
$\frac{{\ddot{R}}}{R}=\frac{\Lambda}{3}-\frac{4\pi{{G}_{N}}}{3}\left(\rho+3P\right)\,,$

and a third equation expressing energy conservation,

(5) $\dot{\rho}=-3H\left(\rho+P\right)\,.$

In these equations, the density $\rho$ and isotropic pressure $P$ express the
stress energy momentum of the perfect fluid:

(6)
${{T}_{\mu\nu}}=-P{{g}_{\mu\nu}}+\left(P+\rho\right){{u}_{\mu}}{{u}_{\nu}}\,.$

The cosmological term is taken to the right-hand side of the Einstein’s
equation and may be interpreted as a contribution to the stress energy tensor
that reduces to minus the pressure multiplying the metric $g_{\mu\nu}$ (the
density and the pressure sum to zero). This represents a quantitative
expression of what de Sitter called a world matter in his debate with
Einstein. Let us note that according to the sign of the pressure one can talk
of a de Sitter world matter ($\Lambda$ positive, pressure negative) or an
anti-de Sitter world matter ($\Lambda$ negative, pressure positive).

###  3.2. Dark matter as an anti-de Sitter world matter

In the measurement of the CMB radiation it is crucial to get rid of the light
that is emitted in the foreground in order to obtain the original map of the
CMB radiation. This can be done using known technics, but once this is done,
one is faced with the problem of the gravitational lensing possibly distorting
the path of light between its emission and its arrival at the detector. To
solve this problem, it has been possible to use a technique that has been
already used to get information about the dark matter present in some very
heavy super clusters of galaxies: such a cluster may induce a gravitational
lensing potential distorting (and possibly multiplying) the image of a galaxy
situated far behind the cluster; correlating the distorted observations one
has been able to produce a map of the dark matter present in the cluster or in
its halo that induces the lensing. The success of this technique has been
considered as a proof of the presence of dark matter at extra galactic scales.
Using this technique for the full sky distribution of the CMB with both the
measurements of the temperature and of the polarization of the radiation, the
Planck experiment has been able to yield two outcomes, essential for the
establishment of the cosmological standard model, on the one hand, the
original map of the CMB, not distorted by the lensing, that can be used as the
input data in simulations, and, on the other a full sky map of the
gravitational lensing potential which is tentatively interpreted in [Cohen-
Tannoudji(2018)] and [Cohen-Tannoudji(2019)] as the (anti-de Sitter) world
matter identified with dark matter.

###  3.3. Simulation, a gedanken cosmological experiment algorithmically
performed

The results of simulation, that can be considered as an algorithmic
performance of the final stage of the cosmological gedanken experiment
[Potter&et-al(2017)], are particularly spectacular. The figure 1 can be
interpreted as showing the complex topology of the spacetime of the dark
universe: a web of dark filaments that are tensionless dark strings freely
moving in a void space (the white regions in the figure) with negative
curvature related to the cosmological constant, whereas the spacetime inside
the filaments has a positive curvature.

Figure 1. A section of the largest virtual universe ever simulated. From
[Potter&et-al(2017)]

###  3.4. dS/AdS quantum elementary systems in Wigner’s sense

####  3.4.1. dS and AdS geometries

Here we abandon the cosmological conception of $\Lambda$ as a part (pressure
or density) of the right side of the Einstein equation to instead adopt the
fundamental constant point of view according to the place of $\Lambda$ should
lie on the left of the Einstein equation, as it was discussed in
[Gazeau&Novello(2011)]. Minkowski, de Sitter (dS) and anti-de Sitter (AdS)
space-times are maximally symmetric. dS and AdS symmetries are one-parameter
deformations
[Bacry$Lévy-Leblond(1968)] of Minkowski symmetry. In terms of the cosmological
constant $\Lambda$ we respectively have (in this dS/AdS context $c$ is
restored)

  * •

dS negative curvature $-\sqrt{\Lambda_{\mathrm{dS}}/3}$ ,

  * •

AdS positive curvature $\sqrt{|\Lambda_{\mathrm{AdS}}|/3}$.

The corresponding kinematical groups are the proper orthochronous Poincaré
group $\mathbb{R}^{1,3}\rtimes\,\mathrm{SO}_{\scriptsize 0}(1,3)$ (or
$\mathbb{R}^{1,3}\rtimes\,\mathrm{SL}(2,\mathbb{C})$) the dS
SO${}_{\scriptsize 0}(1,4)$ (or Sp$(2,2)$) and AdS SO${}_{\scriptsize 0}(2,3)$
(or Sp$(4,\mathbb{R})$) groups.

dS space-time is conveniently represented by the one-sheeted hyperboloid
embedded in the $5$d Minkowski space
$\mathsf{H}_{\mathrm{dS}}\equiv\\{x\in\mathbb{R}^{5};~{}x^{2}=\eta_{\alpha\beta}~{}x^{\alpha}x^{\beta}=-\Lambda_{\mathrm{dS}}/3\\},\,\
\alpha,\beta=0,1,2,3,4,$ where $\eta_{\alpha\beta}=$diag$(1,-1,-1,-1,-1)$.

AdS is represented by the one-sheeted hyperboloid embedded in $\mathbb{R}^{5}$
equipped with the metric:
$\mathsf{H}_{\mathrm{AdS}}\equiv\\{x\in\mathbb{R}^{5};~{}x^{2}=\eta_{\alpha\beta}~{}x^{\alpha}x^{\beta}=|\Lambda_{\mathrm{AdS}}|/3\\}\,,\
\alpha,\beta=0,1,2,3,5,$ where $\eta_{\alpha\beta}=$diag$(1,-1,-1,-1,1)$.

The Lie algebras of groups dS and AdS group are generated by the ten Killing
vectors
$\mathrm{K}_{\alpha\beta}=x_{\alpha}\partial_{\beta}-x_{\beta}\partial_{\alpha}$.

There exists a crucial difference between dS and AdS with regard to the
question of time. While there is no globally time-like Killing vector in dS,
there is one in AdS, namely $K_{50}$. This fact has heavy consequences for
attempting to properly define “energy at rest” in dS, as is shown below.

####  3.4.2. Compared classifications of Poincaré, dS and AdS UIR’s for
quantum elementary systems

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 and AdS respectively:

$\mathrm{K}_{\alpha\beta}\mapsto\mathrm{L}_{\alpha\beta}=\mathrm{M}_{\alpha\beta}+\mathrm{S}_{\alpha\beta}\,,$

with orbital part
$\mathrm{M}_{\alpha\beta}=-\mathrm{i}(x_{\alpha}\partial_{\beta}-x_{\beta}\partial_{\alpha})$
and spinorial part $\mathrm{S}_{\alpha\beta}$ acting on the field components.

The physically relevant UIR’s of the Poincaré, dS and AdS groups are denoted
by $\mathcal{P}^{>}(m,s)$ (“$>$” for positive energies),
$U_{\mathrm{dS}}(\varsigma_{{}_{\text{\scriptsize dS}}},s)$, and
$U_{\mathrm{AdS}}(\varsigma_{\text{\scriptsize AdS}},s)$, respectively. These
UIR’s are specified by the spectral values $\langle\cdot\rangle$ of their
quadratic and quartic Casimir operators. The latter define two invariants, the
most basic ones being predicted by the relativity principle, namely proper
mass $m$ for Poincaré and $\varsigma_{{}_{\text{\scriptsize dS}}}$,
$\varsigma_{\text{\scriptsize AdS}}$ for dS and AdS respectively, and spin $s$
for the three cases (details on their respective ranges are given in
[Gazeau(2020)]).

  * •

For Poincaré the Casimir operators are fixed as

(7)
$\begin{split}\mathrm{Q}^{(1)}_{\text{Poincar\'{e}}}&=\mathrm{P}^{\mu}\,\mathrm{P}_{\mu}={\mathrm{P}^{0}}^{2}-\mathbf{P}^{2}=m^{2}\,c^{2}\,,\\\
\mathrm{Q}^{(2)}_{\text{Poincar\'{e}}}&=\mathrm{W}^{\mu}\mathrm{W}_{\mu}=-m^{2}\,c^{2}\,s(s+1)\hbar^{2},\
\mathrm{W}_{\mu}:=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\mathrm{J}^{\nu\rho}\mathrm{P}^{\sigma}\,.\end{split}$

  * •

For de Sitter,

(8)
$\begin{split}\mathrm{Q}_{\mathrm{dS}}^{(1)}&=-\frac{1}{2}\mathrm{L}_{\alpha\beta}\mathrm{L}^{\alpha\beta}=\varsigma_{{}_{\text{\scriptsize
dS}}}^{2}-\left(s-\frac{1}{2}\right)^{2}+2\equiv\langle\mathrm{Q}_{\mathrm{dS}}^{(1)}\rangle\,,\\\
\mathrm{Q}_{\mathrm{dS}}^{(2)}&=-\mathrm{W}_{\alpha}\mathrm{W}^{\alpha}=\left(\varsigma_{{}_{\text{\scriptsize
dS}}}^{2}+\frac{1}{4}\right)s(s+1)\,,\quad\mathrm{W}_{\alpha}:=-\frac{1}{8}\epsilon_{\alpha\beta\gamma\delta\eta}\mathrm{L}^{\beta\gamma}\mathrm{L}^{\delta\eta}\,.\end{split}$

  * •

For Anti-de Sitter,

(9)
$\begin{split}\mathrm{Q}_{\mathrm{AdS}}^{(1)}&=-\frac{1}{2}\mathrm{L}_{\alpha\beta}\mathrm{L}^{\alpha\beta}=\varsigma_{\mathrm{AdS}}(\varsigma_{\mathrm{AdS}}-3)+s(s+1)\equiv\langle\mathrm{Q}_{\mathrm{AdS}}^{(1)}\rangle\,,\\\
\mathrm{Q}_{\mathrm{AdS}}^{(2)}&=-\mathrm{W}_{\alpha}\mathrm{W}^{\alpha}=-(\varsigma_{\mathrm{AdS}}-1)(\varsigma_{\mathrm{AdS}}-2)s(s+1\,,\
\mathrm{W}_{\alpha}:=-\frac{1}{8}\epsilon_{\alpha\beta\gamma\delta\eta}\mathrm{L}^{\beta\gamma}\mathrm{L}^{\delta\eta}\,.\end{split}$

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
$\Lambda\to 0$ of these representations. In this respect, a mass formula for
dS has been established by Garidi [Garidi(2003)]:

(10)
$m^{2}_{\mathrm{dS}}:=\frac{\hbar^{2}\Lambda_{\mathrm{dS}}}{3c^{2}}(\langle
Q_{\mathrm{dS}}^{(1)}\rangle-2)=\frac{\hbar^{2}\Lambda_{\mathrm{dS}}}{3c^{2}}\left(\varsigma^{2}_{\mathrm{dS}}+\left(s-\frac{1}{2}\right)^{2}\right)\,.$

This definition should be understood through the contraction limit of
representations:

$\mathrm{dS}\,\mathrm{UIR}\longrightarrow\mbox{Poincar\'{e}}\,\mathrm{UIR}\,.$

More precisely, with

(11) $\Lambda_{\mathrm{dS}}\to
0\,\quad\varsigma_{\mathrm{dS}}\to\infty\,,\quad\text{while
fixing}\quad\varsigma_{\mathrm{dS}}\hbar\sqrt{\Lambda_{\mathrm{dS}}}/\sqrt{3}c=m_{\text{Poincar\'{e}}}\equiv
m\,.$

we have

(12)
$U_{\mathrm{dS}}(\varsigma_{\mathrm{dS}},s)\underset{|\varsigma_{\mathrm{dS}}|\sqrt{\Lambda_{\mathrm{dS}}}/\sqrt{3}=\frac{mc}{\hbar}}{\underset{\Lambda_{\mathrm{dS}}\to
0\,,\,|\varsigma_{\mathrm{dS}}|\to\infty}{\longrightarrow}}{c_{>}\mathcal{P}}^{>}(m,s)\oplus
c_{<}\mathcal{P}^{<}(m,s)\,.$

This result was proved in [Mickelsson&Niederle(1972)] and discussed in
[Garidi&et-al(2003)]. One should notice the possible breaking of dS
irreducibility into a direct sum of two Poincaré UIR’s with positive and
negative energy respectively. To some extent the choice of the factors
$c_{<},\,c_{>}$, is left to a “local tangent” observer. In particular one of
these factors can be fixed to 1 whilst the other one is forced to vanish. This
crucial dS feature originates from the dS group symmetry mapping any point
$(x^{0},\mathsf{P})\in\mathsf{H}_{\mathrm{dS}}$ into its mirror image
$(x^{0},-\mathsf{P})\in\mathsf{H}_{\mathrm{dS}}$ with respect to the
$x^{0}$-axis. Under such a symmetry the four dS generators $\mathrm{L}_{a0}$,
$a=1,2,3,4$, (and particularly $\mathrm{L}_{40}$ which contracts to energy
operator!) transform into their respective opposite $-\mathrm{L}_{a0}$,
whereas the six $\mathrm{L}_{ab}$’s remain unchanged.

Concerning AdS a mass formula similar to (10) has been given in
[Gazeau&Novello(2008), Gazeau&Novello(2011)]:

(13)
$\begin{split}m^{2}_{\mathrm{AdS}}&=\frac{\hbar^{2}|\Lambda_{\mathrm{AdS}}|}{3c^{2}}\left(\langle
Q^{(1)}_{\mathrm{AdS}}\rangle-\langle
Q^{(1)}_{\mathrm{AdS}}|_{\varsigma_{\mathrm{AdS}}=s+1}\rangle\right)\\\
&=\frac{\hbar^{2}|\Lambda_{\mathrm{AdS}}|}{3c^{2}}\left[\left(\varsigma_{\mathrm{AdS}}-\frac{3}{2}\right)^{2}-\left(s-\frac{1}{2}\right)^{2}\right]\,.\end{split}$

One here deals with the AdS group representations
$U_{\mathrm{AdS}}(\varsigma_{\mathrm{AdS}},s)$ with
$\varsigma_{\mathrm{AdS}}\geq s+1$ (discrete series and its lowest limit), and
their contraction limit holds with no ambiguity

(14)
$U_{\mathrm{AdS}}(\varsigma_{\mathrm{AdS}},s)\underset{\varsigma_{\mathrm{AdS}}\sqrt{|\Lambda_{\mathrm{AdS}}|/3}=\frac{mc}{\hbar}}{\underset{\Lambda_{\mathrm{AdS}}\to
0\,,\,\varsigma_{\mathrm{AdS}}\to\infty}{\longrightarrow}}\mathcal{P}^{>}(m,s)\,.$

Now, the contraction formulae (12) and (14) give us the freedom to write

(15) $m_{\mathrm{dS}}=m_{\mathrm{AdS}}=m\,,$

which agrees with the Einstein position that the proper mass of an elementary
system should be independent of the geometry of space-time, or equivalently
there should not exist any difference between inertial and gravitational mass.

Let us now disclose a property of AdS which is essential for our
interpretation of dark matter in our universe. Since the invariant
$\zeta_{\mathrm{AdS}}$ is the lowest value of the discrete spectrum of the AdS
time generator we define the positive rest energy as

(16) $E_{\mathrm{AdS}}^{\mathrm{rest}}:=\hbar
c\sqrt{\frac{|\Lambda_{\mathrm{AdS}}|}{3}}\,\zeta_{\mathrm{AdS}}\,.$

It results from Equation (13):

(17)
$E_{\mathrm{AdS}}^{\mathrm{rest}}=\left[m^{2}c^{4}+\hbar^{2}\omega^{2}_{\text{\scriptsize
AdS}}\left(s-\frac{1}{2}\right)^{2}\right]^{1/2}+\frac{3}{2}\hbar\omega_{\text{\scriptsize
AdS}}\,,$

with frequency $\omega_{\text{\scriptsize
AdS}}:=\sqrt{\frac{|\Lambda_{\mathrm{AdS}}|}{3}}\,c$. Hence, to the order of
$\hbar$, an AdS elementary system in the Wigner sense is a deformation of both
a relativistic free particle with rest energy $mc^{2}$ and a $3$d isotropic
quantum harmonic oscillator with ground state energy
$\frac{3}{2}\hbar\omega_{\mathrm{AdS}}$ and with excited states which, apart
from degeneracy, are spaced at equal energy intervals of
$\hbar\omega_{\mathrm{AdS}}$. A complete proof of this feature in the $1+1$
AdS case is given in [Gazeau&Renaud(1993)].

We do not find such a limpid result with dS. Nevertheless let us formally
define

(18) $E_{\mathrm{dS}}^{\mathrm{rest}}:=\hbar
c\sqrt{\frac{\Lambda_{\mathrm{dS}}}{3}}\,\zeta_{\mathrm{dS}}\,,$

which can assume any real value. The counterpart of (17) reads:

(19)
$E_{\mathrm{dS}}^{\mathrm{rest}}=\pm\left[m^{2}c^{4}-\hbar^{2}c^{2}\frac{\Lambda_{\mathrm{dS}}}{3}\left(s-\frac{1}{2}\right)^{2}\right]^{1/2}\,.$

There is a noticeable simplification in both cases for spin $s=1/2$ :

(20) for dS : $\displaystyle\quad E_{\mathrm{dS}}^{\mathrm{rest}}=\pm
mc^{2}\,,$ (21) for AdS: $\displaystyle\quad
E_{\mathrm{AdS}}^{\mathrm{rest}}=mc^{2}+\frac{3}{2}\hbar\omega_{\mathrm{AdS}}\,.$

The choice $E_{\mathrm{dS}}^{\mathrm{rest}}=mc^{2}$ should be privileged for
obvious reasons. Moreover, in the massless case and spin $s$, we have

(22) for dS : $\displaystyle\quad
E_{\mathrm{dS}}^{\mathrm{rest}}=\pm\mathrm{i}\hbar\sqrt{\frac{\Lambda_{\mathrm{dS}}}{3}}c\left(s-\frac{1}{2}\right)\,,$
(23) for AdS: $\displaystyle\quad
E_{\mathrm{AdS}}^{\mathrm{rest}}=\hbar\sqrt{\frac{|\Lambda_{\mathrm{AdS}}|}{3}}c(s+1)\,.$

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 and is strictly positive for any spin, and in particular for
spin 1 massless bosons we get

(24) $E_{\mathrm{AdS}}^{\mathrm{rest}}=2\hbar\omega_{\mathrm{AdS}}\,.$

One should add that in the AdS massless cases with $s>0$ Gupta-Bleuler & gauge
structures have to be introduced in the description of quantum states. The
spectrum of the time generator $L_{05}$ is still of the harmonic type, but
there are possible different degeneracies. Moreover the concept of helicity in
AdS has to be reconsidered in terms of conformal symmetry. Details are given
in [Fronsdal(1975)] [Binegar&et-al(1983)] [Gazeau&et-al(1989)]. One important
point to notice is that the quantum states in the massless cases
$\zeta_{\mathrm{AdS}}=s+1$, are described in holographic terms of vector-
valued functions on the three-dimensional Shilov boundary of the Cartan
classical domain $\mathcal{R}_{IV}$ of the fourth type [Hua(1963)] whereas the
massive cases $\zeta_{\mathrm{AdS}}>s+1$ are described in terms of holomorphic
functions in $\mathcal{R}_{IV}$. The latter is diffeomorphic to the left group
coset Sp$(4,\mathbb{R})$/K, where K is the maximal compact subgroup
$\mathrm{S}(\mathrm{U}(1)\times\mathrm{SU}(2))$. Its Shilov boundary is
diffeomorphic to $[0,\pi]\times\mathbb{S}^{2}$ (“Lie sphere”), or equivalently
to the null cone in $\mathbb{R}^{5}$ equipped with the $(+,-,-,-,+)$ metric.
The latter can be viewed as the future horizon of the AdS space time.

The existence of a strictly positive at rest energy for massless systems in
AdS should not misinterpreted. For instance, the energy in (24) might be
viewed as a kind of “mass gap”. However, the gluons, like all AdS elementary
systems for which $\zeta_{\mathrm{AdS}}=s+1$ and so
$\langle\mathrm{Q}_{\mathrm{AdS}}^{(1)}\rangle=2(s^{2}-1)$, are rigorously
massless and propagate on the light cone, as proved for instance in the
appendix of [Flato&etal(1986)]. Even though the ground state energy that they
acquire in that AdS environment may be assimilated to that mass gap or/and
energy gap, they remain conformally coupled, and do not experience any decay.

####  3.4.3. From the elementary quantum context to the quantum cosmological
context

All what has been done in the present section in the elementary quantum
context can be transposed in the cosmological context by coming back to the
standard conception considering the cosmological term as a part of the right
hand side of the Einstein’s equation. The phenomenological description of the
matter content of the universe in terms of a perfect fluid characterized by a
density and a pressure involves, in a four dimensional Minkowskian spacetime,
a thermodynamical interpretation of the Friedman Lemaître differential
equations that assimilates the boundaries in the far future (point $\omega$ in
Figure 2 below, in which the Hubble radius $L(a)={{H}^{-1}}(a)$ is plotted
versus the scale factor $a(t)$ in logarithmic scale) and the remote past
(point $\alpha$ in Fig. 2) implied by the Hubble expansion to _event horizons
with quantum properties_ [Padmanabhan&Patel(2003)]. The future event horizon
occurs in the region where the dark energy (attributed to CC) dominates, which
leads us to call it a _de Sitter horizon_ ; and, for reasons that will appear
clearer below we call the past event horizon, an _anti-de Sitter horizon_.

The methodology underlying this phenomenological description is the one of the
_effective theories_ according to which, if there are parameters very large or
very small with respect to the quantities of physical interest (with the same
dimensions), one can integrate out the very small and/or very large parameters
and obtain a simpler, approximate description (said semi-classical) of the
phenomena in terms of a family of effective theories depending only on finite
but _variable_ effective parameters (said running or comoving): so, in our
interpretation, the de Sitter and anti-de Sitter world matter densities are
meant to be _effective co-moving world matter densities_.

##  4\. Matching the standard models of particle physics and cosmology

###  4.1. Our interpretation of the assets of $\Lambda$CDM

####  4.1.1. From time dependent densities to effective co-moving densities

The way how we interpret the assets of $\Lambda$CDM in terms the co-moving de
Sitter and anti de Sitter world matter densities is illustrated by Figure 2.

Figure 2. Hubble radius $L(a)={{H}^{-1}}(a)$ ($c=1$) is plotted versus the
scale factor $a(t)$ in logarithmic scale (from [Cohen-Tannoudji(2018)]).

The cosmic evolution is schematized on the thick line, on which the cosmic
time, that is proportional to the logarithm of the scale factor, is made
implicit, which allows somehow to solve the “problem of time in cosmology” by
replacing all dimensioned quantities depending on the local time $t$, by
“effective co-moving densities” that are scaled by the scale factor depending
on a global time, denoted $\tau$, which is said _thermal_
[Connes&Rovelli(1994)] because it depends through the Unruh’s constant
$U=\dfrac{h}{k_{B}c}$ on the temperature, at a power corresponding to their
dimensions.

In particular, this means that the dS and AdS curvatures discussed above in
the quantum elementary context have to be replaced, in the quantum
cosmological context by effective co-moving curvatures. For instance, the
Hubble radius between $\beta$ and $\epsilon$ behaving like $a^{2}$ must be
rescaled by a factor $a^{-1}$ because it is a length. The boundary of the
Hubble $3$-sphere is a _co-moving horizon_. So, _the comoving Hubble radius
behaves like the comoving radius of the universe_ , which means, in terms of
densities, that the number of degrees of freedom in the bulk equals the number
of degrees of freedom on the boundary. It turns out that this “holographic”
relation can be extended to the whole region between $\epsilon$ and $\psi$ in
which pressure-less matter dominates over radiation, in such a way that
holography is at work in the full expansion region from $\beta$ to $\psi$.

####  4.1.2. Our interpretation of the flatness sum rule

All quantum fluctuations exit from the co-moving horizon in the primordial
inflation phase, enter it in the expansion phase and re-exit it in the late
inflation phase. No information-carrying quantum fluctuation with a wavelength
smaller than ${{\lambda}^{-}}$ on the left of the _past informational_
$\alpha$ (this is the reason why we called it above an anti-de Sitter
horizon), or with a wavelength larger than ${{\lambda}^{+}}$ on the right of
the _future informational, or de Sitter event horizon_ $\omega$ enters the co-
moving horizon. Note that the scenario of a bouncing phase taking place at the
point $\alpha$ has been proposed in recent works co-authored by one of us (see
the review [Bergeron&etal(2020)] and references therein).

The interpretation of the holographic relation, that is at work in the full
expansion phase from point $\beta$ to point $\psi$ is particularly clear at
point $\psi$ that marks the transition from the expansion phase to the re-
inflation phase at which the function $R(t)$ presents an inflexion point
($\ddot{R}=0$) which, through the second Friedman equation (4) leads to equate
the total bulk energy, or total active mass, with the contribution of the
cosmological constant (CC). It is clear, since the pressure associated with
$\Lambda$ is negative, that this cannot be realized without a contribution
with a positive pressure, that is with an anti-de Sitter world matter
$\rho_{\text{AdS}}^{\text{ind}}(\psi)$ which exactly cancels at point $\psi$
the contribution of CC. Such an anti-de Sitter world matter can be interpreted
as the constant of integration resulting of integrating out the wave lengths
smaller than ${{\lambda}^{-}}$, namely beyond the anti-de Sitter horizon. When
applied in the full expansion phase, the cancellation of CC by this anti-de
Sitter world matter amounts to replace in our quantum cosmology, the local
time $t$ by the global time $\tau$. One could say that considering these two
times amounts to a complexification of the time, and that $t$ and $\tau$ are
complex conjugate variables: if the densities in our quantum cosmology are
analytic functions depending on the global time $\tau$ they do not depend on
its complex conjugate, namely the local time. We shall come back to this idea
of the complexification of time in our contribution to the incoming book
[Lestienne&Harris(2021)] on _Time in Science_.

More generally, the flatness sum rule that expresses the vanishing of the
spatial curvature [Cohen-Tannoudji(2019)] equates the sum of the visible
energy density $\rho_{\text{vis}}$ (the baryonic $\rho_{\text{b}}$ and
radiative $\rho_{\text{R}}$ energy densities), the dark energy density and the
dark matter energy density, which amounts to nothing but the total active mass
in the effective comoving dark universe with a radius equal to its Hubble
radius, to the so called critical density $\rho_{\text{c}}=\dfrac{3H^{2}}{8\pi
G_{\text{N}}}$. The latter is the energy density at the boundaries in the far
past and in the far future of the Hubble horizon in the absence of any
integration constant and any spatial curvature. More precisely, this flatness
sum rule reads as:

(25)
$\rho_{\text{vis}}+\rho_{\mathrm{DM}}+\rho_{\text{DE}}-\rho_{\text{c}}=0\,,$

with

(26)
$\rho_{\text{vis}}=\rho_{\text{b}}+\rho_{\text{R}}\,,\quad\rho_{\text{DE}}=\frac{\Lambda}{8\pi
G_{N}}\,.$

Our interpretation is inspired by the seminal work of Brout, Englert and
Gunzig [Brout&Englert&Gunzig(1978)] which states

> Cosmology, because it is concerned with the variation of $g_{\mu\nu}$ within
> a distribution of matter and not without, is described - at least in the
> mean - by only that part of $g_{\mu\nu}$ which is its determinant that may
> be represented by a scalar field $\phi$ in Minkowski space .

This leads us to equate in Eq. (25) the critical density $\rho_{\text{c}}$ to
the energy density $\rho_{\phi}$ of the so-called dilaton, i.e., the covariant
(comoving) quantum field $\phi$, representing the determinant of the Friedman,
Robertson, Walker (FRW) metric of the effective comoving dark universe.
According to our methodology of effective field theory, $\phi$ has the
equation of state $W_{\phi}=P_{\phi}/\rho_{\phi}=-1/3$. Thus
$\rho_{\text{c}}=\rho_{\phi}=-3P_{\phi}$ and this insures the vanishing of the
total active mass of the vacuum, the zero point of energy: with

(27)
$\rho_{\text{vis}}+\rho_{\mathrm{DM}}\equiv\rho_{\mathrm{M}}=\frac{1}{2}\rho_{\text{DE}}=-P_{\phi}\,,$

Eq. (25) becomes, with all densities and pressures (including
$\rho_{\text{DE}}$) being rescaled at time $\tau$,

(28)
$\rho_{\text{vis}}+\rho_{\mathrm{DM}}+\rho_{\mathrm{DE}}=-3P_{\phi}=\rho_{\phi}\,.$

This means that the dark matter density and the dark energy sum to what must
be added to the visible matter density in order, for the total matter-energy
density, to satisfy the principle of the relativity of inertia
$\rho_{\mathrm{M}}+P_{\phi}=0$, or the vanishing of the spatial curvature
$\rho_{\phi}+3P_{\phi}=0$.

An important remark is in order here: it may look puzzling that, whereas the
equations of states (EoS) of both dark matter and dark energy are equal to
$-1$, the EoS of the effective comoving dark universe is equal to $-1/3$. But,
in [Melia(2017)] F. Melia has solved this puzzle: the EoS of energy densities
(that are not true scalar energy densities) are frame dependent; when the
active mass does not vanish, a comoving frame does not, in general, coincide
with a free fall frame, (that may happen only at one particular position), but
“in FRW, the comoving and free-falling frames are supposed to be identical at
every spacetime point.”

####  4.1.3. The primordial inflation and the minimal “beyond the standard
model” (BSM) assumption

In $\Lambda$CDM, the primordial inflation phase occurs at a Hubble radius of
about $10^{3}$ to $10^{4}$ Planck’s lengths, which is clearly a domain of
physics beyond the standard model (BSM) and supposedly relying on quantum
gravitational effects. It is generally admitted that, although it does not
resolve completely the cosmological singularity problem (the singularity is
sent behind an event horizon) the primordial inflation phase helps correcting
the well-known defects of the “simple big bang model”, namely the absence of
monopoles, the vanishing of the spatial curvature and the particle-horizon
problem. It ends with the emission of particles with a mass of about $2.5$
meV, which could be neutrinos, and which, as is frequently assumed, we
associate with the cosmological constant and thus to the dark energy. The
scale at which it occurs has long been associated with the grand unification
symmetry breaking or supersymmetry breaking, but the fact that it corresponds
to the zero point of the energy strongly suggests that it has rather to be
associated with the matter/antimatter symmetry breaking: the right hand side
of Eq. (27), half of the dark energy density, is nothing but the missing dark
energy, the one of the anti-matter that has been “integrated out”. We remind
that if matter would reduce to visible matter (which is supposed to be
pressure-less) there would be no negative pressure to insure the principle of
the relativity of inertia. It is the addition of the dark matter density that
may insure the validity of this principle. We shall see below that this
feature is an essential key point of our paper.

To link the scales of the primordial inflation, of the cosmological constant,
of the neutrino masses and of the matter/antimatter symmetry breaking is
indeed very appealing. To explore the possible consequences of such an
association, one must discuss the problem of the neutrino masses per the Brout
Englert Higgs (BEH) mechanism. It is well known that the standard model using
the BEH mechanism is compatible with massless neutrinos. In fact, right-handed
neutrinos which would be necessary to generate mass through the Yukawa
coupling of the BEH boson to the right-handed and left-handed neutrinos, have
quantum numbers which make of them _SM sterile_ particles (zero weak isospin,
zero charge and zero weak hypercharge). So, the SM is completely compatible
with massless neutrinos. If neutrinos are massive, as they seem to be, their
mass is thus highly likely a signal of BSM physics. The simplest assumption is
to assume that there do exist sterile right-handed neutrinos. The problem is
that if one gives the neutrinos Dirac masses through the Yukawa couplings to
the BEH boson, one does not understand why these masses are much smaller than
the Dirac masses of the charged fermions. The way out of this difficulty comes
from the fact that the right-handed neutrinos can have a Majorana mass by
their own. If the Majorana mass is exceptionally large, one can manage, with a
“seesaw” mechanism to get, by diagonalizing the mass matrix, a physical
neutrino which would be a linear combination of the normal left-handed
neutrino plus a small anti-right-handed neutrino component. With a Majorana
mass of the order of the primordial inflation scale, one can have neutrino
masses in the milli-eV range in possible agreement with the results of
neutrino oscillations experiments. Furthermore, this mechanism has the
advantage that through their Yukawa couplings, the sterile righthanded
neutrinos can decay into standard model particles (a BEH boson plus a lepton)
thus providing a mechanism of lepton number non-conservation (leptogenesis)
(see for instance [Ramond(2005), Buchmuller&et-al(2005)]). Now, through the
so-called “sphaleron” mechanism, the breaking of lepton number can lead to the
breaking of baryon number (baryogenesis) at the primordial inflation scale,
thus satisfying one of the Sakharov’s conditions for the origin of matter
[Sakharov(1967)B]. We thus adopt this assumption which can be considered as
the minimal BSM assumption about the initial condition of quantum cosmology.

###  4.2. The dark matter induced by QCD

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”, $\psi$ to
point $\delta$ that marks the transition from the QCD quark gluon plasma to
the colorless hadronic phase, or, “top down” from point $\beta$ and through
point $\gamma$, the electroweak symmetry breaking per the BEH mechanism, to
point $\delta$ 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 $\rho_{\text{AdS}}^{\text{ind}}$ as a
comoving density that, when evaluated at point $\delta$, would plays the role
of the anti-desitterian world matter, induced by QCD, to cancel the
contribution of the comoving CC at point $\delta$. Now, it turns out that
following an idea of Sakharov [Sakharov(1967-68)] and the work of Adler
[Adler(1982)] such a contribution can be rigorously evaluated (or at least
estimated) [Brindejonc&Cohen-Tannoudji(1995), Brindejonc(1998)]. 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:

(29)
$-\frac{1}{2\pi}\frac{\Lambda_{\mathrm{ind}}}{G_{\mathrm{ind}}}=\frac{\int\mathrm{d}\\{\Phi\\}\,e^{\mathrm{i}S\left[\\{\Phi\\},\eta_{\mu\nu}\right]}T(0)}{\int\mathrm{d}\\{\Phi\\}\,e^{\mathrm{i}S\left[\\{\Phi\\},\eta_{\mu\nu}\right]}}\,,$

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
$M(g,\mu)=\mu\exp(-1/b_{0}g^{2})$ where $\mu$ is the renormalization scale and
$b_{0}=\left(11N_{c}-2N_{f}\right)$ 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 $a(\theta)$ where $\theta$ 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 di-gluons through what is called in [Adler(1982)] the
gluon pairing amplitude to the trace anomaly reads

(30) $\left\langle T^{\mu}_{\
\,\mu}\right\rangle_{0}=-\frac{1}{8}\left[11N_{c}-2N_{f}\right]\,\left\langle\frac{\alpha_{s}}{\pi}\left(F^{a}_{\mu\nu}F^{a\mu\nu}\right)^{r}\right\rangle_{0}\,.$

In the following quote from [Cohen-Tannoudji(2019)] it was argued:

> _The minus sign in the right hand side shows that when the constant
> $b_{0}=\left(11N_{c}-2N_{f}\right)$ is positive, all the QCD condensates
> contribute negatively to the energy density, which means that the QCD world-
> matter is globally an anti-de Sitter world-matter (dominance of an anti-de
> Sitter world-matter over a smaller de Sitter world-matter)._
>
> _The multiplicative factor $b_{0}$ allows reading, thanks to the well-known
> property that boson and fermion loops contribute in quantum field theory
> with opposite signs, see Figure 3, the relative contributions of the
> components of the QCD vacuum to the full world-matter:_
>
>   * •
>
> _the bosonic (gluon) loops, proportional to_ $N_{c}$, contribute to the
> anti-de Sitter world matter that represents the contribution of the bulk of
> free gluons to the total active mass in the effective dark universe, and
>
>   * •
>
> _the fermionic (quark) loops, proportional to_ $N_{f}$, contribute to the
> normal de Sitter world matter, which, per our interpretation, represents the
> kinetic energy density of the quarks which have survived to the global
> annihilation of fermions and antifermions, namely the constituents of the
> baryonic matter.
>
>

In this quotation of a previous work appears the expression of free gluons
that we will not retain in the present paper because it is misleading: the
gluons in the quark gluon plasma that can be considered as free before the
hadronization transition, have to be “integrated out” (see Eq. (29)) to form,
in the colorless hadronic phase, the non-interacting world matter induced by
QCD that will be interpreted below (4.3) as Bose-Einstein condensate of di-
gluons.

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
hadronic phase occurs in the region of expansion in which we use the
methodology of effective theories, we assimilate the full content of the
universe at point $\delta$ to an “effective dark universe” for which the
radius of the universe is equal to the Hubble radius. This means that we have
(by thought) sent the baryonic matter at the Hubble horizon namely made of its
energy density a de Sitter world matter. This is the key point of [Cohen-
Tannoudji(2019)]: the term in Eq. (30) proportional to $N_{f}$ is a de Sitter
world matter that represents, at point $\delta$, the kinetic energy of the
quarks, called “valence quarks” that constitute the baryonic matter, whereas
the term proportional to $N_{c}$ is an anti de Sitter worldmatter that
represents the active mass of the gluons, namely the dark matter. At point
$\delta$, $N_{c}$ is equal to 3, and $N_{f}$, which is not the number of quark
flavors, but rather the number of fermions that constitute a nucleon, is also
equal to 3. Eq. (30) thus allows us to conjecture the value of the ratio
Dark/Visible to be equal to 11/2, at point $\delta$ but also today since pure
numbers need not to be rescaled. This is the main outcome of [Cohen-
Tannoudji(2019)] that we now confront with the results of the Planck
experiment: in Table 2 of [PlanckVI(2018)] these results are compared with the
expectations from the so-called “base LambdaCDM” standard model. In the last
column of this table, we can read the 68% limits on the parameters based on
“Planck TT,TE,EE,+lowE+lensing”: for $\Omega_{\Lambda}$ we read $0.6889\pm
0.0056$, for $\Omega_{\mathrm{M}}$ (that stands for
$\Omega_{\mathrm{vis}}+\Omega_{\mathrm{DM}}$) we read $0.3111\pm 0.0056$. Now,
according to our model, we expect

(31)
$\Omega_{\Lambda}-\Omega_{\mathrm{vis}}=\Omega_{\Lambda}-\frac{2}{11}\Omega_{\mathrm{M}}=2\Omega_{\mathrm{M}}\Rightarrow\Omega_{\mathrm{M}}=\frac{11}{24}\Omega_{\Lambda}\approx
0.3157\,,$

to be favourably compared with $0.3111\pm 0.0056$.

####  4.2.1. Baryons as “chromo-magnetic” monopoles

A superconductor analogy was used in [Nielsen&Olesen(1979)] by Nielsen and
Olesen who proposed a suggestive model of the QCD vacuum involving unconfined
chromo-magnetic monopoles moving freely along magnetic flux lines. The
interpretation of baryons as color magnetic monopolesb)b)b)As a special
tribute to Georges Lochak (1930-2021), French physicist known for his work on
magnetic monopoles. had been proposed by Ed. Witten [Witten(1979)] in the
following quote (where $N$ has to be replaced by $N_{c}$):

> _Indeed, the baryon mass is of order $N$, which can be written as $1/(1/N)$.
> But $1/N$ is the “coupling constant” of the strong interactions, which
> characterizes the interaction among mesons. $1/N$ plays in QCD roughly the
> role that $\alpha$ plays in spontaneously broken gauge theories of the weak
> and electromagnetic interactions. The fact that the baryon mass is of order
> $1/(1/N)$ is analogous to the fact that the Polyakov-’t Hooft monopole mass
> is of order $1/\alpha$._

.

####  4.2.2. Magnetic flux lines as dark matter filaments

In [Nielsen&Olesen(1979)] Nielsen and Olesen have argued that “one gains
energy by separating a monopole and an anti-monopole”. It is thus reasonable
to interpret the color magnetic flux lines (a pure QCD effect) as filaments of
a world matter (that is a component of the universe with no interaction other
than gravitational) that is filaments of dark matter connecting monopoles to
anti-monopoles. Now, since we can assume that anti-monopoles have been
integrated out, in giant black holes at the center of galaxies or galaxy
clusters, we expect that in simulations of the distribution of galaxies, the
filaments must close at the points where are located the heavy black holes.
And this is precisely what is seen in Figure 1.

###  4.3. Bose Einstein condensation in the cosmological context

Since 1999 there is experimental evidence (RHIC, LHC) of the quark-gluon
plasma as a super-liquid which is produced in ultra-relativistic heavy ion
collisions, see for instance [Pasechnik&Šumbera(2017), Andronic&et-al(2018)]
and the recent comprehensive historical account [Rafelski(2020)]. Measurements
indicate that quarks, antiquarks and gluons flow independently in this liquid.
On the other hand the universe at its quark epoch, i.e. from $10^{-12}$s to
$10^{-6}$s, with temperature $T>10^{12}$K, was uniformly filled with QGP which
once the Universe cooled below evaporated into a gas of hadrons. This
corresponds to the point $\delta$ in Figure 2. As explained above, an
effective AdS with curvature provided by $\Lambda$ is present at this period.

Returning to our approach to elementary systems in dS or AdS space-times,
Equation (21) tells us that the energy at rest of a fermion in an AdS
background decomposes into a “visible” mass part, like in Minkowski, and a
“dark” part which is like the ground state energy of a quantum three-
dimensional isotropic harmonic oscillator with frequency equal to
$\sqrt{\dfrac{\left|{{\Lambda}_{\text{AdS}}}\right|}{3}}c$. This feature led
one of us in [Gazeau(2020)] to infer that at the point $\delta$ , i.e., at the
hadronization phase transition, “chemical freeze-out” temperature
$T_{cf}\gtrsim 1.8\times 10^{12}$K, the ratio “dark/visible”
$r:=\dfrac{3}{2}\dfrac{\hbar\omega_{\mathrm{AdS}}}{mc^{2}}$ for light quarks
$u$ (mass $m_{u}\approx 2.2$ MeV/$c^{2}$) and $d$ ($m_{d}\approx 4.7$
MeV/$c^{2}$) are given by $r(u)\approx 108$ and $r(d)\approx 49$ respectively.
In Reference [Gazeau(2020)] it was suggested that dark matter originated from
this supplementary mass granted to (anti-)quarks by the ADS environment. In
the present paper we instead explain the existence of dark matter as holding
its origin from the gluonic component of the QGP. On the same footing, we
tentatively explain dark matter by asking a simple question: what becomes the
huge amount of gluons after the transition from QGP period to hadronization?

Equation (24) tells us that the energy at rest of a spin 1 massless boson in
an AdS background is purely “dark” and is twice the elementary quantum
$\hbar\omega_{\mathrm{AdS}}$. Hence the QGP gluons in the AdS background at
the point $\delta$ acquire an effective mass $2\hbar\omega_{\mathrm{AdS}}$.
The latter is qualitatively determined through the equipartition
$k_{B}T_{cf}\approx\hbar\omega_{\mathrm{AdS}}$. Hence,
$2\hbar\omega_{\mathrm{AdS}}/c^{2}=144\times m_{u}\approx 317$ MeV/$c^{2}$.
One should notice that this gluonic effective mass is about $4/3$ times the
effective mass acquired by quarks and antiquarks in that QGP-AdS environment.

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 a large number, say $N_{G}$, of non-interacting entities that are
not individual free gluons, but rather decoupled gluonic colorless systems,
which are assumed to form a grand canonical Bose-Einstein ensemble. As said
above, in 4.1.2, the holographic relation that holds in the expansion phase,
from point $\beta$ to point $\psi$ in Figure 2, cannot be satisfied without an
anti-de Sitter world matter density that can compensate (see Eq. (30)) the
contribution of CC in Eq. (4). The simplest purely gluonic system of this
type, susceptible to form a Bose-Einstein condensate is a di-gluon whose
contribution is the one of the gluon pairing amplitudes of Eq. (30). The di-
gluons can be called “dark matter quasi-particles”, which, with an AdS rest
mass equal to $\sqrt{\Lambda}$ form a Bose-Einstein condensate because their
Compton wavelength as well as their mean relative distances equal with the
Hubble radius. Let us remind the reader that, according to the methodology of
effective theories, all dimensioned quantities such as the rest mass of a
quasi-particle are rescaled, i.e. depend on the global thermal time $\tau$
(see above 4.1.2).

Actually, the “effective dark universe” which, as said above, is supposed to
provide the cosmological standard model with a quantum vacuum or a ground
state, cannot be thought of as an “empty spacetime” in which some objects
would move, but rather as a medium in which occur vacuum polarization events,
what Gürsey [Gürsey(1963)] calls scintillation eventsc)c)c)The mass
scintillation model imagined by Gürsey, is comparable with the steady state
cosmology of Bondi [Bondi&Gold(1948)] in which the creation, at constant
density of matter- energy, induces the expansion of the universe., namely
events each consisting in the virtual creation of a particle-antiparticle
pair, followed, a short time later, by its annihilation. If the particles of
the pair are fermions, such event would contribute to a negative curvature
(normal de Sitter) world matter, a sort of a _baryonic Fermi-Dirac sea_ , at
the exterior of the Hubble horizon, and if they are bosons, the event would
contribute to a positive curvature (anti-de Sitter) world matter, a _gluonic
Bose-Einstein condensate_ , at the interior of the Hubble horizon.

Therefore we consider an assembly of these $N_{G}$ dark matter di-gluons
viewed as quasi-particles in an AdS environment with individual energies
$E_{n}=E_{\mathrm{AdS}}^{\mathrm{rest}}+n\hbar\omega_{\mathrm{AdS}}$ where
$E_{\mathrm{AdS}}^{\mathrm{rest}}$ is the expression (17) taken at $m=m_{G}$
(can be zero or negligible) and $s=0$, and degeneracy $g_{n}=(n+1)(n+3)/2$
[Fronsdal(1975)]. Consequently, those remnant entities are analogous to
isotropic harmonic oscillators in $3$-space. They are assumed to form a grand
canonical Bose-Einstein ensemble whose the chemical potential $\mu$ is fixed
by the requirement that the sum over all occupation probabilities at
temperature $T$ yields [Grossmann&Holthaus(1995), Mullin(1997)]

(32)
$N_{G}=\sum_{n=0}^{\infty}\frac{g_{n}}{\exp\left[\frac{\hbar\omega_{\mathrm{AdS}}}{k_{B}T}\left(n+\nu_{0}-\mu\right)\right]-1}\,,\quad\nu_{0}:=\frac{E_{\mathrm{AdS}}^{\mathrm{rest}}}{\hbar\omega_{\mathrm{AdS}}}\,.$

Since this number is very large one expects that this Bose-Einstein gas
condensates at temperature

(33)
$T_{c}\approx\frac{\hbar\omega_{\mathrm{AdS}}}{k_{B}}\left(\frac{N_{G}}{\zeta(3)}\right)^{1/3}\approx
1.18\times 10^{-3}\times\sqrt{|\Lambda_{\mathrm{AdS}}|}\,N_{G}^{1/3}$

to become the currently observed dark matter. The above formula involving the
value $\zeta(3)\approx 1.2$ of the Riemann function is standard for all
isotropic harmonic traps (see for instance [Grossmann&Holthaus(1995)],
however, one should not be misled by the latter term: there is no harmonic
trap here, it is the AdS geometry which originates the harmonic spectrum on
the quantum level). To support this scenario, and as explained in [Cohen-
Tannoudji&Dalibard&Laloë(2005)], it has been shown, thanks to the physics of
ultra-coldd)d)d)As a side remark, one could propose to name the cosmological
standard model $\Lambda$UCDM (UCDM for Ultra Cold Dark Matter) atoms, that
Bose Einstein condensation can occur in non-condensed matter but also in gas,
and that this phenomenon is not linked to _interactions_ but rather to the
_correlations implied by quantum statistics._

Although we do not precisely know at which stage beyond the point $\delta$ in
Figure 2 does take place the gluonic Bose Einstein condensation, let us see if
Eq. (33) yields reasonable orders of magnitude by putting $T_{c}$ equal to the
current CMB temperature, $T_{c}=2.78$K, and
$|\Lambda_{\mathrm{AdS}}|\approx\frac{5.5}{6.5}\times\frac{11}{24}\times\Lambda_{\mathrm{dS}}=0.39\times\Lambda_{\mathrm{dS}}$,
with $\Lambda_{\mathrm{dS}}\equiv$ present CC $\Lambda=1.1\times 10^{-52}$m-2.
We then derive from (33) the following estimate on the number of di-gluons in
the condensate:

(34) $N_{G}\approx 5\times 10^{88}\,.$

This estimate already seems reasonable since the gluons are around $10^{9}$
times the number of baryons, and the latter is estimated to be around
$10^{80}$. Keeping this number in Eq. (33) allows to estimate the scaling
factor defined by

(35) $\sigma_{c}:=\frac{\Lambda_{\mathrm{dS}}}{T^{2}_{c}}\approx 1.85\times
10^{6}\times N_{G}^{-2/3}\approx 1.36\times 10^{-53}\,.$

Hence, having $T_{c}$ of the order of “Matter-dominated era” temperature, say
$10^{4}$K, i.e., at the point $\varepsilon$ in Fig. 2, yields the following
value for $\Lambda_{\mathrm{dS}}$:

(36) $\Lambda_{\mathrm{dS}}(\varepsilon)\approx 1.36\times
10^{-44}\,\mathrm{m}^{-2}\,.$

Some years ago, it has been shown that Bose-Einstein condensation of dark
matter can solve the core/cusp problem in the rotation curves of dwarf
galaxies [Boehmer&Harko(2007), Harko(2011)] with dark matter particles of mass
in the meV range e)e)e)We would like to acknowledge the anonymous referee who
let us know these references. In these motivating articles the exact nature of
the BEC particles is not made precise, but strong interaction between them is
assumed, and dark matter is viewed as a non-relativistic, Newtonian Bose-
Einstein gravitational condensate gas, whose density and pressure are related
by a barotropic equation of state. Some observed properties of dwarf galaxies
appear to fit well with this scenario. More recently, the interpretation of
dark matter in terms of a Bose-Einstein condensate has drown interest in the
framework of the so called “fuzzy dark matter” (FDM) model [Hu&et-al(2000)].
Originated from the idea that the dark matter particle is an ultralight
particle [Hui&et-al(2017)], the _axion_ , this model intends to solve the
small scale shortcomings of the cold dark matter (CDM) [Duffy&Bibber(2009)]
model (“cuspy” dark matter halo profiles and an abundance of low mass halos).
In the FDM model, the mass of the would be dark matter particle must of the
order of $10^{-22}$ eV in order for its wave nature to be significant at
astrophysical scales. For testing the ability of such a model to account for
observations, it was necessary to use simulations. Such high precision
simulations were done in [Schive&et-al(2014)] and they confirm the expected
order of magnitude of $10^{-22}$ eV for the mass of the dark matter particle.
Whereas such small masses seem highly unrealistic for particles to be
considered in particle physics, we think that our scalar covariant quantum
“dilaton” field $\phi$, which is not a “dark matter particle”, but rather,
using a terminology of condensed matter physics, a “dark matter quasi-
particle, or a collective excitation” viewed as a di-gluon, can have a
temperature-dependent mass gap of this order of magnitude.

##  5\. Discussion

In this concluding discussion section, we would like, on the one hand, to see
in which way the present paper could fill the gaps between our two former
approaches and, on the other hand, to propose new arguments in favor of their
common aim, namely the aim of reconciling the standard models of particle
physics and cosmology. The starting point of the approach of GC-T was, as
explained above, to associate dark matter with a gravitational lensing
potential and thus, to assimilate it to an anti-de Sitter world matter. This
approach was lacking a firmer theoretical ground, which, we think has been
provided by the approach of J-PG. The idea of this approach is that, apart
from the ordinary mass, elementary quantum systems can acquire in an anti-de
Sitter space time a supplement of mass, of purely geometric origin, that could
be a candidate, in the cosmological context, that is, as explained in
subsection 3.2, when the cosmological term in the Einstein’s equation is put
back in the right hand side in terms of world matter densities, to constitute
the dark matter.

Actually, the link between the two approaches can be provided by what we call
the “ _dilaton_ ” field $\phi$, that is a scalar _covariant_ quantum field
which is given by _non-perturbative_ effects of the gauge theories of the
standard model, at each cosmic event of emergence of masses, a “Higgs like”
potential (with a Mexican hat shape) that can be interpreted as a world-matter
energy density. This is indeed true at the electroweak symmetry breaking at
point $\gamma$ in figure 2, but also at the QCD hadronization transition at
point $\delta$, in Figure 2. Let us note that, in [Duffy&Bibber(2009)], the
authors attribute to the _axion_ the potential provided by the non-
perturbative QCD effects, whereas we attribute it to the _dilaton field_ ,
which, as explained above looks much more reasonable.

In particle physics, the frontier of the standard model is essentially due to
non-perturbative effects such as instantons or monopoles that can be
encountered in the cosmological context [Weinberg(1996)]. Fortunately, as
explained in 4.1.2, it turns out that cosmology at extra galactic scales may
be, because it depends only on the determinant of the metric, easier to deal
with than at smaller scales. The question to know whether the filament
structure that appears in the distribution of dark matter at extra galactic is
also present at smaller scales, is still an open question. For instance, it is
often claimed that, at galactic scales, dark matter is rather distributed in
the halo of ordinary matter. But we think that this feature cannot be objected
as an argument against our approach, because our interpretation of dark matter
as a Bose-Einstein condensate, precisely aimed to correct the defects of the
conventional cold dark matter model, has been some how comforted, as shown
above, by the fuzzy dark matter paradigm. On the other hand, it seems that a
very recent work, based on deep learning, has shown the existence of a local
web of dark matter at galactic scales [Hong&et-al(2021)].

### Acknowledgments

The authors thank Dr. Vincent Brindejonc (Thalès) and Prof. Maxim Khlopov
Director of Virtual Institute of Astroparticle Physics, for insightful
comments and suggestions.

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