Fakultät für Physik und Astronomie

Ruprecht-Karls-Universität Heidelberg

im Studiengang Physik

vorgelegt von

Janning Martin Hermann Meinert

aus Hamburg

Kondensate ultraleichter Axionen und

eine Verknüpfung leptonischer Skalen

zu dunkler Materie

Die Masterarbeit wurde von Janning Meinert

ausgeführt am

Institut für Theoretische Physik der Universität Heidelberg

unter der Betreuung von

Herrn PD Dr. Ralf Hofmann

sowie von

Herrn Prof. Dr. Matthias Bartelmann

Institut für Theoretische Physik

Department of Physics and Astronomy

University of Heidelberg

Master thesis

in Physics

submitted by

Janning Martin Hermann Meinert

born in Hamburg

Condensates of ultralight axions

and a link of leptonic scales

to dark matter

This Master thesis has been carried out by Janning Meinert

at the

Institut for Theoretical Physics

under the supervision of

PD Dr. Ralf Hofmann

Prof. Dr. Matthias Bartelmann

Institute for Theoretical Physics

Wir bestimmen die Masse ultraleichter Axionen, um die explizite U(1)$_A$ Symmetrie-Brechungs-Skale $\Lambda$ bei einer Peccei-Quinn Skale von der Größenordnung der Planck-Masse zu bestimmen.
Dabei nehmen wir an, dass der dominante Beitrag zur Masse einer Galaxie mit geringer Oberflächenhelligkeit nur von einer Axionspezies im Sinne von fuzzy dark matter (Klumpen) bestimmt wird. Für Rotationskurven-Fits an galaktische Rotationskurven benutzen wir deshalb das Soliton-Navarro-Frenk-White-Modell, welches gemäß der Lösung des Poisson-Schrödinger-Systems einen Kondensat-Kern plus korrelierte Axionen im Halo annimmt. Zusätzlich betrachten wir drei häufig verwendete Massendichteprofile: Navarro-Frenk-White-, Pseudo-Isothermal- und Burkert-Modell. Eine Masse $ m_a $ von $0.675 \times 10^{- 23} \,$ eV wird extrahiert, welche die Ergebnisse von [1] reproduziert. Dies impliziert eine effektive Yang-Mills-Skala von $ \Lambda \sim 287 \, $ eV, die nur um einen Faktor von $ 15 $ kleiner ist als die Yang-Mills-Skala einer SU(2)-Theorie, die zur Beschreibung der ersten Lepton-Familie vorgeschlagen wurde [2]. Das kosmologische Modell SU(2)$_{\rm CMB}$ legt nahe, dass drei SU(2) Yang-Mills-Theorien, die jeweils für die Entstehung der Lepton-Dubletts (e, $\nu_e$), ($\mu$,$\nu_\mu$) und ($\tau$,$ \nu_\tau$) verantwortlich sind, gleichermaßen zur gegenwärtigen Dichte der dunklen Materie beitragen. Parameter eines isolierten Klumpen, wie der gravitative Bohr-Radius oder die Virialmasse, werden ausschließlich durch die Planck-Masse und die entsprechende Leptonmasse bestimmt. Wenn ein Großteil der in einer Galaxie enthaltenen dunklen Masse durch e-Klumpen repräsentiert ist, könnte möglicherweise eine Mischung aus $\tau-$ und $ \mu$-Klumpen das Vorhandensein massiver kompakter Objekte in galaktischen Zentren erklären, und $\tau$-Klumpen könnten im Zusammenhang mit Kugelsternhaufen und der Halo-Masse stehen. Eine naive Abschätzung eines gravitativen $\tau$-Klumpen Kollaps könnte eine theoretische Erklärung für die Massen-Lücke zwischen stellaren und super massiven schwarzen Löchern liefern.

We determine the mass of ultralight axions in order to determine the explicit U(1)$_A$ symmetry breaking scale $ \Lambda $ at a Peccei-Quinn scale of the magnitude of the Planck mass. We assume that the dominant contribution to the mass of a galaxy with low surface brightness is only determined by one axionic species in the sense of fuzzy dark matter (lumps). For rotation curve fits to galactic rotation curves, we therefore use the Soliton-Navarro-Frenk-White model, which assumes a condensate core plus correlated axions in the halo according to the solution of the Poisson-Schrödinger system. In addition, we consider three commonly used mass density profiles: Navarro-Frenk-White, pseudo-isothermal and Burkert model. An axion mass $m_a$ of $0.675\times10^{-23}\,$eV is extracted, which reproduces the results of [1]. This implies an effective Yang-Mills scale of $\Lambda \sim 287\,$ eV, which is only a factor of $15$ smaller than the Yang-Mills scale of an SU(2) theory that is used to describe the first lepton family [2]. The cosmological model SU(2)$_{\rm CMB} $ suggests that three SU(2) Yang-Mills theories, each for the formation of the lepton doublets (e,$\nu_e$), ($\mu$,$\nu_\mu$), and ($\tau$,$\nu_\tau$) are equally responsible for contributing to the current density of dark matter. Parameters of an isolated lump, such as the gravitational Bohr radius or the virial mass, are determined solely by the Planck mass and the corresponding lepton mass. If the dominant constituent of the dark mass contained in a galaxy is represented by e-lumps, a mixture of $ \tau- $ and $ \mu$-lumps could possibly explain the presence of massive compact objects in galactic centers, and $ \tau$-lumps could be related to globular clusters and the halo mass. A naive estimate of gravitational $ \tau$-lump collapse could provide a theoretical explanation for the mass gap between stellar and super massive black holes.

Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.

Heidelberg, den 09.06.2021     []4cm0.4pt

CHAPTER:  INTRODUCTION

Thinking late at night in Egypt, see [3]

• What is the motivation of this thesis?

• What are fibre bundles?

• What are Lie groups?

• How does the chiral anomaly induce a mass?

§ INTRODUCTION

Both the Standard Model of particle physics and the Standard Model in cosmology, $\Lambda$CDM are highly successful in their descriptions of nature [4, 5, 6, 7]. Yet, old problems like the nature of dark matter and, more recentlyk dark energy still remain. While both are implemented in the Standard Model in cosmology, they have no representation in the Standard Model of particle physics[Unless neutrinos turn out to be able to account for dark matter. However, due to thermalisation in the (hot) early Universe it is believed that neutrinos can only contribute to hot dark matter [8], which is currently disfavoured by experimental evidence [9].]. This problem becomes ever more  pressing as new problems in $\Lambda$CDM emerge such as the Hubble constant crisis [10], the discrepancy of cosmologically local and model dependent Hubble constant determinations; the missing baryon problem [11], which refers to a higher baryon density than expected by $\Lambda$CDM and Big Bang nucleosynthesis (BBN) than locally observed; small-scale problems such as the missing satellites problem [12] and the core-cusp problem [13];

and unexplained anomalies in the Cosmic Microwave Background (CMB) [14].

Many particle theories for explaining dark matter have been considered such as weakly interacting massive particles (WIMPs) [15, 16], QCD axions [17, 18], sterile neutrinos [19, 20], and other approaches such as primordial black holes [21, 22], modified gravity [23, 24] and many more. In this thesis we focus on ultralight Planck scales axions [25] coupled to an SU(2) theory which describes thermalized photon gases, the so called SU(2)$_{\rm CMB}$ [26], and its implications for the dark sector. The main motivation behind this approach is that SU(2) Yang-Mills thermodynamics incorporate a non-trivial structure of the vacuum [27]. This particular aspect of an SU(2) Yang-Mills theory could potentially solve two problems at once:

i) The vacuum structure of the Standard Model of particle physics is trivial. This leads to a vacuum expectation value (VEV) which is many orders of magnitude higher, $\rho_{\rm QFT} \sim (10^{28} \,\text{eV} )^4$ [28] than expected by astronomical observations $\rho_{\rm obs.} \sim ( 10^{-3} \,\text{eV} )^4$ [29, 28, 30]. This discrepancy is known as the cosmological constant problem [31]. While an SU(2) Yang-Mills theory for the photon has a too small VEV of $\rho_{\rm SU(2)} \sim (6 \times 10^{-5} \,\text{eV} )^4$ [27], coupled to a Planck scale axion it is $\rho_{\rm SU(2),M_P} \sim 0.7\times\rho_c\sim (5.92 \times 10^{-3} \,\text{eV} )^4$ [32] which is in relatively good agreement with cosmological observation.

ii) The U(1) description of the photon has one thermal phase, the photon gas. In contrast, an SU(2) Yang-Mills theory description of the photon has three thermal phases: The deconfined, preconfined and confined phase [27]. A transition between the first two phases could explain the Gaussian offset in the CMB line temperature as observed by the Arcade 2 experiment and previous ground based radio experiments [33]. Electric monopoles[In a pure gauge theory those monopoles are magnetic, however, here they are interpreted in dual way.] are assumed to emerge in the deconfined phase [34]. This non-trivial ground state would dynamically break the U(1) symmetry of the photon[The pure SU(2) description of the photon is only valid for high redshifts. This symmetry is dynamically broken down to U(1) [27], today one can assume that SU(2)$_{\rm CMB} \underset{\mathrm{today}}{=}$ U(1)$_Y$ [32].] rendering it Meissner massive [34].

Moreover, anomalies in the CMB may also be explained by SU(2)$_{\rm CMB}$ [35]. A fit of SU(2) theory to the CMB [26] retrieves a Hubble constant of $H_0 = 74.24 \pm 1.46$ (km/s)/Mpc which agrees with the locally obtained values [10], a reionisation redshift $z_{re} \sim 6.23^{+0.41}_{-0.42}$ which agrees with results from studying quasar spectra [36, 37] and a baryon density $\omega_{b,0} = 0.0173 \pm 0.0002$ which might mitigate or resolve the missing baryon problem [26]. Compared to $\Lambda$CDM's baryon density of $\omega_{b,0} = 0.0225 \pm 0.00016$ [6] the SU(2) baryon density is lower. The baryon density of the SU(2) fit lies within the bounds of earlier BBN baryon densities  $\omega_{b,0} =  0.019 \pm 0.0024$ [38], however, disagrees with more recent BBN results $\omega_{b,0} =  0.022305 \pm 0.000225$ [39]. Another implication of the CMB fit is a transition between dark energy and dark matter at a redshift of about $z \sim 53$ [26], in order to obtain a sufficiently high dark matter density. This so called process of depercolation goes hand in hand with earlier attempts of connecting dark matter and dark energy, i.e. quintessence, compare  [25, 40, 41, 42, 43].

The main purpose of this thesis is to extract a Planck scale axion mass by fitting rotation curves of low luminosity galaxies from the SPARC library [44]. The first chapter of the thesis will use the mathematically rigorous concept of fibre bundles to introduce the chiral anomaly in an informal way. The second chapter introduces some aspects of SU(2)$_{\rm CMB}$ cosmology. The third chapter will link the chiral anomaly with axions and give a brief introduction to dark matter and rotation curves. In the fourth and following chapters, rotation curve fits will be used in order to extract an ultralight Planck scale axion mass. This mass will then determine an explicit symmetry breaking scale $\Lambda$. We assume that the dominant contribution to the mass of a galaxy with low surface brightness is only determined by one axionic species in the sense of fuzzy dark matter (lumps). Those lumps can be assumed to be a non-thermal and non-relativistic boson condensate which acts as a superfluid [32]. For rotation curve fits to galactic rotation curves, we therefore use the Soliton-Navarro-Frenk-White model, which assumes a condensate core plus correlated axions in the halo according to the solution of the Poisson-Schrödinger system. In addition, we consider three commonly used mass density profiles: Navarro-Frenk-White, pseudo-isothermal and Burkert model. An axion mass $m_a$ of $0.675\times10^{-23}\,$eV is extracted, which reproduces the results of [1]. This implies an effective Yang-Mills scale of $\Lambda \sim 287\,$ eV, which is only a factor of $15$ smaller than the Yang-Mills scale of an SU(2) theory that is used to describe the first lepton family [2]. In the same way as the photon, the electron is also described with an SU(2) Yang-Mills theory in this thesis, SU(2)$_e$. Therefore, the factor of $15$ can be interpreted as the difference between the confining phase in which the axion condensate exists now, and the deconfining phase of the electron for which the electron scale $\Lambda_e$ was calculated [2].

The central result of this thesis is that each leptonic SU(2) Yang-Mills theory, SU(2)$_{e}$, SU(2)$_{\mu}$, and SU(2)$_{\tau}$ is linked to the current dark matter content of the Universe and contributes equally to it. In contrast, the axion field which is associated to SU(2)$_{\rm CMB}$ has not depercolated yet (formed lumps) and is thus spatially homogeneous. It can be interpreted as dark energy [32], however, both claims include strong assumptions and cannot be proved as of today. One of the core assumptions is that all four axion fields are created at the Planck-scale. This may be motivated by the VEV as mentioned earlier. The conclusion after linking the leptonic sector to three SU(2) Yang-Mills theories is to add those groups to the gauge group SU(2)$_{\rm CMB}$ which might describe thermalized photon propagation. The model now has the following structure: SU(2)$_{\rm CMB} \times$SU(2)$_{e} \times$SU(2)$_{\mu} \times$SU(2)$_{\tau}$. Parameters of an isolated lump, such as the gravitational Bohr radius or the virial mass, are determined solely by the Planck mass and the corresponding lepton mass. If the dominant constituent of the dark matter contained in a galaxy is represented by e-lumps, a mixture of $ \tau- $ and $ \mu$-lumps could possibly explain the presence of massive compact objects in galactic centers, and $ \tau$-lumps could be related to globular clusters. A naive estimate of gravitational $ \tau$-lump collapse could provide a theoretical explanation for the mass gap between stellar and super massive black holes. Unless stated otherwise, we always work in natural units $c=\hbar=k_B=1$. We use Einstein's sum convention and sum over indices where Greek letters go from $\mu = 0,1,2,3$ and Latin letters start at $1$. The Feynman slash notation is used: $\slashed{\partial}= \gamma^\mu \partial_\mu$.

Definition: Definitions and additional information will be given in a grey box

§ FROM FIELDS TO FIBRES

In the last century the cornerstones of present-day fundamental physics have emerged. Prominently among them is the formulation of gauge theories and in particular Yang-Mills theory, which poses a generalisation of Maxwell's electrodynamics to the dynamics governed by non-Abelian gauge groups. The key quantity of interest in gauge theories is the vector potential $A_\mu$ which is easiest to describe in the fiber bundle formalism. Before focusing on gauge groups and in particular on the SU(2) group in the next chapter, the purpose of this chapter is to provide physical intuition to the bundle formalism. For a more and precise formulation of fibre bundles, instantons, gauge and Yang-Mills theories consider reading [45, 46]. For thermal quantum SU(2) and SU(3) Yang-Mills theories and calorons – finite temperature instantons – consider [47].

Michael Faraday was arguable one of the first physicist to use a field description for electricity. The easiest example of a field is a scalar field. Fig. <ref> a) shows a temperature map, where each point is associated with a scalar, in this case a temperature.

An example for a scalar field is a temperature map a). A vector field is usually used to describe airflow, not only the direction but also the velocity of the wind b). Curvature of a Riemannian manifold can be described by a tensor field c).

Another example of a field is a vector field, where each point on the base manifold is associated with a given a vector, see figure <ref> b). A little bit more abstract use of the fields is given in general relativity, where the curvature of spacetime is described by  a tensor field $R_{\mu\nu}$, see c) for the two-dimensional case.

The first theory which included a gauge freedom in the definition of its basic field variable $A_\mu$ was Maxwell's formulation of electromagnetism, see Section <ref>. Simply put, a gauge field $A_\mu$ has more degrees of freedom than the
physical fields of the theory. For Lie-groups (Section <ref>), the symmetry given by the group structure is continuous.

Before giving examples of gauge fields, it is instructive to introduce fibre bundles first. A fibre is a manifold $E$ which is attached to a point of a base manifold $M$, see figure <ref>. We will use four-dimensional Minkowski spacetime as a base manifold from now on.

Fibre bundle: Let $\pi: E \rightarrow M$ be a smooth surjection. Then the triple $(E,M,\pi)$ is called a fibre bundle if there
exists a manifold $F$ such that the following holds. Every $m \in M$ admits an open neighbourhood $U$ and a diffeomorphism $\chi : \pi^{-1}(U) \rightarrow U \times F$ fulfilling ${\rm pr}_U \circ \chi =
\pi$. The manifold $F$ is called the typical fibre of $\pi$. This definition has been taken from <cit.>.

Examples of fibre bundles: a) identical fibres with the manifold $F$ are glued to a base manifold $M$,  b) if each point of the base manifold $M$ associates with the complex number line it is called "C-line bundle". c) A physically relevant bundle is a U(1) bundle. In the case of Maxwell's theory a U(1) gauge group describes the freedom to locally change a phase along $F=S^1$.

Principal bundle: A principal bundle, is a fiber bundle $\pi: P \rightarrow M$ together with a continuous right action $P \times G \rightarrow P$ such that $G$ preserves the fibers of $P$, where $G$ is any topological group (also called principal $G$-bundle). The tuple $(P, G, M, \Psi, \pi)$ is called a principal bundle if for every $m \in M$ there exists an open neighbourhood $U$ of m and a diffeomorphism $X : \pi^{-1}(U) \rightarrow U \times G$ such that $X$ intertwines $\Psi$ with the $G$-action on $U \times G$ by translations on the factor $G$, and ${\rm pr}_U \circ X (p) = \pi(p)$ for all $p \in \pi^{-1}(U)$.
Compare <cit.>

An abstract version of a fibre bundle is given in <ref> a), it is the collection of multiple, identical fibres of the manifold $R$ which are glued to a base manifold $M$. A concrete example of a fibre bundle is given in <ref> b), each point is given a complex number line. This is called "C-line bundle". Physically relevant fibre bundles usually use smooth manifold as a fibre, see figure <ref> c). In fact, the physically most relevant bundles use the manifolds of Lie-groups as fibres. Those bundles are also called principle bundles, because the group structure is represented in the fibre itself. This means that each fiber of the principle bundle is homeomorphic to the group $G$ itself.

Let $(P, G, M, \Psi, \pi)$ be a principal bundle. A section of $P$ is a smooth mapping

$s: M \rightarrow P$ such that $\pi \circ s = \mathbb{1}$.
This definition has been taken from <cit.>.

Describing fundamental particle physics with gauge theory assumes that the interaction is mediated by a gauge potential $A_\mu$. Geometrically, a gauge potential represents a local connection. This connection lives on a principal fibre bundle over Minkowski spacetime, compare [46]. A connection generalizes infinitesimal motion on a smooth manifold and is thus the generalisation of the directional derivative of objects on a manifold, compare [48]. In order to aqcuire some intuition of this formalism of describing nature, imagine a dice. The orientation of the dice can be described with a position $S^2$ sphere, see Fig. <ref>. The section of the $S^2$-bundle, from one point in space to another should be smooth. From now on we will refer to the rather mathematical term section as connection $A_\mu$, since it is the term more commonly used by physicist.

The rotation of a dice can be described by a point on a $S^2$ sphere. The photos of the dices were taken by Brett Jordan.

What does this abstract way of introducing fibres and their sections have to do with physics and gauge fields? Gauge fields are sections through principal fibre bundles.
For example, the section of a U(1) bundle can be interpreted as the photon field $A_{\mu}$.

§ LIE GROUPS AND LIE ALGEBRAS

In this thesis we will focus on the postulate that the SU(2) Lie group describes the photon propagation rather than a U(1)$_Y \times$ U(1)$_{\rm SU(2)}$ group and its implications for the dark sector. Lie groups and Lie algebras are of special interest in gauge theories, because Lie algebras generate Lie groups and their commutation relation induces the quantisation of gauge fields not only in quantum mechanics but also in quantum field theory (canonical quantisation). Therefore, we will briefly introduce them here.

Two elements of a non-abelian group, like SU(2) for example, do not commute. However, both groups have in common that they are continuous groups. A continuous group $G$ of order $n$ can be parametrized by $n$ real parameters. If the set of parameters lives in a differentiable manifold $M^n$, the group G is called a Lie group.

Every element $g$ of $G$ can be described by the exponential map $e^{i\omega^a t^a}$. Where $t^a$ are the generators and  $\omega_a$ are finite parameters. If the group structure group is defined by a linear trafo, matrix is given by a matrix, as it is the case for SU(2), the exponential map is given by the series, compare [49]

\begin{align}
e^{i\omega^a t^a} \, = \, \sum_{n=0}^{\infty} \frac{i^n}{n!} \, \left( \omega^a t^a \right)^n \,.
\label{defExpMap}
\end{align}

Elements of $G$ which are close to the identity can be written as

\begin{align}
g \, = \, \mathbf 1 + i \omega^a t^a\,,
\label{defInfg}
\end{align}

Let $g_1,g_2 \in G$ be two elements of the group $G$ which are close to the identity. Using the equation above (<ref>) we can write them as $g_1=\mathbf 1 + i \omega_1^a t^a$
and $g_2=\mathbf 1 + i \omega_2^a t^a$. Multiplying both up to first order gives

\begin{align}
g_1 g_2 = \mathbf 1 + i (\omega_1^a + \omega_2^a) t^a + O(\omega^2)\,.
\label{LieComm}
\end{align}

which commutes $[g_1,g_2]=O(\omega^2)$. Now we can demand that any two generators of a Lie group $t^a$ and $t^b$ constitute another generator $t^c$:

\begin{align}
[t^a, t^b] \, = \, i f^{abc} t^c\,.
\label{defCommLie}
\end{align}

The Lie bracket guarantees that a composition of two group elements can be represented by the exponential map by virtue of the Baker-Campbell-Hausdorff formula. The reason why one demands in Lie groups G that the commentator of two generators is again a generator is that the exponential representation must also apply to the product of two group elements. A visualisation of the exponential map is given in Fig. <ref>.

The Lie bracket of two generators $[t^a,t^b]$ corresponds to a tangential manifold $TM$ at the identity [50]. With the exponential (yellow) and logarithmic map (orange), any point on the tangential manifold $TM$ can be projected on to the fibre $F=U(1)$ and back to the base manifold.

Where $f^{abc} \in \mathbb R$ is the structure constant of the Lie algebra $\mathfrak{g}$ with respect to the basis $t^a$. The Lie algebra $\mathfrak{g}$ of the group $G$ is defined in general

\begin{align}
[\cdot, \cdot]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}, (x,y)\rightarrow [x,y] \,.
\end{align}

The rank $n$ of the Lie algebra $\mathfrak{g}$ is defined as the number of mutually commuting and linear independent generators. In the following we will focus on the non-abelian Lie group SU(2):

SU(2) = {A ∈GL(2, ℂ) | AA̅^t =1, det  A =1} ,

where $GL(n,K)$ is the general linear group of rank $n$ over a body $K$. One can reformulate this statement as

SU(2) = {
α    β

-β̅     α̅
: α, β∈ℂ, | α|^2+| β|^2=1
} .

Now $\alpha$ can be expressed as $\alpha=x_1+i x_2$ and $\beta= x_3+i x_4$ with $x_i \in \mathbb{R}$. With this we can identify SU(2) as being diffeomorphic to $S^3 = \{ x_1^2+x_2^2+x_3^2+x_4^2 = 1 \} \subset \mathbb{R}^4$, compare [51].

§ GAUGE FIELDS

The three fundamental forces of the Standard Model, strong, weak and electromagnetism are described with gauge fields. In a gauge theory, the Lagrangian is locally invariant under certain transformations. The Lagrangian has spurious degrees of freedoms, which can be cancelled by choosing an explicit gauge. Historically, electrodynamics was the first theory which was described with a gauge symmetry.

The dynamics of the photon field $A_\mu$ in the presence of an external source are described by the Maxwell Lagrangian density $\mathcal{L}_{M}$

ℒ_M ≡-1/4 F^μνF_μν-e j^μA_μ

where $F_{\mu\nu}$ is the field strength or Faraday tensor, $e$ is the coupling constant of electromagnetic interaction, and  $j^\mu$ is the current density. The field strength tensor $F_{\mu\nu}$ can be defined by the action of the commutator $[D_\mu,D_\nu]$ on an complex scalar $\phi$

F_μν ϕ=i/e[D_μ,D_ν]ϕ ,

where $D_\mu$ is the covariant derivative in the direction of $\mu$, which is defined as an ordinary derivative and an additional contribution from the gauge potential $A_\mu$

D_μ= ∂_μ-i e A_μ .

The familiar electromagnetic vector fields E and B are recovered from the field strength tensor by the identities

\begin{align}
E^i & \equiv -F^{0i}\,, \\
B^i & \equiv -\frac{1}{2} \epsilon^{ijk} F_{jk}\,,
\end{align}

where $\epsilon^{ijk}$ is the Levi-Civita-Symbol and $F_{jk}$ is the electric part of the Faraday tensor
$F_{\mu\nu}$. The Aharonov-Bohm effect implies that the gauge potential $A_\mu$ is the more fundamental description than the field strength tensor $F_{\mu\nu}$.

The Aharonov-Bohm effect can be observed in the interference pattern change of an electron double slit experiment [52]. Without an electric current the interference pattern is unaffected (black trajectories of the electron in Fig. <ref>) while it is shifted for a shielded current (blue trajectories of the electron in Fig. <ref>). A shielded current induces a magnetic field which should classically have no influence on the electrons. The common interpretation is that the vector potential has a more fundamental meaning than its associated force field.

The Aharonov-Bohm effect in an electron double slit experiment. Electrons are emitted at a source $S$, pass a screen with two holes and create an interference pattern on a second screen. If a magnetic field $B$ (blue circle) is induced by a shielded current, a deviation from the normal interference pattern (indicated by black trajectories) is observed (indicated by blue trajectories) in the form of a phase change is observed.

Putting Eq. (<ref>) into Eq. (<ref>) shows their relationship

F_μν = ∂_μA_ν- ∂_νA_μ .

With this expression it is easier to see that the fieldstrength $F^{\mu\nu}$ and the Maxwell Lagrangian <ref> are invariant under U(1) transformation $A^\mu \longrightarrow {A^{\mu}}'=A^\mu + \partial^\mu \alpha(x)$:

\begin{align*}
F_{\mu\nu}' & = \partial_\mu A_\nu'- \partial_\nu A_\mu' \\
& = \partial_\mu (A_\nu + \partial_\nu \alpha(x))- \partial_\nu (A_\mu + \partial_\mu \alpha(x)) \\
& = \partial_\mu A_\nu - \partial_\nu A_\mu
+\partial_\mu\partial_\nu \alpha(x)
-\partial_\nu\partial_\mu \alpha(x)\,.
\end{align*}

The last two terms cancel because the interchangeability of second-order derivatives.
For the transformed Lagrangian $\mathcal{L}_{M}'$ we therefore have

\begin{align}
{\mathcal{L}_{M}}' & = -\frac{1}{4}\,{F^{\mu\nu}}'{F_{\mu\nu}}'-e\,j^\mu {A_\mu}'\\
& =-\frac{1}{4}\,F^{\mu\nu}F_{\mu\nu}
-e\,j^\mu A^\mu-e\,j^\mu \partial^\mu \alpha(x)
\end{align}

The last term does not affect the action $S[A^\mu]$ as long as j or $\nabla \alpha$ vanish on the boundary of the integral.

One way of visualization is offered by the bundle formalism as discussed in section <ref>, see Fig. <ref>. Here the bundle structure is given by the U(1) group, and the base manifold is chosen to be Minkowski spacetime. In the case of electromagnetism the abundant degrees of freedom are local phases, i.e. polarization. A specific phase can be represented by a point on a $S^1$ circle representing the U(1) group. The gauge potential $A_\mu$, which can be interpreted as photon field is the connection from one U(1) fibre to another.

The connection of a U(1) bundle is the gauge potential $A_\mu$, it can be interpreted as photon field. The base manifold $M$ should be associated with Minkowskian spacetime, in the case of pure electromagnetism.

Meissner massive photons form inside a medium, e.g. a superconductor of the type II after a spontaneous symmetry braking of the U(1) symmetry. Under the critical temperature this U(1) symmetry is spontaneously broken because two entangled electrons form Cooper pairs. See figure blow, Fig. <ref> for flux tubes (left) which lead to levitating superconductors (right) in a phenomena which is called flux pinning.

left: Images of vortices in a thin YBCO film which was taken by a SQUID scanning Microscop below the critical temperature, compare [53]. right: Flux pinning in a superconductor type II. Inside the flux tubes, the photon is massive [54].

If one associates the gauge field of a local U(1) fibre with the local polarisation, as suggested by Fig. <ref>, it is possible to introduce the so called Meissner massive photons in an intuitive way. The informal argument is that a photon has two polarisations orthogonal to the direction of its propagation. Since the speed of light $c$ is only attained by massless particles, and it is the absolute velocity of the Universe, it must not have an additional polarization while traveling with the velocity $c$. A third polarization, as depicted on the right-hand side of Fig. <ref> would allow to temporarily transport information faster and slower than the speed of light. In order to gain a third, longitudinal polarisation the photon needs to slow down. This is equivalent with gaining a small mass. Note however, that a third polarisation breaks the U(1) symmetry dynamically. This argumentation can be turned around, arguing that a small mass induces a third polarisation and breaks the U(1) symmetry. Hopefully, this will help in section <ref> to understand how the chiral anomaly induces masses for axions. In short, the chiral anomaly describes exactly that, a non conservation of the U(1)$_A$-symmetry after quantisation and thus a small induced mass.

For a trivial ground state, a section / gauge connection through an U(1) bundle describes a photon field which has two degrees of freedom. An additional degree of freedom is introduced if the U(1) symmetry is dynamically broken, e.g. by a medium, Cooper pairs or electric monopoles in a non-trivial ground state. This induces a so called Meissner mass.

In reality, this argument is a little more complicated. According to the Goldstone theorem, a massless boson of spin zero is created by a spontaneous symmetry breaking of a global symmetry [55]. Originally this concept was applied in superconductors by Yoichiro Nambu [56], this is why they are also often referred to as Nambu-Goldstone bosons.

However, if a gauge symmetry is broken instead of a global symmetry the would-be Goldstone boson gets "eaten" by the gauge bosons of the broken generators. This particle physicist formulation of eating Goldstone bosons is equivalent to saying that the most natural way of interpreting the additional longitudinal gauge mode is to associate it with a mass. This mechanism is also called the Higgs mechanism and renders the $W^{\pm}$ and the $Z$ boson massive when the electroweak symmetry SU(2)$\times$U(1) is broken down to U(1) [57]. A depiction of the dynical symmetry breaking in this context is shown in Fig. <ref>.

The broken U(1) symmetry in the Higgs mechanism. While a Goldstone boson is created by a spontaneous symmetry breaking a), it is massless and obtains a mass after dynamical symmetry breaking b). Note, that the Higgs potential is three dimensional and only a slice has been shown in one plain, while the U(1) symmetry (red) is shown in an orthogonal plain.

§ CHIRAL ANOMALY

In physics the non-conservation of classical symmetries is often referred to as an anomaly. The chiral anomaly is the failure of the chiral symmetry of massless free fermions after quantization, see [58, 59]. The massless free fermion field $\phi$ is described by the equation of motion

Dϕ=0 ,

where $\phi$ can be interpreted as a section through the vector bundle[the vector bundle is actually $\mathbb{Z}$/2-graded, please consider [60] and [61] for more details.] $V \rightarrow M$ over a Riemannian manifold $M$, and where $D$ is a generalized Dirac operator, compare [61].

A very brief and abstract introduction into the chiral anomaly can be given by the chiral involution operator $\Gamma$. The operator for the chiral involution $\Gamma$ can be constructed by subtracting the identity on even sections
of $V$ with the identity on odd sections of $V$ [61]. In the case of a U(1) bundle, subtracting all sections through even elements of the fibre minus all sections through odd elements is equivalent to the difference between a small left-handed rotation followed by a small right-handed rotation (or vice versa). In this context, the chiral involution operator $\Gamma$ can be thought of as a measure for the equivalence of left- and right-handedness, compare Fig. <ref>. The operator $\Gamma$ preserves the equations of motion because it anti-commutes with $D$ (since $D$ is odd for the $\mathbb{Z}$/2-grading). It then follows because of Eq. (<ref>) that

D(Γϕ)=0 .

Since $\Gamma$ preserves the equations of motion it generates a U(1) symmetry of the classical theory. This symmetry is called chiral or axial symmetry. While it is a symmetry of classical physics it is broken after quantisation, this is known as axial anomaly and induces a mass similarly to the Meissner massive photon as discussed before.

Helicity, sometimes called "handedness", refers to the sign of the spin projected onto the momentum vector. A negative sign is "left"-handed and a positive sign is called "right"-handed. For massless particles chirality is equivalent to helicity, for massive particles (e.g. after a broken U(1) symmetry) that is not the case.In general, the Chirality of a particle refers to whether the particle transforms in a right- or left-handed representation of the Poincaré group. In cases where the representation has both left and right handed components, such as a spinor field (section through a spinor bundle) projection operators that treat one handedness at a time can be constructed.

Chirality and helicity are the same for massless (e.g. intact U(1)) particles but differ for massive particles. The chiral operator $\Gamma$ is zero in the massless case. The visual clue is the same path lengths by the left handed and right handed paths along U(1) in Fig. <ref>. Similarly, the chiral involution is non zero for a broken U(1) fibre bundle.
In other words, the chiral symmetry is broken for massive particles.
Of course this statement can be turned around by saying, breaking the chiral symmetry (or axial current) makes a particle massive.

Chirality and helicity are the same for massless (e.g. intact U(1)) particles but differ for massive particles (e.g. broken U(1)).
The chiral involution operator $\Gamma$ is a measure for equivalence of left- and right-handedness. In the case of massless particles (left) both paths are equivalent. Hence chirality is the same as helicity. After breaking the axial symmetry by making the particle massive (right) it matters
whether the right- or left-handed path is taken.

Another way of illustrating the chiral anomaly is by directly computing that while the vector current $V^a_\mu$ is conserved, the axial current $A^a_\mu$ is not. Therefore, let the massless Lagrangian for the up- and down quark be

ℒ = i Ψ̅ ∂ Ψ ,

where the isospinor notation for the fermions, $\Psi$ = (u, d) is used and $\slashed{\partial}$ is defined as $\partial_\mu \gamma^\mu$.

(i) For the vector-current, consider the following transformation:

Λ_V: Ψ⟶e^-iσ/2ΘΨ≃(1-iσ/2Θ) Ψ ,

where $\sigma$ refers to the Pauli matrices. The Lagrangian is invariant under this transformation

i Ψ̅ ∂ Ψ⟶i Ψ̅ ∂ Ψ- i Θ (
Ψ̅ i ∂ σ/2 Ψ-Ψ̅ σ/2 i ∂  Ψ)
=i Ψ̅ ∂ Ψ ;

the associated Noether current is called vector current and is given by

V^a_μ=Ψ̅ γ_μσ^a/2 Ψ .

(ii) For the axial-current, consider the transformation:

Λ_A: Ψ⟶e^-i γ_5σ/2ΘΨ≃(1-iγ_5σ/2Θ )Ψ ,

due to the anti-commutation relations of the gamma matrices, the Lagrangian is also invariant under $\Lambda_{\textrm A}$ and the associated axial-current is

A^a_μ=Ψ̅ γ_μγ_5 σ^a/2 Ψ .

The Lagrangian of massless fermions is invariant under $\Lambda_{\textrm V}$ transformations, as well as $\Lambda_{\textrm A}$ transformation. This is another way of formulating ciral symmetry. The associated group structure is SU(2)$_{\textrm V}\times$SU(2)$_{\textrm A}$. If a mass term $m \bar{\Psi}  \Psi$ is added to the Lagrangian we get

ℒ = i Ψ̅ ∂ Ψ+m Ψ̅  Ψ ,

it remains invariant under the transformation $\Lambda_{\textrm V}$, however, not under the transformation $\Lambda_{\textrm A}$

Λ_A: m Ψ̅Ψ⟶m Ψ̅Ψ- 2 i  m Θ(
Ψ̅ σ/2 γ_5 Θ Ψ) ,

and thus the axial current is not conserved for massive quarks, compare [62]. The group structure of chiral-symmetry transformations is given by SU(2)$_{\textrm V}\times$SU(2)$_{\textrm A}$. If this symmetry is spontaneously broken down to SU(2)$_V$, three massless Goldstone modes are created, the pions $\pi^-,\pi^0,\pi^+$. In reality, those particles are of course massive and hence also called pseudo-Goldstone bosons. The chiral anomaly is thoroughly tested and experimentally verified by measuring the lifetime of neutral pions $\pi^0$. A short summary is given in the info-box below and the corresponding Fig. <ref>.

Neutral pion decay: Yet another way of introducing the chiral anomaly using Feynman diagrams was originally pursued by Adler [63, 64],  Bell and Jackiw [58]. One can estimate the lifetime of the neutral pion $\pi^0$ by calculating the matrix element as illustrated below on the left-hand side. The unexpectedly short lifetime of the neutral pion is caused by the chiral anomaly which enables another decay channel into to photons [65]. The Feynman diagram of the "decay" of the divergence of the axial current into two photons is shown on the right-hand side below.

Two Feynman diagrams are shown: On the left the decay of the neutral pion into two photons. Its decay is predominantly implemented via the axial current anomaly [59, 65]. This diagram is shown in the right.

However, the main application of the axial anomaly is the axion. As it also enables the axion to acquire a mass, see section <ref>.

CHAPTER:  SU(2)CMB

Comparison of two CMB angular power spectrum fits: The SU(2)CMB model fit (turquoise) and a $\Lambda$CDM fit (green, dotted). The 2015 Planck data points have been used for the fit, compare [7].

• What are the current challenges of $\Lambda$CDM?

• What is SU(2)CMB?

• How could SU(2)CMB mitigate current tensions?

• What are the implications of SU(2)CMB for the dark sector?

§ TENSIONS WITH THE LCDM MODEL

The Cosmological Standard Model is the basis of modern cosmology. To determine all parameters of $\Lambda$CDM at a high accuracy, cosmological distance scales can be calibrated by high-redshift data (inverse distance ladder, global cosmology), coming from precision observations of the Cosmic Microwave Background (CMB) or from large-scale structure surveys probing Baryon Acoustic Oscillations (BAO). Alternatively, low-redshift data (direct distance ladder, local cosmology) can be used by appeal to standard or standardisable candles such as cepheids, TRGB stars, supernovae Ia, and supernovae II. However, depending on whether distances are calibrated with inverse distance ladder data or with local distance ladder anchors (using standard or standardisable candles such as cepheids, TRGB stars, and supernovae Ia), tensions may arise [66]
in some of the  $\Lambda$CDM parameters (e.g., $H_0$ [67, 68, 69, 70] and $\sigma_8-\Omega_m$ [71, 72, 73]. Moreover, it can be shown that those tensions cannot be resolved by a modification of $\Lambda$CDM at low redshift [74]. Figure <ref> shows a time-line of this discrepancy.
Local distance ladder measurements such as S$H_0$ES [69] are shown in blue, whereas model dependant early-Universe data obtained by WMAP are shown in red. One measurement based on Sirens is included in green.

Compilation of Hubble Constant predictions and measurements taken from the recent literature:
This graphic is from [75] and adopted thereby the graphic from [76];
Local distance ladder measurements such as S$H_0$ES [69] are shown in blue, whereas model dependant early-Universe data obtained by WMAP are shown in red.
The first standard sirens measurement GW170817 is shown in green [77].
Forecasts of CMB Stage IV, standard sirens and distance ladder with full GAIA and HST [78] are also depicted.

Another discrepancy may occur in the $\sigma_8$ and $\Omega_m$ relation as shown in Fig. <ref>, the graphic was adapted from [79]. A relatively new tool in Cosmology is Cosmic shear [80]. Cosmic shear is created by large-scale structure in the Universe which distort images of distant galaxies due to weak gravitational lensing.
Properties on large scales as well as the geometry of the Universe can be concluded by measuring galaxy shape correlations. Note, however, that in the following study a spatially flat Universe based on $\Lambda$CDM has been adopted.
The so called Cosmic shear bananas compare the relationship between $\sigma_8$ and $\Omega_m$ as it can be seen on the left hand side of Fig. <ref> and $S_8$ and the total mass density $\Omega_m$ on the right hand side. Thereby is $S_8$ defined as $S_8=\sigma_8\sqrt{\Omega_m/0.3}$ and $\sigma_8$ is the standard deviation of matter density fluctuations in spheres with a comoving radius of 8 Mpc/h. The Planck data results are shown in bright red. Depending on the redshift calibration, discrepancies up to $3\sigma$ may arise. The turquoise stars in the figure below, Fig. <ref> indicate roughly the values of $\sigma_8$ and $\Omega_m$ according to the SU(2)CMB fit. Their values are shown in Fig. <ref>. Note that the stars in Fig. <ref> are drawn in by hand and are not plotted and that the shear model relies on $\Lambda$CDM.

Cosmic shear constraints for the joint distributions of $\sigma_8$ and $\Omega_m$ (left), as well as $S_8=\sigma_8\sqrt{\Omega_m/0.3}$ and $\Omega_m$ (right). The 68% and 95% credible regions are shown for COSEBIs (orange), band powers (pink) and the 2PCFs (cyan). Planck (2018, TT, TE, EE+lowE) results are shown in bright red. The turquoise star indicates roughly the values of $\sigma_8$ and $\Omega_m$ according to SU(2)CMB in Fig. <ref>. Note that the stars are drawn in by hand and are not plotted. This graphic was adapted from [79].

One way of addressing these tensions is to postulate that photon propagation is governed by an SU(2) rather than a U(1) gauge principle. The idea is to describe the CMB as a gas of thermal photons supplemented by a thermal ground state and two invisible vector modes or excitations V$_\pm$ [81].
Thus called the SU(2)$_{\rm CMB}$. Furthermore, this SU(2) gauge theory can be coupled to an ultralight Planck scale axion field. Such an axion is a viable candidate for quintessence, i.e. dynamical dark energy, being associated with today's cosmological acceleration [82, 25].

A direct fit of SU(2)$_{\rm CMB}$ to the CMB [26] retrieves a Hubble constant of $H_0 = (74.24\pm1.46)$ km ${\rm s}^{-1}$ ${\rm Mpc}^{-1}$ which matches well with the locally measured Hubble constant of $H_0 = (73.24\pm1.74)$ km ${\rm s}^{-1}$ ${\rm Mpc}^{-1}$ [10]. Furthermore, model-independent extraction of the co-moving sound horizon at baryon-velocity freeze-out from the relation $r_s\cdot H_0={\mbox const}$ obtains a similarly good match with the local Hubble constant [83]. The co-moving sound horizon is based on low-z observation [1, 10]. Thereby, SU(2)$_{\rm CMB}$ seems to resolve the current discrepancy between the $\Lambda$CDM concordance model [7] and the local measurements of $H_0$ [10].Another open question of modern cosmology poses the missing baryon problem: $\Lambda$CDM fits to the CMB fits and the primordial obtain a baryon density of $100\, \omega_{b,0} = 2.225 \pm 0.016$ [7], which is in a 1.9-$\sigma$ range of the weighted average from several Deuterium-to-Hydrogen ratios (D/H) in Big-Bang Nucleosynthesis (BBN) of $100\, \omega_{b,0} = 2.17 \pm 0.024$ [84]. However, $^7$Li observations contradict predictions based on BBN; the 4–5-$\sigma$ mismatch is called the lithium problem [85]. In SU(2)$_{\rm CMB}$ fits to the CMB a low value of $100\,\omega_{b,0} = 1.73 \pm 0.02$ is obtained which could provide a theoretical solution the missing baryon problem.
As of present, the low baryon density of SU(2)$_{\rm CMB}$ contradicts BBN. Note that the (anti)-screening function $G$ [49] is not implemented in the CLASS code when fitting the CMB multipole spectrum. This could mitigate the offset at low multipoles.

One of the original motivations for SU(2)$_{\rm CMB}$ was to explain the offset from the low-frequency CMB line temperature as measured by ARCADE 2’s [86], right hand side of Fig. <ref>, and earlier radio-surveys. Based on SU(2) Yang-Mills thermodynamics the excess at low frequencies could be explained by evanescent, non-thermal photon fields of the CMB. In this context, the excess is interpreted as a phase-boundary effect [34].

The normalized spectral intensities of CMB modes (thick line) at the critical temperature $T_c$ and of calibrator modes (dotted) at various temperatures on the left, compare [34]. The temperature offset in the microwave range as measure by ARCADE 2 on the right, [33].

The interpretation of Fig. <ref> in the context of SU(2)$_{\rm CMB}$ cosmology is that the SU(2) theory which describes the photon is dynamically broken down to an U(1) symmetry. The idea is that at a critical temperature $T_c \sim 2.73$ electric monopoles emerge which dynamically break the U(1) symmetry and introduce a massive mode which leads to a photon mass. This is the same mechanism as in Superconductors Type II, Meissner massive photons. The only difference is that here instead of a material, e.g. the superconductor, the ground state of the vacuum changes adopts the new structure: A (anti)-monopole condensate. Furthermore, the integrated SU(2) anomaly generates a dipolar modulation which could explain large-angle anomalies in the CMB  [35]. This SU(2) anomaly can be falsified terrestrially by low frequency black body spectroscopy, as proposed in [49].

§ SU(2) YANG-MILLS THEORY

An SU(2) Yang-Mills theory can be described by a Lagrangian which is invariant under any local SU(2) transformation.[This is valid for any SU(N) Yang-Mills theory.] The pure Yang-Mills SU(2) Lagrangian $\mathcal{L}_{\rm YM}$ is given by

ℒ_YM = -1/4F^aμνF^a_μν= -1/2 tr F^μνF_μν .

The pure Yang-Mills SU(2) Lagrangian is invariant under Lorentz transformations, gauge symmetry, is CP-invariant and renormalizeable [87, 88]. Here, pure refers to a Yang-Mills theory which contains only gauge fields without matter fields. Nonetheless, gauge bosons can still become massive by dynamical symmetry breaking via the Higgs mechanism[Note, that you need a transforming scalar field for that.].
As in Eq. (<ref>), the field strength tensor $F_{\mu\nu}$ can be defined by the action of the commutator $[D_\mu,D_\nu]$ of the fundamental covariant derivative on a fundamental scalar $\phi$

F_μν ϕ=i/g[D_μ,D_ν]ϕ .

As previously in Maxwell electrodynamics, the covariant derivative $D_\mu$ for the SU(2) gauge is given in the fundamental representation by

D_μ =  1_2×2  ∂_μ- i g A_μ .

In contrast to the previous U(1) theory, calculating the commutator in Eq. (<ref>) in order to obtain the fieldstrength with the covariant derivative Eq. (<ref>)
results in an extra term $- i g [A_\mu,A_\nu]$:

F_μν = ∂_μA_ν- ∂_νA_μ- i g [A_μ,A_ν] ,

This term which is proportional to $g$ represents the self-interactions of the gauge field and is due to the non-abelian nature of the SU(2) theory. If the constraint of CP-invariance is relaxed, another term proportional to $F_{\mu\nu}^a \tilde{F}^{a \mu\nu}$ can be added to the Lagrangian. Here, $\tilde{F}^{a \mu\nu}$ is the dual fieldstrength tensor of the SU(2) theory which is defined by $\tilde{F}^{a \mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\mu\nu}$. (Anti)self-dual solutions of the Yang-Mills equations are particularly interesting, because they correspond to minimae of the Yang–Mills action [46]. By integrating the SU(2) Yang-Mills fieldstrength  $\mathcal{L}_{\rm YM}$, we obtain the SU(2) Yang-Mills action $\mathcal{S}_{\rm YM}$ which is given by

𝒮_YM = ∫d^4 x  ℒ_YM = -1/4 ∫d^4 x F^aμνF^a_μν .

In the next section, SU(2) theory will be applied to derive a modified temperature-redshift relation.

§ MODIFIED TEMPERATURE-REDSHIFT RELATION

SU(2)CMB is a high-redshift modification which connects to the standard cosmological model $\Lambda$CDM at low redshifts.

Redshift $z$ is similarly to the Doppler-effect the deviation of emitted to received wavelength. It is defined as

λ_obs/λ_emit=1+λ_obs-λ_emit/λ_emit=1+z ,
where $\lambda_{emit}$ is the wavelength at the time of emission and $\lambda_{obs}$ is the wavelength at the time of observation, compare <cit.>.

A spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe is assumed for the background evolution. Let the energy density of the deconfining phase of SU(2) Yang-Mills thermodynamics be $\rho_{\rm YM}$ and pressure $P_{\rm YM}$. Then, if we assume energy conservation
the following equation has to be considered:

dρ_YM /d a = -3/a(ρ_YM+P_YM) ,

where $a$ is the cosmological scale factor, normalised such that today $a(T_0) = 1$ and $T_0 = 2.725$ K, the current temperature of the CMB, compare [26]. If you solve for $a$, the solution of Eq. (<ref>) is

a=exp ( -1/3 log(s_YM(T)/s_YM(T_0)  ) ) ,

where $\mathpzc{s}_{\rm YM}$ denotes the entropy density [26]

s_YM≡ρ_YM+P_YM/T .

In figure <ref> you can see the multiplicative deviation
$\mathpzc{S}(z)$ from linear scaling which is given as

S(z)=(ρ_YM(0)+P_YM(0)/ρ_YM(z)+P_YM(z) )^1/3

As you can see, $T \approx 0.63$ for $T\gg T_0$. With this scaling function $\mathpzc{S}(z)$, we are able to define the temperature as a function of $z$:

T(z)= (1/4)^1/3 T_0  (z + 1),

and modify the temperature dependant radiation densities $\Omega_{\rm YM}(z)$ for the Yang-Mills plasma and $\Omega_{\nu}(z)$ for the neutrinos. This factor, $(1/4)^\frac{1}{3}$ is one of the main consequences of using an SU(2) group in order to describe the CMB photons instead of a U(1) group.

The function $\mathpzc{S}(z)$ of Eq. (<ref>) which indicates the deviation from the linear $T-z$ relation. The curvature at low z is due to the breaking of scale invariance in the deconfining SU(2) Yang-Mills plasma for $T \sim T_0$. Notice the rapid approach towards $(\frac{1}{4})^{1/3} \approx 0.63$ with increasing $z$, compare [26].

§ DARK SECTOR OF SU(2) YANG-MILLS THERMODYNAMICS

In [26] a dark sector was introduced as a deformation to $\Lambda$CDM which describes a
sudden transition from dark energy to dark matter at a redshift $z_{\rm p}$. Such a transition is required to
reconcile high-$z$ cosmology, which is changed compared to $\Lambda$CDM
as a result of an SU(2)-induced temperature-$z$ relation
[89] with $\Lambda$CDM at low $z$.
For the entire dark sector we have

(z+1)^3,      z<z_p

(z_p+1)^3,     z≥z_p

The density parameters are defined as:

Ω_r(z)= Ω_YM(z)+Ω_ν(z)

Ω_YM(z)= 4 π^2/30 T(z)^4/ρ_c
Ω_ν(z)= 21/8 (16/23)^4/3 π^2/30 T(z)^4/ρ_c

with $T_0=2.725\,$K$\cong 2.348 \times 10^{-4}\,$eV. The critical density $\rho_{c}$ is defined as

ρ_c=3 H_0^2/8 π M_p^2=1.76×10^-9  eV^4

In the simplified model, we assume only one depercolation at $z_p$. This means that we consider primordial dark matter $\omega_{\rm pdm}$ always as a part of the matter density $\Omega_m(z)$:

Ω_m(z)= ω_b + ω_pdm  /$\omega_{\rm b}$+$\omega_{\rm edm}$+ω_Λ+$\omega_{\rm pdm}$

for the dark energy density we have

Ω_Λ(z)=ω_Λ/$\omega_{\rm b}$+$\omega_{\rm edm}$+ω_Λ+$\omega_{\rm pdm}$ (z_p+1)^3 ,

For emergent dark matter we assume

(z+1)^3,      z<z_p

(z_p+1)^3,     z≥z_p

Assuming that curvature $\Omega _{k}$ is zero, $H(z)$ is then calculated as following:

H(z) = H_0 √(
Ω_r(z) +Ω_Λ+Ω_m(z))

Fitting the temperature fluctuations collected by the Planck collaboration (as seen in Fig. <ref>), we get for the initial densities $\omega_\Lambda$, $\omega_{\rm b}$, $\omega_{\rm edm}$ and $\omega_{\rm pdm}$ the values in Fig. <ref>. The SU(2)CMB fit as well as the $\Lambda$CDM model are compared in Fig. <ref>.

SU(2)CMB model fit (turquoise) in comparison to a $\Lambda$CDM fit (green). The 2015 Planck data points have been used for the fit [7].

How can we deduce the energy content, and more over the specific contributions to the total energy of dark matter, dark energy, baryons from temperature fluctuations at the beginning of the universe, as seen in Fig. <ref>?
The temperature deviations from the background temperature, which is about 2.725 K today, arise from small density and radiation-pressure variations of matter in the early before the universe turned transparent. Fitting the CMB is a highly non-trivial and complicated task, many coupled Boltzmann equations have to be solved simultaneously.

Best-fit cosmological parameters of the SU(2)CMB model in comparison to $\Lambda$CDM, compare [26].

Programs such as the Cosmic Linear Anisotropy Solving System (CLASS) [90] are often employed when solving these equations. Roughly speaking, the first peak in Fig. <ref> corresponds to the particle clouds which collapsed only once. It can give information about the total energy content of the universe. The second peak corresponds to the particle clouds which were able to collapse twice and the height of the peak gives information about the baryon content of the universe. All other peaks are linked to the dark matter content of the universe, compare [91].

Recombination describes the point in time when the Universe turned transparent for photons[Note, that gravitational waves were able to propagate freely before as well as neutrinos.]. According to $\Lambda$CDM this happened at a redshift $z \sim 1090$ (see section <ref> for a definition of redshift). The SU(2)CMB fit indicates a higher redshift, $z \sim 1715$. Before the recombination, the Universe was opaque because photons scattered of freely moving electrons. The Universe became transparent after those electrons combined with protons to form hydrogen atoms.

The Cosmic Microwave background emerges from the first photons which are able to freely move long distances in the Universe after recombination. Sound waves in the photon-baryon fluid, so called baryonic acoustic oscillation (BAO) lead to small density fluctuations in the fluid which translates into small ($\sim \mu$K) temperature fluctuations in the CMB. Those osculations can be are linear-combination of multipoles as seen in Fig. <ref>. An intuitive way of interpreting the multipoles is shown in Fig. <ref>, in the first panel (top,left) a randomly generated dipole is shown ($l=1$), followed by a quadrupole ($l=2$), an octupole ($l=3$), ($l=3\,\cdots\,12$).

Randomly generated multipoles from $l=1$ to $l=12$, top to bottom, left to right. The dipole ($l=1$) starts in the top corner left. Fitting the observed CMB temperature fluctuation enables one to fit those multipoles and conclude the total, energy density of the universe as well as the baryon and dark matter mass contributions, see Fig. <ref>. The Fig. was made by Ville Heikkilä, compare https://astro.uni-bonn.de/$\sim$kbasu/ObsCosmo/Slides2019/CMB_Part2.pdf.

Baryonic Acoustic Oscillation (BAO) are density waves in the photon-baryon fluid before recombination. Until the decoupling of matter and radiation, those sound waves were created by pressure and gravitational instability. Their maximum wavelength is the horizon size $r_H$ at the time of decoupling which can still be detected today by the anisotropies in the CMB spectrum and by the  distribution of galaxies in space.

After this very brief overview of the CMB angular spectrum we can now visualize the density parameters of SU(2)$_{\rm CMB}$, see Fig. <ref>. The normalized energy composition at different redshifts $z$ is shown alongside the size of the universe $r_H=1/H(z)$ in Mpc. The most notable difference to $\Lambda$CDM model is the spike at $z=53$ which is induced by the Heaviside function of Eq. (<ref>). At this point dark energy spontaneously depercolates into dark matter.
As one of the main results of this thesis we will argue in chapter <ref> how to link the leptonic sector with three axions species. A fourth, the one corresponding to the photon SU(2) theory, has not yet depercolated. Consequently, not only one point of depercolation at $z=53$ but three are also indicated in the graphic. We get those high redshifts simply by comparing the size of the Universe $r_H$ with the gravitational Bohr radius of the axion species which we link to the electron scale $\Lambda_e$. This will be discussed in more detail in section <ref>.

Cosmological model SU(2)$_{\rm CMB}$ of [26] (with parameter values fitted to the TT, TE, and EE CMB Planck power spectra and taken from column 2 of Table 2 of that paper) in terms of relative density parameters as functions of redshift $z$. $\Omega^\prime_\Lambda$ stands for dark energy, $\Omega^\prime_{m}$ for total matter (baryonic and dark), and $\Omega^\prime_{r}$ for radiation (three flavours of massless neutrinos and eight relativistic polarisations in a CMB subject to SU(2)$_{\rm CMB}$). The dotted red line represents the Hubble radius of this model. The redshifts of $e$-lump, $\mu$-lump, and $\tau$-lump depercolations are indicated by vertical lines intersecting the $z$-axis. Only e-lump depercolation is taken into account explicitly within the cosmological model SU(2)$_{\CMB}$ since at $z_{p,\mu}=40,000$ and $z_{p,\tau}=685,000$ the Universe is radiation dominated.

CHAPTER:  DARK MATTER AND ROTATION CURVES

A weak lensing image reaveals dark matter distribution which is shown in the coloured contours. Credit: Satoshi Miyazaki

• What is Dark matter?

• What motivates its existence?

• What could it be?

• What are axions?

§ INTRODUCTION

The idea of dark matter has been around for well over one-hundred years [92]. In 1933 the Swiss-American astronomer Fritz Zwicky found large discrepancies between estimates of dispersion velocities and measurements for the Coma galaxy cluster [93]. Therefore, he came to the conclusion that dark matter might be more abundant than ordinary matter.

The second key moment in the history of dark matter is the development of a spectroscope by Kent Ford in the 1960s which enabled him and Vera Rubin to measure rotation curves [92]. Rotation curves of galaxies plot the circular velocity of visible stars or gas in a galaxy over their corresponding radii to the center of the galaxy. Kent and Rubin observed the rotation curve of the galaxy M31 in 1970 [94]. To their surprise, the circular velocity did not show a reciprocal dependence on the radius $r$ as expected from Kepler's law, but was flat to the largest radius observed. This discrepancy between visible mass, based from hydrogen surface density profiles by e.g. Rogstad and Shostak in 1972 [95], and the total galaxy mass implied by rotation curves is nowadays one of the most convincing arguments for the existence of dark matter. This discrepancy is visualized for the galaxy UGC01230 in the figure <ref>, below.

Further experimental evidence for dark matter came from measurements of rotation curves. Instead of a declining velocity, as expected from Kepler's laws  (red, $\propto 1/\sqrt{r}$), the rotation curve has a relatively constant slope (blue, plotted Burkert profile of galaxy UGC01230). This suggest that galaxies are heavier than expected based on baryonic matter. More matter is inside galaxies than we see: matter which does not interact with light and hence is dubbed dark matter.

Before focusing on axions in general and Planck-scale axions in particular, the main dark matter candidate of this thesis, we would like to briefly mention relatively new advances (within the last two decades) in search for dark matter. Namely, the cosmological standard model $\Lambda$CDM fit to the Planck satellite data, which as the name suggest accounts for cold dark matter, and the bullet cluster which is widely assumed to be a direct proof of the existence of dark matter.

The Lambda Cold Dark Matter (LCDM or $\Lambda$CDM) model incorporates additionally to the Cold Dark Matter model – as published in 1984 by George Blumenthal, Joel Primack, and Sandra Faber [96] – the cosmological constant $\Lambda$, which is associated with dark energy. The model, which is commonly also referred to as the standard cosmological model, has only the assumptions of spatial isotropy and homogeneity of the cosmos on the right scales for all observers and general relativity. Friedmann-Lemaître-Robertson-Walker Cosmology contains three major components for the present epoch, the aforementioned dark energy, cold dark matter and baryonic matter. A fit to the Cosmic Microwave Background as seen in Fig. <ref> reveals their respective densities as seen in the table below. The $\Lambda$CDM fit densities are shown in Fig.<ref>.

Arguably the strongest argument for the existence for dark matter comes from weak gravitational lensing. As shown on the right-hand side of Fig. <ref>, there are two former galaxy clusters which are now, after a collision, part of the galaxy cluster 1E 0657-56 [97]. This galaxy cluster is also known as the “Bullet Cluster” because of the shape of the galaxy cluster on the right-hand side in Fig. <ref> resembles a bullet. The hot gas (red in Fig. <ref>) was detected by the Chandra telescope in the X-ray regime. The dark matter content in this galaxy cluster which is indicated in blue in Fig. <ref>, was reconstructed using weak gravitational lensing.

Gravitational lensing describes the distortion of light passing by a strong source of spacetime distortion. Such distortions can be caused by galaxy clusters, massive objects in galactic centers and black holes. They are predicted by General Relativity and can be used to estimate masses of the sources of those spacetime distortions. Special cases such as the multiplying of a light source is called Einstein's cross, as depicted below in Fig. <ref>.

A gravitational lensing image (white rectangle) shows the quadrupling of the quasar QSO 2237+0305 (which started at a source S and is observed by an observer O) by a nearby galaxy, compare [98]. Sometimes the multiplying is referred to as Einstein cross.

So far, the bullet cluster represents the strongest evidence for the existence of dark matter: The dominant mass portions of the merging galaxy clusters is ahead of the baryonic mass in their trajectories. This can  easily be explained by dark matter, which does not interact with the visible matter other than by gravitational interaction. They fly right through the hot gas lumps. This spatial offset of the baryonic mass distribution can not be explained by modified theories of gravity [99].

An image by the Chandra telescope of the bullet cluster, 1E 0657-558, overlayed with a weak gravitational image. The white bars in both pictures indicates a distance of 200 kpc, visible light is shown on the left picture, the hot gas detected by X-rays (red) and the dark matter distribution (blue) is shown on the right [99].
The dark matter distribution was reconstructed using gravitational lensing which indicate the mass distribution determined by weak gravitational lensing  (green lines) [97, 99].

§ AXIONS

Although dark matter in some form or another has been around for at least one-hundred years, there are plenty possible candidates covering 80 orders of magnitude (in eV), compare figure <ref>.
Unfortunately, it is not possible to give a comprehensive overview of all of them in
this framework. We will focus on axions in general and on Planck-scale axions in particular (left green corner in figure <ref>).

Sketch (not to scale) of the huge range of possible DM models that have been conceived. They span many orders of magnitude
in mass, with DM represented by very distinct phenomena, ranging from new elementary particles to black holes, compare figure 1 in [100].

The idea of axions emerged as a byproduct of an attempt to solve the strong CP-problem in QCD. The strong CP-problem arises due to the non-trivial vacuum structure of non-Abelian gauge theories: Although the Standard model SU(3)$\times$SU(2)$\times$U(1) has no thermal phases, instantons and other topologically charged field configurations generate a non-trivial vacuum structure of the gauge theory [101] which allows for (effective) CP-violating interactions of the strong and the weak force [102]:

ℒ_CP viol.= θ_strong  α_s/8 π F^μν_aF_aμν+
θ_weak  α_2/8 π W^μν_aW_aμν .

Here $F^{\mu\nu}_a$ and $W^{\mu\nu}_a$ are the non Abelian gauge field strengths and $\widetilde{F}_{a\mu\nu}$, $\widetilde{W}_{a\mu\nu}$ their duals; $\alpha_s$ is the coupling constant of the strong force as $\alpha_2$ is the coupling constant of the weak force. The weak vacuum angle $\theta_{\textrm weak}$ can be rotated to zero since the electroweak theory is chiral [103]. The CP-problem arises from the angle $\theta_{\textrm strong}$, which cannot be rotated away in the same fashion. More precisely, $\theta_{\textrm strong}$ gets modified due to the chiral rotation to minimize $\theta_{\textrm weak}$. The 'physical' angle is $\bar{\theta}= \theta_{\textrm strong} + \text{Arg det M}$ where M is the quark mass matrix [102]. Therefore, the CP-violating term reads:

ℒ_QCD⊃θ̅  α_s/8 π F^μν_aF_aμν .

If $\bar{\theta}$ is of order one, the strong force would strongly violate CP-symmtery. However, measurements of the electric dipole moment of the neutron show no indications fur such a CP-violation [102]: They imply a very small $\bar{\theta}$ of order $\sim 10^{-10}$. So the strong CP-problem boils down to the question of why $\bar{\theta}$ is so small?

The idea by Helen Quinn and Roberto Peccei was to remove $\bar{\theta}$ dynamically by introducing a new, global chiral symmetry to the Standard Model, the so called, U(1)$_{\textrm PQ}$ symmetry. Similar to the electroweak theory with gauge group SU(2)$\otimes$U(1) where the chiral nature allows to rotate $\theta_{\textrm weak}$ away, the new chiral symmetry U(1)$_{\textrm PQ}$ allows to relax $\bar{\theta}$ to zero by dynamically and explicitly breaking it [104, 105, 102].
Frank Wilczek [106] and Steven Weinberg [107] pointed out how the (pseudo) Nambu-Goldstone boson, the so axion, of dynamical U(1)$_{\textrm PQ}$ breaking couples to $F^{\mu\nu}_a\widetilde{F}_{a\mu\nu}$.

§ WHAT ARE ROTATION CURVES?

Rotation curves plot the averaged velocity of visible matter, i.e. stars which rotate around a galactic center, over their distances from the galactic center. The discrepancy between the classically expected rotation curve (blue) and the actually measured ones (green) in figure <ref> can be explained by postulating dark matter. Therefore, if one accepts the premise that dark matter exists, we can swap the argumentation and use rotation curves as a tool to look at the dark matter content of a given galaxy.

As a warm-up we first look at a vacuum solution of Einstein's equations, the Schwarzschild solution <ref>. We compare an exact solution of the orbit equation in this background by utilizing elliptic functions with a post-Newtonian approximation, section <ref>. In the next chapter we will focuse on the nonrelativistic limit and fit rotation curves to data from the SPARC library [44].

§ ROTATION CURVES FOR THE SCHWARZSCHILD SOLUTION

Based on an exact solution to the orbit equation for the bound trajectory of a test mass in the Schwarzschild background, obtained in [108] in terms of a Weierstrass elliptic function, we numerically compute rotation curves (mean tangential velocity $\bar{v}_t$ as a function of proper distance $s_r$ to the horizon) in dependence of orbital eccentricity $\epsilon$. As it should, for $s_r\gg r_s$ the Newtonian behaviour $\bar{v}_t\propto 1/\sqrt{s_r}$ is reproduced.

In his paper [108], G. Scharf obtained exact Schwarzschild geodesics in terms of elliptic functions. Building on his work, the aim of the present section is to derive rotation curves, that is, the dependence of the mean tangential velocity (averaged over one pseudo orbit) of a test mass on the proper distance to the Schwarzschild horizon, including the strong-field regime close to this horizon.

This could be of observational interest when the effects of mass distributions close to the gravitational center of a given spiral galaxy need to be disentangled from the strong-field effects [109]. Recall that, based on an estimate of luminous matter located in the central region, the observation of a flattening of rotation curves in spirals far away from the center serves as  an indication for the existence of a dark-matter halo [110, 97].

The Schwarzschild metric in canonical coordinates is given as [111]:

\begin{equation}
ds^2=\frac{r-r_s}{r} c^2 dt^2-\frac{r}{r-r_s} dr^2-r^2(d\theta ^2+sin^2\theta\, d\phi^2)
\label{SchSch}
\end{equation}

where the Schwarzschild radius $r_s$ is given as $r_s = {2 \,GM}/{c^2} $. With $G$ being the gravitational constant, $M$ the mass, and $c$ the speed of light. Usually, when discussing relativistic effects in the orbit of a test particle in the background of eq. <ref>, one expands the metric coefficients in powers of $r/r_s$. Thanks to the knowledge on the exact solution $r(\phi)$ to the orbit equation ([108], eq. 2.11)

\begin{equation}
\frac{dr}{d\phi}= \sqrt{\frac{E^2-m^2}{L^2}\,r^4+\frac{m^2}{L^2}\,r_s r^3-r^2+r_s r}	\equiv \sqrt{f(r)}
\label{orbit}
\end{equation}

orbital parameters such as the perihelion shift could be obtained up to quadratic order in $r/r_s$ very economically by expanding this solution. Here $E$ and $L$ are constants of motion, energy and angular momentum and $m$ is the mass of the test-particle.
Here, we would like to exploit this knowledge to derive exact rotation curves in dependence of eccentricity $\epsilon$.

The approach based on Weierstrass functions was used in [108]. According to ([108], eq. 2.13) of this work, the radial coordinate $r(\phi)$ of a test particle in the background of the geometry governed by eq. <ref> is

\begin{equation}
r(\phi)= r_1 + \frac{f'(r_1)}{4\wp(\phi; g_2, g_3)-f''(r_1)/6}
\label{eq:refname1}
\end{equation}

Where $f(r)$ is a quartic of the form ([108], eq. 2.16):

\begin{equation}
f(r) = a_0 \, r(r - r_1)(r - r_2)(r - r_3)
\end{equation}

$a_0$ can related to the energy $E$, angular momentum $L$ and the mass of the test-particle $m$ as $a_0 = (E^2-m^2)/L^2$ ([108], eq. 2.18). The radial points $r_1$, $r_2$, and $r_3$ are zeros of this quartic where $r_1$ can be interpreted as perihelion, $r_2$ as aperihelion, and $r_3$ can be expressed as a combination of $r_1$, $r_2$ and $r_s$ ([108], eq. 2.21), as well as the invariants $g_2$ and $g_3$ ([108], eq. 2.22 f.)

The first and second derivatives of the quartic are given as:

\begin{align}
f'(r_1) &= -\frac{r_1 (r_1 - r_2)\cdot(r_2 - r_3)}{r_2 r_1 + r_2 r_3 + r_3 r_1} \\ \vspace{1cm}
f''(r_1) &= \frac{r_1 (r_1-r_2) + r_1 (r_2-r_3) + (r_1-r_2)\cdot(r_2- r_3)}{r_2 r_1 + r_2 r_3 + r_3 r_1},
\label{eq:derivatives}
\end{align}

We get to the final form of $r(\phi)$ only depending on the perihelion $r_1$, $r_2$ and the angle $\phi$. However, it is convenient to define $r_2$ via the eccentricity $\epsilon$ as $r_2 = r_1 (1-\epsilon)/(1+\epsilon)$. In order to obtain a rotation curve we need to define the tangential velocity $\Bar{v_t}$ of the test-particle averaged over one pseudo orbit, as a function of the averaged distance $\Bar{s}$. The tangential velocity is:

\begin{align*}
v_t = r(\phi(\tau)) \, \frac{d\phi}{d\tau},
\end{align*}

where $r(\phi)$ is a solution of the orbit equation and $\phi(\tau)$ is a solution of the geodesic equations in dependence of the affine parameter $\tau$. For readability this dependence is notationally suppressed in the following. Then the averaged tangential velocity $\bar{v_t}$ is obtained by:

\begin{align}
\bar{v_t} &= \frac{1}{T} \int_{0}^{T} d\tau \, v_t\\
&=  \frac{r_s}{T} \int_{0}^{2\pi} d \phi \, r(\phi)
\label{eq:vbar}
\end{align}

With $T$ the eigentime of the orbit. The averaged distance $\Bar{s}$ is:

\begin{align*}
\bar{s} &= \frac{1}{2 \pi} \int_{0}^{2 \pi} d\phi \, s_r(r(\phi))
\end{align*}

$s_r$ is the average proper distance to the horizon, which is defined as:

\begin{align*}
s_r(y) &= \int_{r_s}^{y} \frac{dy^\prime}{\sqrt{1-\frac{r_s}{y^\prime}} }
%= y-r_s\left(\frac{1}{2}-\log\left(4 \frac{y}{r_s} \right) \right)-\frac{3 r_s^2}{8 y}+O\left(\left(\frac{r_s}{y} \right)^2 \right),
= y \sqrt{1-\frac{r_s}{y}}+\frac{r_s}{2} \log \left(\frac{1+\sqrt{1-\frac{r_s}{y}}}{1-\sqrt{1-\frac{r_s}{y}}} \right) \\
%&= y-r_s\left(\frac{1}{2}-\log\left(4 \frac{y}{r_s} \right) \right)-\frac{3 r_s^2}{8 y}+O\left(\left(\frac{r_s}{y} \right)^2 \right),
\label{eq:refname4}
\end{align*}

$\bar{v}(\bar{s})$ can now be calculated by inverting $\bar{s}(r(\phi))$ and plugging it in equation <ref>. The only thing missing is $T$ which
needs to be found by normalization, which here means comparing the general expression

\begin{equation}
T=  \frac{1}{L} \int_{0}^{2\pi} d\phi \, r(\phi)^2, \hspace{0,5cm} L=m \sqrt{ \frac{r_2 r_1 + r_2 r_3 + r_3 r_1}{r_2 + r_1 + r_3} \cdot r_s}
\ \end{equation}

where $L$ is the angular momentum ([108], eq. 2.8 & 2.19), with the physical orbital periods. In this case, normalization gets rid of the factor $2 \pi\, m$. Plugging $T$ in eq. (<ref>) gives:

\begin{align}
\bar{v_t} &=2 \pi\, m \sqrt{\frac{r_2 r_1 + r_2 r_3 + r_3 r_1}{r_2 + r_1 + r_3} \, \cdot r_s} \frac{\int_{0}^{2 \pi} d \phi \, r(\phi) }{\int_{0}^{2 \pi} d\phi \, r(\phi)^2}
\end{align}

Because of the perihelion motion we have to integrate over $4 \omega$ instead of $2\pi$, where $\omega$ is the real half-period of the $\wp$-function which is given by ([108], eq. 3.12):

\begin{align*}
\omega = \pi  \left( 1+\frac{3 r_s}{4}\, \frac{r_1 + r_2}{r_1 r_2} + O(r_s^2)\right)
\end{align*}

Note, that here the post-Newtonian approximation has already been used. Together with the normalisation our tangential velocity finally is:

\begin{align}
\bar{v_t} &= \sqrt{\frac{r_2 r_1 + r_2 r_3 + r_3 r_1}{r_2 + r_1 + r_3}\cdot r_s} \frac{\int_{0}^{4\omega} d \phi \, r(\phi) }{\int_{0}^{4\omega} d\phi \, r(\phi)^2}
\end{align}

Calculating the averaged tangential velocity $\Bar{v_t}$ as a function of the averaged distance $\Bar{s}$ gives us the rotation curve of a test-particle. Figure <ref> shows the rotation curve in natural units ($G=1, c=1$) for different eccentricities $\epsilon \in \{0-0.9\}$.

Rotation curve for eccentricity from for different eccentricities $\epsilon \in \{0-0.9\}$ (red to green), $r_s=1$ for an exact solution of the Schwarzschild geodesics with elliptical functions.

§ ROTATION CURVES FOR A POST-NEWTONIAN APPROXIMATION

Recall that the Weierstrass functions was used in the previous section and originally in [108] in order to obtain a function for the orbit, compare <ref>

\begin{equation}
r(\phi)= r_1 + \frac{f'(r_1)}{4\wp(\phi; g_2, g_3)-f''(r_1)/6}
\end{equation}

The first and second derivative $f'(r_1),f''(r_1)$ can be expressed as <cit.>

\begin{align}
f'(r_1) &= 2 r_1 \left(\frac{\epsilon}{1+\epsilon}-\frac{r_s}{r_1}\frac{3\epsilon+\epsilon^2}{(1+\epsilon)^2}\right) \\ \vspace{1cm}
f''(r_1) &= 2  \left(\frac{1-5\epsilon}{1+\epsilon}+3\frac{r_s}{r_1}\frac{1-4 \epsilon -\epsilon^2}{(1+\epsilon)^2}
\right)\,,
\end{align}

The Weierstrass function $\wp$ is given in terms of theta functions in order to account for the  relativistic corrections for $r(\phi)$, compare <cit.>

\begin{equation}
\wp(\phi)= e_2+\frac{\pi^2}{4 \omega^2}\left(
\frac{\theta_1'(0)}{\theta_3(0)}\frac{\theta_3(\phi)}{\theta_1(\phi)}
\right)^2\,,
\end{equation}

where $\phi$ is given by $\phi=\frac{\pi}{2\omega}\varphi$ and the theta functions are given by <cit.>

\begin{align}
\theta_1(z,q) & = 2 q^{1/4} (\text{sin}(z)-q^2 \text{sin}(3 z)+ q^6 \text{sin}(5 z) - \dots)\\
\theta_2(z,q) & = 2 q^{1/4} (\text{cos}(z)+q^2 \text{cos}(3 z)- q^6 \text{cos}(5 z) + \dots)\\
\theta_3(z,q) & = 1+2 q (\text{cos}(2 z)+q^3 \text{cos}(4 z)- q^8 \text{cos}(6 z) + \dots)\\
\theta_4(z,q) & = 1-2 q (\text{cos}(2 z)-q^3 \text{cos}(4 z)+ q^8 \text{cos}(6 z) - \dots)\,.
\end{align}

In this context, $q$ is called a Nome [108] and can be expressed as <cit.>

\begin{equation}
q = k^2/16+ 8(k^2/16)^2+\dots\,.
\end{equation}

Since $k$ is small the series is converging <cit.>. This leads to the following Weierstrass function $\wp$ [ibid.]

\begin{equation}
\wp(\phi)=  \frac{f''(r_1) }{24} + \frac{1+\epsilon- 2 \epsilon \text{sin}^2(\phi)}{4 (1+\epsilon) \text{sin}^2(\phi)} \left(
1+\frac{r_s}{r_1}\frac{1}{1+\text{cos}(2 \phi)}\left(
\right)
\right)+\mathpzc{O}(\frac{r_s}{r_1})^2
\end{equation}

Substituting this into <ref>, one obtains the orbit given by

\begin{equation}
\text{r}(\theta) = \frac{r_1 (\epsilon +1)}{\epsilon  \cos \left(\frac{2 \theta }{\alpha}\right)+1}+\frac{ r_s \left(2 \epsilon  \sin ^2\left(\frac{\theta }{\alpha}\right)\right) }{\epsilon  \cos \left(\frac{2 \theta }{\alpha}\right)+1}\left(\frac{-\frac{2 \epsilon  (\epsilon +3) \sin ^2\left(\frac{\theta }{\alpha}\right)}{\epsilon +1}+\frac{1}{2} \epsilon  \left(1-\cos \left(\frac{\theta }{\alpha}\right)\right)+3}{\epsilon  \cos \left(\frac{2 \theta }{\alpha}\right)+1}-\frac{\epsilon +3}{\epsilon +1}\right)
\end{equation}

Rotation curve for eccentricity for different eccentricities $\epsilon \in \{0-0.9\}$, $r_s=1$ in a post-Newtonian approximation.

Where $\alpha$ is defined as:

\begin{align*}
\alpha = 2 \left(\frac{3 r_s (r_2+r_1)}{4 r_1 r_2}+1\right)
\end{align*}

Which results in the rotation curves as seen in figure <ref>. Interestingly, for high eccentricities around $\epsilon = 0.9$ , the rotation curve seems to be steeper, i.e. crossing over RCs with less eccentricity at the center of the galaxy. However, this effect might be explained by insufficient numerical accuracy close to the center for high eccentricities.

CHAPTER:  AXION MASS EXTRACTIONS

Frequency distribution of 80 axion masses $m_{a,e}$, extracted from the Burkert-model fits of $M_{200}$ to the RCs of galaxies with a $\chi^2/\text{d.o.f.}<1$. The maximum of the smooth-kernel distribution (solid, blue line) is $m_{a,e}=(0.65\pm0.4)\times10^{-23}$ eV (red, vertical line).

• Which axion mass do we extract?

• What are properties of Planck scale axions?

• Which model fits most galaxies?

• Do the models agree with each other?

§ INTRODUCTION

Despite the outstanding success of the standard model of Cosmology which is based on cold dark matter (CDM) in explaining large-scale structures, cold dark matter has difficulties predicting small-scale structures [114]. Such small-scale problems are the prediction of more satellite dwarf galaxies around a host galaxy then currently observed, this is also known as the missing satellite problem[Similarly to the missing baryon problem and the missing intermediate massive black holes, frequently a detection is proposed and afterwards refuted. The author is currently not aware of a resolution in either one of those three problems.] [115]. The so called cusp-core problem is another small-scale problem of cold dark matter [116]: The dark matter density is predicted to increase steeply towards the center of a galaxy with $\rho \sim r^\alpha$ and $\alpha \in \{-1,-1.5\}$ [ibid.]. This behaviour is called a cuspy slope, however, a flatter profile is often observed [114]. Such flat behaviour could be explained by a core or possible also by baryonic physics [ibid.]. Nonetheless, the latter has not yet been observed in dwarf galaxies where the problem arises to begin with and seems to contradict the surprisingly high star formation efficiency in the center of galaxies (which is often referred to as the young star problem). The difference, between a cuspy and a core profile can be observed in Fig. <ref>. The core-like behaviour is shown in the red line, a characteristic for the Soliton Navarro–Frenk–White profile (SNFW), while a cuspy slope can be observed in orange for the unmodified Navarro–Frenk–White profile (NFW).

One framework which aims to resolve the aforementioned problems is so called fuzzy dark matter. The idea is that fuzzy dark matter consists out of non-thermal, non-relativistic ultralight bosons which form a Bose-Einstein condensate and behave like cold dark matter on large scales, i.e. larger than the de Broglie wavelength of usually 1 kpc [117]. However, on small scales their wavelike dynamics behaves different compared to cold dark matter, and one might argue that they fit better to the observed rotation curves. Typical values of axion masses in this framework lie between $m_a \sim 10^{-21}$ eV and $m_a \sim 10^{-22}$ eV [118]. In this chapter we will reproduce Matos et al. result of  $m_a \sim 5 \times 10^{-24}$ eV [119]. However, several studies doubt the feasibility of a dark matter model with a lower dark matter mass than $m_a \le 10^{-21}$ eV [117, 120]. Especially Ly$\alpha$ forest power spectra suggest that such a small mass would be indistinguishable from cold dark matter [120, 121]. At first glance this would be unappealing since this would imply that fuzzy dark matter loses the desired feature of a core like behaviour. However, we argue that in the here presented three component dark matter model this constraint is not valid. The galactic core might form from two different axion types which correspond to the screened leptonic Yang-Mills scales in association to the muon and tau mass. Their masses of $m_{a,\mu} \sim 2.9 \times 10^{-19}$ eV and $m_{a,\tau} \sim 8.2 \times 10^{-17}$ eV are well above the constraint. Moreover, the scenario where a $\tau$-lump condensate gravitationally collapses, consumes the $\mu$-lump and becomes the central massive object of a given galaxy is still feasible. In this case the core of a galaxy is given by a complex succession of dark matter physics. This might also mitigate the cold dark matter behaviour on small scales by the e-lump with a mass of only $m_{a,e} \sim 6.7 \times 10^{-24}$ eV, since an explicit core already provides the desired core-like rotation curve behaviour.

One way of extracting a mass estimate for dark matter due to fitting of rotation curves was discussed in the previous chapter. In this chapter we determine a Planck scale axion mass in the context of SU(2)$_{\rm CMB}$ cosmology, which was introduced in chapter <ref>. Therefore, we use low surface brightness galaxies of the SPARC catalogue [44]. Low surface brightness galaxies are especially interesting for us, because they can assumed to be dominated by dark matter. Rotation curves are fitted with four different models starting from section <ref>:

* The Soliton Navarro–Frenk–White profile (SNFW)

* The Navarro–Frenk–White profile (NFW)

* The pseudo-isothermal model (PI)

* and the Burkert model

Some results of this and the following chapter will be published soon. In particular, a preprint of the SNFW profile and a comparison to the Burkert model can be found under [122].

§ PROPERTIES OF A PLANCK-SCALE AXION

In this section, we would like to convey some basic facts about selfgravitating Planck-scale axion condensates: lumps. We start with the observation in [123] that ultralight bosons necessarily need to occur in the form of selfgravitating condensates in the cores of galaxies. Halos of axion particles were formed around these cores, because they were separated in the course of nonthermal depercolation. Such a halo reaches out to a radius of $\sim r_{200}$ where its mass density starts to fall below 200 times the critical cosmological energy density of the spatially flat FLRW Universe. A key concept in describing such a fuzzy-dark-matter (FDM) system, e.g. a lump, is the gravitational Bohr radius $r_B$ defined as

r_B≡M_P^2/M m_a^2 ,
where $M$ is the mass of the lump which should coincide with the virial mass, say $M_{200}$. We use two FDM models of the galactic mass density $\rho(r)$ to describe low-surface-brightness galaxies and to extract the axion mass $m_a$: The Soliton-NFW model, see [124] and references therein, and the Burkert model [125, 126]. Let

denote the reduced Compton wavelength and

the mean distance between axion particles within the spherically symmetric lump of mean dark-matter mass
density $\bar{\rho}_{\rm }$. One has

ρ̅_∼M/4π/3r_B^3 .
A finite-extent lump self-consistently forms if the reduced Compton wavelength $\lambdabar_C$ - the correlation length in the condensate of free axion particles at zero temperature - is not very far from the gravitational Bohr radius $r_B$. More precisely, we can state that

r_B/_C=κ(δ) ,
where $\kappa$ is a smooth dimensionless function of its dimensionless argument $\delta\equiv m_a/M_P$ with the property that $\lim_{\delta\to 0}\kappa(\delta)<\infty$. This is because the lump mass $M$,
which enters $r_B$ via Eq. (<ref>), is itself only a function of the two mass scales $m_a$ and $M_P$ for an isolated selfgravitating condensate of otherwise free particles of mass $m_a$. If $\delta$ turns out to be sufficiently smaller
than unity, which will always be the case in what follows, we can treat the right-hand side of Eq. (<ref>) as a universal constant. In practice, we will in Secs. <ref> and <ref> derive from the match of dark-matter halos of low surface-brightness galaxies with heuristic mass density models that $\kappa\sim 314$.

Eq. (<ref>) together with Eqs. (<ref>), (<ref>), and (<ref>) imply for the mass $M$ of the lump

M=1/κM_P^3/Λ^2 .
Eq. (<ref>) is important because it predicts that the ratios of lump masses are solely determined by the squares of the ratios of the respective Yang-Mills scales or, equivalently [2], ratios of charged lepton masses $m_e$, $m_\mu$, and $m_\tau$. One has

M_τ/M_μ   =   (m_τ/m_μ)^2∼283 ,

M_μ/M_e   =   (m_e/m_μ)^2∼2.3×10^-5 ,

M_τ/M_e   =   (m_e/m_τ)^2∼8.3×10^-8 .Moreover, Eqs. (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>)
fix the ratio $\xi\equiv\frac{d_a}{\lambdabar_C}$ as

ξ=(4π/3)^1/3(κΛ/M_P)^4/3 .
Since $\Lambda\ll M_P$ we have $\xi\ll 1$, and therefore a large number of axion
particles are covered by one reduced Compton wave length. This assures that Planck-scale
axions always occur in condensed form. A thermodynamical argument for the necessity of axion condensates throughout the Universe's expansion history is given in Sec. <ref>. In [123] the non-local and non-linear
(integro-differential) Schrödinger-equation, obtained from a linear Schrödinger equation and a Poisson equation for the gravitational potential [127] and governing the lump, was analysed. An excitation of such a lump in terms of its wave-function $\psi$ containing radial zeros
was envisaged in [123]. Here instead, we assume the lump to be in its ground
state parameterised by a phenomenological mass density
$\rho(r)\propto |\psi|^2(r)>0$. Finally, Eq. (<ref>) together with Eqs. (<ref>) and (<ref>) yield for the
gravitational Bohr radius

r_B=κM_P/Λ^2 .

In this section, we extract the axion mass $m_{a,e}$ from observed RCs of low-surface-brightness galaxies which fix the lump mass $M_e$ and a characterising length scale – the gravitational Bohr radius $r_{B,e}$. This, in turn, determines the Yang-Mills scale $\Lambda_e$ associated with the lump. We analyse RCs from the
SPARC library [44].

§ ANALYSIS OF ROTATION CURVES (RCS)

To investigate, for a given galaxy and RC, the underlying spherically symmetric
mass density $\rho(r)$ it is useful to introduce the orbit-enclosed mass
M(r) = 4π∫_0^rdr'  r'^2ρ_(r') .

Assuming virialisation, spherical symmetry, and Newtonian gravity the orbital
velocity $V(r)$ of a test mass (a star) is given as

\begin{align}
V(r) = \sqrt{\frac{G M(r)}{r}}\,,
\label{eq:Vcirc}
\end{align}

where $M(r)$ is defined in Eq. (<ref>), and $G\equiv M_P^{-2}$ denotes Newton's constant. The lump mass
$M$ is defined to be $M_{200}\equiv M(r_{200})$. Here $r_{200}$ denotes the virial radius defined such that

\begin{align}
\label{NFWrange}
\rho_{\rm }(r_{200})=200\,\frac{3\,M_P^2}{8\pi}H_0^2\,,
\end{align}

where $H_0$ is the Hubble constant, and $\lambda_{\rm deB}=\lambda_{\rm deB}(r)$ indicates the de-Broglie wavelength of an axionic particle for $r_e<r<r_{200}$ where the NFW model applies. Note that within the core region $r<r_e$ the correlation length in the condensate is given by the reduced Compton wave length $\lambdabar_C=1/m_a$.

We analyse the RCs from the
SPARC library [44]. For a given galaxy, the value of $r$ in Eq. (<ref>) is assumed to coincide with the largest radius $r_B$ at which the orbital velocity $V$ (circular orbits) was determined observationally. This is only a weak constraint on generality because
of the logarithmic dependence on $r$ of the integral in Eq. (<ref>).

§ SOLITON NAVARRO–FRENK–WHITE PROFILES

For an extraction of $m_{a,e}$ and therefore the associated Yang-Mills scale governing the mass of a lump according to Eq. (<ref>), we use the Soliton-Navarro-Frenk-White (SNFW) model. The consistency of the with Eq. (<ref>) gained knowledge about the Bohr radius is tested with virial mass extractions of the NFW, PI and the Burkert model. The mass-density profile of the NFW-part of the SNFW-model is
given as [128]

\begin{align}
\rho_{{\rm NFW}}(r) = \frac{\rho_s}{\frac{r}{r_s}(1+\frac{r}{r_s})^2}\,,
\label{eq:RhoNFW}
\end{align}

where $\rho_s$ associates with the central mass density, and $r_s$ is a scale radius which represents the onset of the asymptotic cubic decay in distance $r$ to the galactic center. Note that profile $\rho_{\rm NFW}$ exhibits an infinite cusp as $r\to 0$ and that the orbit-enclosed mass $M(r)$ diverges logarithmically with the cutoff radius $r$ for the integral in Eq. (<ref>). In order to avoid the cuspy behavior for $r\to 0$, an axionic Bose-Einstein condensate (soliton density profile) is assumed to describe the soliton region $r\le r_e$. From the ground-state solution of the Schrödinger-Poisson system for a
single axion species one obtains a good analytic description of the soliton density profile as [119]

\begin{align}
\rho_{{\rm sol}}(r) = \frac{\rho_c}{(1+0.091(r/r_c)^2)^8}\,,
\label{eq:SolitonNFW}
\end{align}

where $\rho_c$ is the core density [129]. On the whole, the fuzzy dark matter profile can than be approximated as

\begin{align}
\rho_{{\rm SNFW}}(r) = \Theta(r_\epsilon-r)\rho_{{\rm sol}}+ \Theta(r-r_\epsilon)\rho_{{\rm NFW}}\,.
\label{eq:SolitonNFWwhole}
\end{align}

Using Eqs. (<ref>), (<ref>), and (<ref>),
we obtain the orbital velocity $V_{\rm SNFW}$ of the
SNFW model <cit.> which is fitted to observed RCs.

\begin{equation} \label{eq:SolitonNFWvelocity}
\begin{split}
V_{{\rm SNFW}}(r) & =  \Theta(r_\epsilon-r)\,V_{{\rm sol}}(r) \\
& + \Theta(r-r_\epsilon)
\left(
V_{{\rm sol}}(r_\epsilon)-V_{{\rm NFW}}(r_\epsilon)+V_{{\rm NFW}}(r)
\right)\,.
\end{split}
\end{equation}

Where $V_{{\rm sol}}(r)$ is given in Eq. (<ref>)

\begin{equation} \label{eq:Vsoliton}
\begin{split}
V_{{\rm sol}}(r) & =   \frac{r_c^3 \rho_c}{(r^2 + 10.9 r_c^2)^7} (1.9 r^{13} r_c + 1.4 \times 10^2 r^11 r_c^3 + 4.4 \times 10^3 r^9 r_c^5\\
& +   7.5 \times 10^4 r^7 r_c^7 + 7.5 \times 10^5 r^5 r_c^9 +
4.3\times 10^6 r^3 r_c^{11} - 3.4\times 10^6 r r_c^{13} \\
&   + (5.8\times 10^{-1} r^{12} +  4.5 \times 10 r^{12} r_c^2 + 1.4\times 10^3 r^{10} r_c^4 +  2.7\times 10^4 r^8 r_c^6 \\
& + 2.9\times 10^5 r^6 r_c^8 +  1.9\times 10^6 r^4 r_c^{10} +
7.2\times 10^6 r^2 r_c^{12} +  1.1\times 10^7 r_c^{14})\\ & \arctan{[(3\times 10^{-1} r)/ r_c]})\,,
\end{split}
\end{equation}

and $V_{{\rm NFW}}(r)$ is given by Eq. (<ref>) as

\begin{equation} \label{eq:VNFW}
\begin{split}
V_{{\rm NFW}}(r) & =  \frac{(0.99 r_c^{16} r_\epsilon (r_\epsilon +  r_s)^2 (r_s/(r + r_s) + \log{[r + r_s]}) \rho_c)}{(r_c^2 +  0.091 r_\epsilon^2)^8}\,.
\end{split}
\end{equation}

Fig. <ref> shows an example for the characteristic density profile $\rho_{{\rm SNFW}}(r)$ of Eq.  (<ref>), and the orbit velocity $V_{{\rm SNFW}}(r)$ of Eq. (<ref>) for the galaxy NGC0300. As intended, the up to $r_\epsilon$ the density profile is relatively constant. The typical $r^{-3}$ behaviour is given by the NFW density profile.

An example for the characteristic density profile $\rho_{{\rm SNFW}}(r)$, Eq. (<ref>), of galaxy NGC0300 is given on the left-hand side. On the right-hand side the corresponding orbit velocity $V_{{\rm SNFW}}(r)$,
Eq. (<ref>), of the same galaxy is fitted to the data as obtained by the SPARC library [44]. The scale radius $r_s$ is left indicated by a vertical red, dotted line and right with a green arrow.
The transition radius from soliton to the NFW radius at $r_\epsilon$ is left indicted by the blue dotted region (left) and on the right hand side with a yellow arrow. The core radius $r_c$ is indicated by the orange arrow).

This determines the parameters $r_\epsilon$, $r_s$, and $\rho_c$. The density $\rho_s$ relates to these fit parameters by demanding continuity of the SNFW mass density at $r_\epsilon$ [119]. As a result, one has

ρ_s(ρ_c,r_c,r_ϵ,r_s)=ρ_c (r_ϵ/r_s)(1+r_ϵ/r_s)^2/(1+0.091(r_ϵ/r_c)^2)^8 .

Examples of good fits with $\chi^2/\text{d.o.f.}<1$, see Tab. 2, are shown in Tab. 1. The derived quantity $m_{a,e}$ is extracted from the following equation  [129]

ρ_c ≡1.9×10^9 (m_a,e/10^-23 eV)^-2 (r_c/kpc)^-4 M_⊙kpc^-3

The other derived quantities $r_{200}$ and $M_{200}$ are obtained by employing Eqs. (<ref>) and (<ref>) with $M(r=r_{200})\equiv M_{200}$, respectively. In Fig. <ref> a frequency distribution of $m_{a,e}$ is shown, based on a sample of 17 best fitting galaxies, see Collage 1 for the fits to the RCs. The maximum of the smooth-kernel-distribution (solid line) is at

m_a,e=(0.72±0.5)×10^-23 eV .

Collage 1. Best fits of SNFW to RCs of 17 SPARC galaxies. The arrows indicate the Bohr radius of the e-lump, $r_{B,e}$ (red), the core radius of the soliton $r_{c}$ (orange), the transition radius from the soliton model to the NFW model $r_{e}$ (yellow), and the scale radius of the NFW model $r_{s}$ (green). The dagger$^\dagger$ indicates galaxy NGC6195.

left: Frequency distribution of axion mass $m_{a,e}$ as extracted from the SNFW model for 17 best fitting galaxies. The maximum of the smooth-kernel-distribution (solid, blue line) is at $m_{a,e}=(0.72\pm 0.5) \times10^{-23}$ eV  (red, vertical line).
right: The same axion masses in a mass-luminosity plane.

The frequency distribution of axion mass $m_{a,e}$ as extracted from the SNFW model for 17 best fitting galaxies is shown in the left side of Fig. <ref>. On the right side those axion masses are shown in the surface-brightness plane. In Fig. <ref> a frequency distribution of $M_{200}$ is shown for the 17 best-fitting galaxies on the left side, the right side depicts the distribution of these galaxies in the $M_{200}$ – surface-brightness plane. The maximum of the smooth-kernel-distribution is at $M_{200}=(6.3\pm 3)\times 10^{10}\,M_{\odot}$. With Eq. (<ref>) this implies a mean Bohr radius of

\begin{align}
\label{maBurkert}
\nonumber r_{B,e}&=\frac{M_p^2}{(6.3\pm4) \times 10^{10}\,M_{\odot}\, ((0.72\pm 0.5) \times 10^{-23}\, {\rm eV})^2 }\\
&=(0.26\pm 0.1)\,{\rm kpc}\,.
\end{align}
This value of $r_{B,e}$ is used in the following models in order to determine together with the model dependant virial masses an axion mass.

On the right-hand side of Fig. <ref>, the individual virial masses $M_{200}$ are depicted in the mass-luminosity plane.
Table 1 shows the luminosity, axion mass $m_a$, $r_{200}$, virial mass $M_{200}$, central density $\rho_0^{\rm NFW}$, scale radius $r_s$, transition radius $r_e$, core radius $r_c$, and $r_e/r_c$ of those 17 best fitting galaxies in the SNFW model.

The fitting constraints arise due to the many degrees of freedom. It is easy for two radii, e.g. $r_e$ and $r_c$ to take on the same value and "stick together", effectively decreasing the degrees of freedom of the underlying function again. Oftentimes, parameter as $r_{200}$ also took on unrealistic high values without any constraint. To avoid this undesired fitting behaviour, we have chosen those constraints. They are purely heuristic. Other fitting constraints we tried with less success were:

\begin{align*}
r_{200} < 200\,{\rm kpc}, r_e, r_c< 10\,{\rm kpc},r_e/r_c>0.1\\
200\,{\rm kpc} > r_{200} > r_e > r_c > 1\,{\rm kpc},r_e/r_c>0.1\\
\end{align*}

left:Frequency distribution of the virial mass $M_{200}$ in units of solar masses $M_{\odot}$ from the 80 best-fitting galaxies in the Burkert  model. The maximum of the smooth-kernel-distribution (solid line) is at $M_{200}=(6.5\pm3)\times 10^{10}\,M_{\odot}$.
right: Extracted virial masses $M_{200}$ in units of solar masses $M_{\odot}$ from sample of the 17 best-fitting galaxies in the NFW model.

Table 1. Fits of RCs to NFW model: Galaxy name, Hubble Type, luminosity, $\chi^2/\text{d.o.f.}$, axion mass $m_a$, $r_{200}$, virial mass $M_{200}$, central density $\rho_0$, scale radius $r_s$, transition radius $r_e$, core radius $r_c$, and $r_e/r_c$. The fitting constraints are heuristic and motivated by the results of [119], $r_{200} < 200$ kpc, $r_e, r_c<$ 6 kpc, and $r_e/r_c>0.1$.

§ NAVARRO–FRENK–WHITE PROFILES

The Navarro–Frenk–White (NFW) profile is a spherical density profile of dark matter which was developed by Julio F. Navarro, Carlos S. Frenk and Simon D.M. White around 1995 [131]. Their density profile leans on their findings of N-body simulation of cold dark matter. One of their findings is that dark matter halos are not well described by isothermal spheres. Therefore, they adopted the logarithmic slopes as proposed by Hernquist in 1990 for elliptical galaxies [132],

\begin{align}
\rho_{{\rm Hernquist}}(r)  \propto  \frac{1}{r(1+\frac{r}{r_s})^3}\,,
\end{align}

Their mass-density profile of is only slightly modified. It is
given as [128]

\begin{align}
\rho_{{\rm NFW}}(r) = \frac{\rho_s}{\frac{r}{r_s}(1+\frac{r}{r_s})^2}\,,
\end{align}

where $\rho_s$ associates with the central mass density, and $r_s$ is a scale radius which represents the onset of the asymptotic cubic decay in distance $r$ to the galactic center.
Note that profile $\rho_{\rm NFW}$ exhibits an infinite cusp as $r\to 0$ and that the orbit-enclosed mass $M(r)$ diverges logarithmically with the cutoff radius $r$ for the integral in Eq. (<ref>). Using Eqs. (<ref>), (<ref>), and (<ref>),
we obtain the orbital velocity of the
NFW model <cit.> as
V_NFW(r)=√(4 πG/r   ρ^NFW_0   r_s^3 (log(r/r_s+1)-r/r_s (r/r_s+1))) .

The RC model of Eq. (<ref>) is fitted to observed RCs to extract the values of the parameters $r_s$ and $\rho^{\rm NFW}_0$. Fig. <ref> shows an example for the characteristic density profile $\rho_{{\rm NFW}}(r)$ of Eq. (<ref>), and the orbit velocity $V_{{\rm NFW}}(r)$ of
Eq. (<ref>) for the galaxy F579-V1. The scale radius $r_s$ (red dotted line) indicates the transition from a $r^-1$ to a $r^-3$ behaviour of the density profile.

An example for the characteristic density profile $\rho_{{\rm NFW}}(r)$, Eq. (<ref>), of galaxy F579-V1 is given on the left-hand side. On the right-hand side the corresponding orbit velocity $V_{{\rm NFW}}(r)$,
Eq. (<ref>), of the same galaxy is fitted to the data as obtained by the SPARC library [44]. The scale radius $r_s$ (red dotted line) indicates the transition from a $r^-1$ to a $r^-3$ behaviour of the density profile.

According to Eq. (<ref>), the total enclosed lump mass $M$ of this model is given as

M_NFW(r) =4 πρ_s r_s^3 (r_s/r_s+r+log(r_s+r/r_s)-1) .

A comparison between the NFW and the SNFW density profiles can be seen in Fig. <ref> for galaxy NGC0300.

An example for the characteristic density profile $\rho_{{\rm SNFW}}(r)$, Eq. (<ref>), of galaxy NGC0300 is given in red. The red dotted, horizontal line indicates the description by the soliton density profile <ref>, the red line represents the NFW part <ref>. For the same galaxy the NFW density profile was plotted in orange. The scale radii $r_s$ are indecated by vertical dotted lines in the corresponding colour. Notice the intended discrepancy in the galactic center.

The left-hand side of Fig. <ref> shows the frequency distribution of the virial mass $M_{200}$ in units of solar masses $M_{\odot}$ from the 34 best-fitting galaxies in the NFW model. The maximum of the is at $M_{200}=(6.5\pm 3)\times 10^{10}\,M_{\odot}$. On the right-hand side of Fig. <ref>, the corresponding virial masses $M_{200}$ in units of solar masses $M_{\odot}$ are shown. Table 2 shows the luminosity, axion mass $m_a$, $r_{200}$, virial mass $M_{200}$, central density $\rho_0^{\rm NFW}$, scale radius $r_s$ of those 34 best fitting galaxies in the NFW model.

left: Frequency distribution of the virial mass $M_{200}$ in units of solar masses $M_{\odot}$ from the 34 best-fitting galaxies in the NFW model. The maximum of the smooth-kernel-distribution (solid line) is at $M_{200}=(5.7\pm 3)\times 10^{10}\,M_{\odot}$.
right: Extracted virial masses $M_{200}$ in units of solar masses $M_{\odot}$ from sample of the 34 best-fitting galaxies in the NFW model in the virial mass-luminosity plane.

In order to obtain an estimate for the axion mass $m_a$ from each fit, we use the relation between the Bohr radius $r_{B}$ and the axion mass $m_a$ of Eq. (<ref>) and the Bohr radius of $r_{B}=0.26$ kpc as determined in the SNFW fits together with the individually calculated virial mass $M_{200}$ as depicted in figure <ref>. Fig. <ref> shows the frequency distribution of extracted axion masses $m_{a,e}$ from the NFW model for 34 best fitting galaxies. The maximum of the smooth-kernel-distribution is at $m_{a,e}=(0.44\pm 0.3) \times10^{-23}$ eV.

left: Frequency distribution of axion mass $m_{a,e}$ as extracted from the PI model for 34 best fitting galaxies  and the Bohr radius $r_B=0.26$ kpc. The maximum of the smooth-kernel-distribution (solid, blue line) is at $m_{a,e}=(0.48\pm 0.25) \times10^{-23}$ eV (red, vertical line). right: The same axion masses in a mass-luminosity plane.

Table 2. Fits of RCs to NFW model: Galaxy name, Hubble Type, $\chi^2/\text{d.o.f.}$, luminosity, axion mass $m_a$, $r_{200}$, virial mass $M_{200}$, central density $\rho_0^{\rm NFW}$, scale radius $r_s$.

§ THE PSEUDO-ISOTHERMAL MODEL

In 2003, Jimenez et al. [133], showed that a pseudo-isothermal (PI) profile fits well to many types of galaxies, not only dwarf galaxies as in the Burkert model.

For the PI model one assumes a mass-density profile of the form [134]

ρ_PI(r) = ρ_0^PI/1+(r/r_s)^2
where $\rho_0^{{\rm PI}}$ refers to the central mass density. Again, there is a scale
radius $r_s$ which separates the core of the galaxy (ascending RC) from the region of quadratic asymptotic decay in $r$ where the RC is nearly constant. Note that the orbit-enclosed mass $M(r)$ now depends linearly on large $r$ such that our convention to set $r=r_B$ as a cutoff for the integral in Eq. (<ref>) is less controlled than in the NFW case. In analogy to the NFW model we obtain the rotational velocity of the PI model by using Eq. (<ref>), (<ref>), and (<ref>) <cit.>:
\begin{align}
V_{\rm PI}(r)=\sqrt{4 \pi  G \, \rho_0^{\rm PI} \, r_s^2 \left( 1-\frac{r_s}{r} \arctan \left(\frac{r}{r_s}\right) \right)}\,.
\label{eq:PI}
\end{align}

Fig. <ref> shows an example for the characteristic density profile $\rho_{{\rm PI}}(r)$ of Eq. (<ref>), and the orbit velocity $V_{{\rm PI}}(r)$ of
Eq. (<ref>) for the galaxy NGC0300. The scale radius $r_s$ (red dotted line) indicates the transition from a $r^{-1}$ to a $r^{-3}$ behaviour of the density profile.

An example for the characteristic density profile $\rho_{{\rm NFW}}(r)$, Eq. (<ref>), of galaxy NGC0300 is given on the left-hand side. On the right-hand side the corresponding orbit velocity $V_{{\rm NFW}}(r)$,
Eq. (<ref>), of the same galaxy is fitted to the data as obtained by the SPARC library [44]. The scale radius $r_s$ (red dotted line) indicates the transition from an almost constant to a $r^{-2}$ behaviour of the density profile.

The orbit-enclosed mass $M_{\rm PI}$(r) of this model reads

\begin{align}
\label{amPI}
M_{\rm PI}(r) = 4 \pi  \rho_0 r_s^3 \left( \frac{r}{r_s} - \arctan \left(\frac{r}{r_s} \right) \right)
\end{align}

The axion mass $m_a$ is extracted by setting $r=r_B$ in Eq. (<ref>) and presuming that $r_B$ coincides with the largest radius of the observed RC. This is not a robust convention because of the linear dependence on large $r$ of $M_{\rm PI}(r)$ in Eq. (<ref>).

left: Frequency distribution of the virial mass $M_{200}$ in units of solar masses $M_{\odot}$ from the 54 best-fitting galaxies in the NFW model. The maximum of the smooth-kernel-distribution (solid line) is at $M_{200}=(10.1\pm 5)\times 10^{10}\,M_{\odot}$.
right: Extracted virial masses $M_{200}$ in units of solar masses $M_{\odot}$ from sample of the 34 best-fitting galaxies in the NFW model in the virial mass-luminosity plane.

The left-hand side of Fig. <ref> shows the frequency distribution of the virial mass $M_{200}$ in units of solar masses $M_{\odot}$ from the 54 best-fitting galaxies in the NFW model. The maximum of the is at $M_{200}=(10.1\pm 5)\times 10^{10}\,M_{\odot}$. On the right-hand side of Fig. <ref>, the corresponding virial masses $M_{200}$ in units of solar masses $M_{\odot}$ are shown. Table 3 shows the luminosity, axion mass $m_a$, $r_{200}$, virial mass $M_{200}$, central density $\rho_0^{PI}$, scale radius $r_s$ of those 54 best fitting galaxies in the PI model. In order to obtain an estimate for the axion mass $m_a$ from each fit, we use the relation between the Bohr radius $r_{B}$ and the axion mass $m_a$ of Eq. (<ref>) and the Bohr radius of $r_{B}=0.26$ kpc as determined in the SNFW fits together with the individually calculated virial mass $M_{200}$ as depicted in Fig. <ref>. Fig. <ref> shows the frequency distribution of extracted axion masses $m_{a,e}$ from the PI model for 54 best fitting galaxies. The maximum of the smooth-kernel-distribution is at $m_{a,e}=(0.38\pm 0.2) \times10^{-23}$ eV.

left: Frequency distribution of axion mass $m_{a,e}$ as extracted from the PI model for 54 best fitting galaxies. The maximum of the smooth-kernel-distribution (solid, blue line) is at $m_{a,e}=(0.38\pm 0.2) \times10^{-23}$ eV  (red, vertical line). right: The same axion masses in a mass-luminosity plane.

Table 3. Fits of rotation curves to the PI model: Galaxy name, Hubble Type, $\chi^2/\text{d.o.f.}$, luminosity, axion mass $m_a$, $r_{200}$, virial mass $M_{200}$, central density $\rho_0^{\rm PI}$, scale radius $r_s$.

§ BURKERT MODEL FITS

The Burkert profile was developed by Andreas Burkert and Paolo Salucci [125, 126]. It is an empirical model which is leaned on the pseudo-isothermal profile. The model is has a central core which is characterized by the core radius $r_0$ and by the central density $\rho_0$, the mass-density profile takes the form [125, 126]

ρ_Bu(r) = ρ_0  r_0^3/(r+r_0)(r^2+r_0^2) .

The rotational velocity of the Burkert model by using Eq. (<ref>), (<ref>), and (<ref>) :

\begin{align}
V_{\rm Bu}(r)=\sqrt{
\frac{ \pi\,  G\, \rho_0^{\rm Bu}\,  r_0^3}{r}\,
\left(  2 \arctan{\frac{r}{r_0}} + 2 \log{\frac{r_0}{r+r_0}} + \log{\frac{r_0^2}{r^2+r_0^2}}\right)}\,.
\label{eq:B}
\end{align}

Fig. <ref> shows an example for the characteristic density profile $\rho_{{\rm Bu}}(r)$ of Eq. (<ref>), and the orbit velocity $V_{{\rm Bu}}(r)$ of
Eq. (<ref>) for the galaxy NGC0300. Starting relatively constant, the scale radius $r_s$ (red dotted line) indicates the transition from a $r^{-1}$ to a $r^{-3}$ behaviour of the density profile similarly to the NFW profile.

An example for the characteristic density profile $\rho_{{\rm Bu}}(r)$, Eq. (<ref>), of galaxy NGC0300 is given on the left-hand side. On the right-hand side the corresponding orbit velocity $V_{{\rm Bu}}(r)$,
Eq. (<ref>), of the same galaxy is fitted to the data as obtained by the SPARC library [44]. The scale radius $r_s$ (red dotted line) indicates the transition from a $r^{-1}$ to a $r^{-3}$ behaviour of the density profile.

Collage 2 depict the fits of the Burkert model to the 17 RCs used in the SNFW fits. Table 4 a) indicates that three out of these 17 RCs are fitted with a $\chi^2/{\rm d.o.f.}>1$. Therefore, we resort to a sample of 80 galaxies which fit with  $\chi^2/{\rm d.o.f.}<1$, their results are shown in Table 4 a) and b).

Recall that our strategy is to demonstrate that the extracted axion mass $m_{a,e}$ from the SNFW is consistent with the other three empirical models. Therefore, we calculate a Bohr radius Eq. (<ref>) from that axion mass and together with the virial masses $M_{200}$ extracted from RC fits in the corresponding model we retrieve a model (and SNFW)-dependant axion mass. Overall, an ensemble of 80 SPARC galaxies fits well to the Burkert model.

With a typical virial mass of $M_{200}=(2.9\pm4)\times 10^{10}\,M_{\odot}$, Fig. <ref>, the Burkert model yields a frequency distribution of $m_{a,e}=(0.65\pm 0.4)\times 10^{-23}\,$eV as shown in Fig. <ref>. Thus, the maximum of the smooth-kernel distribution is compatible with that in the SNFW model $m_{a,e}=(0.72\pm 0.5)\times 10^{-23}\,{\rm eV}$. Notice how $M_{200}$ clusters around the value $M_{200}\sim 5\times 10^{10}\,M_{\odot}$.

left: Frequency distribution of the virial mass $M_{200}$ in units of solar masses $M_{\odot}$ from the 80 best-fitting galaxies in the Burkert  model. The maximum of the smooth-kernel-distribution (solid line) is at $M_{200}=(2.9\pm4)\times 10^{10}\,M_{\odot}$.
right: Extracted virial masses $M_{200}$ in units of solar masses from Burkert-model fits to 80 RCs with $\chi^2/\text{d.o.f.}<1$ vs. the respective galaxy's central surface brightness in units of $L_{\odot}/\text{pc}^2$.

In our treatment Sec. <ref> of cosmological and astrophysical implications we appeal to the mean value of $m_{a,e}$-extractions in the SNFW and the Burkert model as

m_a,e=0.675×10^-23  eV .

left: Frequency distribution of 80 axion masses $m_{a,e}$, extracted from the Burkert-model fits of $M_{200}$ to the RCs of galaxies with a $\chi^2/\text{d.o.f.}<1$. The maximum of the smooth-kernel distribution (solid, blue line) is $m_{a,e}=(0.65\pm0.4)\times10^{-23}$ eV (red, vertical line).
right: The same axion masses in a mass-luminosity plane.

Collage 2. Burkert-model fits to the 17 best fitting SNFW-model galaxies. The purple arrow indicates the value of $r_{0}$.

Table 4 a) Burkert model: Galaxy name, Hubble Type, $\chi^2/\text{d.o.f.}$, luminosity, axion mass $m_a$, $r_{200}$, virial mass $M_{200}$, core density $\rho_0$, and core radius $r_0$. In the blue frame the 17 galaxies used for the SNFW fit are highlighted.

Table 4 b) Burkert model with $\chi^2/{\rm d.o.f.}<1$: Galaxy name, Hubble Type, $\chi^2/\text{d.o.f.}$, luminosity, axion mass $m_a$, $r_{200}$, virial mass $M_{200}$, core density $\rho_0$, and core radius $r_0$ (ordered as in the SPARC library).

CHAPTER:  INTERPRETATION OF THE RESULTS

Three axion species in condensate (lump) form suggests that in the halo of the Milky Way can be associated with an e-lump, the central compact massive object could originate from a $\mu$-lump, and that the $\tau$-lumps provide the dark sources for the accretion of stars in globular clusters.

• What is the dynamical symmetry braking scale?

• What is the connection to particle physics?

• Are there any indications about typical lump sizes and masses?

• What does this mean?

§ MODEL COMPARISON

A comparison between the SNFW, NFW, PI and Burkert model density profiles for an explicit galaxy, NGC0300 is shown in Fig. <ref>.

An example for the characteristic density profile $\rho_{{\rm SNFW}}(r)$, Eq. (<ref>), of galaxy NGC0300 is given in red. The red dotted line indicates the description by the soliton density profile <ref>, the red line represents the NFW part <ref>. For the same galaxy the NFW density profile was plotted in orange. The scale radii $r_s$ are indecated by vertical dotted lines in the corresponding colour.

The fitted probability density functions for the axion masses in the SNFW, NFW, PI and Burkert model are shown and compared in Fig. <ref> and Fig. <ref>, alongside their histograms. Note, that there are two ways of extracting a Bohr radius $r_B$ from the SNFW fits for the other three models:
(i) fitting a probability density functions to the axion masses $m_a$ and the virial masses $M_{200}$, determine their maxima and using equation <ref> to calculate the gravitational Bohr radius. This results in a gravitational Bohr radius of $R_{B}=0.26$ kpc, or
(ii) calculating for each galaxy the gravitational Bohr radius with the axion mass $m_a$ and the virial masses $M_{200}$ of that particular galaxy and using the maxima of the probability density fit to all Bohr radii; This results in a gravitational Bohr radius of $R_{B}=0.05$ kpc.

Axion mass probability density fits for the PI (blue), NFW (orange), Burkert (green) and SNFW (red) model for a Bohr radius of $R_{B}=0.26$ kpc (left) next to the histogram of in the same colour coding (right)

Figure <ref> compares the axion mass distribution for the four different profiles, while a gravitational Bohr radius of $R_{B}=0.26$ kpc was assumed for the NFW, PI and Burkert model, while Fig. <ref> shows compares the axion mass distribution under the assumption of $R_{B}=0.05$ kpc. Interestingly, the standard deviation of the PI, NFW and the Burkert fit are smaller for $R_{B}=0.26$ kpc, as well as the spread of the maxima with regard to maximum of the SNFW fit compared to  $R_{B}=0.05$. Furthermore, this seems not to be a binning effect, as the shape and the individual bin counts do not vary with a re-scaling.

Axion mass probability density fits for the PI (blue), NFW (orange), Burkert (green) and SNFW (red) model for a Bohr radius of $R_{B}=0.05$ kpc (left) next to the histogram of in the same colour coding (right)

Figure <ref> illustrates the second approach of retrieving a Bohr radius, based on a direct Histogram fit of all 17 Bohr radii as obtained in the SNFW model. This plot might immediately explain the qualitatively worse prediction, as it clearly takes a non-Gaussian form and the smallest bin is overpopulated. Unfortunately, re-binning does not resolve this problem. Moreover, it is interesting to see that although the maxima of the three heuristic models smaller than the SNFW maxima in Fig. <ref> and bigger in Fig. <ref>, the order does not change: The PI model obtains the smallest axion value, followed by NFW and Burkert.

Probability density fit for all Bohr radii in the SNFW model. The smoth kernel distribution (blue) has a maxima of  $R_{B}=0.05$ kpc (red).

Consequently, a gravitational Bohr radius of $R_{B}=0.26$ kpc will be used in the following. The overlap of the SNFW axion mass with the extracted axion masses in the NFW, PI and Burkert model were a consistency check. In the next chapter we look roughly at some implications of a three component dark matter model with lump types of dark matter.

Comparing different dark matter halo profiles is a good way of testing consistency. However, it is prone to systematic errors. Another test of the validity of the fits is given by an empirically found constant relationship of the central halo surface density $r_0 \times \rho_0$, [135]. Interestingly this constant relationship holds throughout many galaxy types and dark matter halo profiles [135]. Apparently, it is as of right now unclear whether fuzzy dark matter can reproduce this as well. Nonetheless, we are able to reproduce this empirical found relation in the Burkert model as well as in the SNFW model, see Fig. <ref>. Note that the one galaxy in the SNFW sample marked with a dagger$^\dagger$ is galaxy NGC6195. As it is shown previously in Table 1, this galaxy has a steep increase in orbital velocity which implies a high baryonic content of this galaxy. This might be an explanation for the relatively large offset as we assume dark matter dominated galaxies. A counter argument for this explanation is the fact that the constant behaviour of $r_0 \times \rho_0$ is valid for a broad range of galaxy types [135] regardless the baryon density.

Constant relationship of the central halo surface density $r_0 \times \rho_0$ in the SNFW model left, and the Burkert model right. The dagger$^\dagger$ indicates galaxy NGC6195, its high baryonic content may cause the offset.

§ A LINK TO SCREENED YANG-MILLS SCALES IN ASSOCIATION TO LEPTON MASSES

In chapter <ref> a brief summary about SU(2)CMB Cosmology was given. Recall that the SU(2)CMB fit, Fig. <ref>, has a smaller dark matter density than $\Lambda$CDM. This requires a sudden increase of dark matter which can be accomplished by a transition from dark energy into dark matter. The sudden transition[Although the model description of the transition in Eq. (<ref>) is in fact assumed to be instantaneous, which is also represented by the spike in Fig. <ref>, it is reasonable to assume a process which took a few million years. Therefore, sudden should be interpreted in the context of cosmological time scales.]
which is called depercolation suggests that dark energy is an ultra-light pseudo-Nambu-Goldstone boson field as well, see e.g. [25, 136, 137, 42, 138, 41].

The idea is that an axion field $\phi$, which is generated by Planckian physics, develops a small mass due to topological defects of a Yang-Mills theory. More precisely, the mechanism to produce axions is by means of the chiral anomaly which is already shown to be implemented by nature in the $\pi_0 \rightarrow \gamma \gamma$ decay [63, 58]. The chiral anomaly occurs on top of a dynamical breakdown of the global U$_A$(N) symmetry. The U$_A$(N) symmetry is carried by fermions $\psi$. By integrating out the groundstate portion of the gauge field $F^i_{\mu\nu}$, the field $\phi$ acquires a potential of the form [139, 105]

\begin{equation}\centering
\left( 1-\cos\frac{\phi}{f}\right) \Lambda^4_{\rm }\,,
\end{equation}

where $\Lambda_{}$ refers to the explicit symmetry breaking scale and $f$ the spontaneous symmetry breaking scale.
If the chiral pseudo-Goldstone field emerges as a consequence of gravitationally induced, chirally-invariant fermion interactions, it is plausible to associate the spontaneous symmetry breaking scale $f$ in the axion potential with the Planck scale $M_P$ [25, 140]. In the latter, [140], it has already been shown that at the Planck scale gravity and in a de Sitter Universe is able to break chiral symmetry dynamically. With the relation

Λ^2=m_a  f =m_a  M_P ,

it is possible to extract $\Lambda_{}$ for a given axion mass $m_a$. As shown in the previous sections, we extracted axion masses in the range of $m_a \in (0.38-0.72)\times 10^{-23}\,$eV. With an averaged mass between Burkert and SNFW of $m_a \sim 0.675\times 10^{-23}\,$eV, we retrieve an explicit symmetry breaking scale of

Λ_e=287  .

This is by only a factor 15 smaller than the scale $\Lambda_e=m_e/118.6$ ($m_e=511\,$keV the mass of the electron) of an SU(2) Yang-Mills theory proposed in [2] to originate the electron's mass in terms of a fuzzy ball of deconfining phase. This deconfining region is immersed into the confining phase and formed by the selfintersection of a center-vortex loop. Considering an undistorted Yang-Mills theory for simplicity[The chiral dynamics at the Planck scale, which produces the axion field, to some extent resolves the ground states of Yang-Mills theories: axions become massive by virtue of the anomaly because of this very resolution of topological charge density.], the factor of 15 could be explained by a stronger screening of topological charge density – the origin of the axial anomaly – in the confining ground state, composed of round, pointlike center-vortex loops, versus the deconfining thermal ground state, made of densely packed, spatially extended (anti)caloron centers subject to overlapping peripheries [27].

A schematic overview of the proposed connection from dark matter to screened Yang-Mills scales which are in association to lepton masses: The averaged mass between Burkert and SNFW of $m_a \sim 0.675\times 10^{-23}\,$eV leads to an explicit symmetry breaking scale of $\Lambda \sim 290$ eV. In the context of an SU(2) theory of the electron this is in the confined phase. The value for the same scale in the deconfined phase as obtained by [2] is larger by a factor of 15.

Linking the electron scale and the explicit symmetry breaking scale of the axion potential suggest further to marry the whole leptonic sector with three flavours of dark matter. This goes hand in hand with SU(2) fits to the CMB, which found a smaller dark matter density after recombination compared to the $\Lambda$CDM best fit to 2015 Planck data; The ratio between the density parameters of primordial and late-time dark matter ranges between 0.5 and 0.7 [26]. This implies at least one event of rapid (on cosmological scales) depercolation of dark matter. Depercolation is the transition of dark energy into dark matter. If the primordial dark matter density in comparison to the current dark matter density in assumed to be roughly 2/1, this is could be a small hint that three axion species are implemented in nature, with corresponding three events of depercolation. As shown in Fig. <ref>, the first two events  of depercolation happen during the radiation dominated area, the depecolation of e-lumps happened around z=53 where a clear spike in both matter and dark energy content is visible. The connection from dark matter to screened Yang-Mills scales proposed here  which are in association to lepton masses is shown schematically in Fig. <ref>.

§ GALACTIC CENTRAL REGIONS AND THE DARK SECTOR

The link between the masses of the three species of ultralight axions, whose fuzzy condensates form lumps of typical masses $M_e$, $M_\mu$, and $M_\tau$, with the three lepton families via the Planck-scale originated axial anomaly within confining phases of SU(2) Yang-Mills theories is compelling. In particular, $M_e=M_{200}$ can be determined by fitting of direct observation, as done in section  <ref>, while $M_\mu$ and $M_\tau$ are predicted by an appeal to Eqs. (<ref>). Such a scenario allows to address two questions: (i) the implication of a given lump's selfgravity for its stability and (ii) the cosmological origin of a given species of isolated lumps.

Before we discuss question (i) we would like to provide a
thermodynamical argument, based on our knowledge gained about axion and lump masses in terms of Yang-Mills scales and the Planck mass, why Planck-scale axions associated with the lepton families always occur in the form of fuzzy or homogeneous condensates. Namely, the Yang-Mills scales $\Lambda_e$, $\Lambda_\mu=\Lambda_e\, m_\mu/m_e$, $\Lambda_\tau=\Lambda_e\, m_\tau/m_e$ and $\Lambda_\gamma= 2 \pi \,  2.725 \, {\rm K} / 13.87 \sim 10^{-4}$ eV together with Eqs. (<ref>), (<ref>), yield axion masses as

\begin{align}
\label{axionmassemutau}
\nonumber&m_{a,e}\sim 6.75\times10^{-24}\,\mbox{eV} \,,\\
&m_{a,\mu}\sim 2.89\times10^{-19}\,\mbox{eV} \,,\\
\nonumber&m_{a,\tau}\sim 8.17\times 10^{-17}\,\mbox{eV},\\
\nonumber&m_{a,\gamma}\sim 4.12\times 10^{-39}\,\mbox{eV}\,.
\end{align}

The critical temperature $T_c$ for the Bose-Einstein condensation of a quantum gas of free bosons of mass $m_a$ and (mean) number density $n_a\sim M/(m_a \frac43\pi r_B^3)$ is given as

T_c=2π/m_a(n_a/ζ(3/2))^2/3 .
We conclude from Eqs. (<ref>),  (<ref>), (<ref>) and (<ref>) that

\begin{align}
\label{critTs}
\nonumber&T_{c,e}\sim 4.16\times 10^{38}\,{\rm eV}\,,\\
&T_{c,\mu}\sim 7.61\times 10^{47}\,{\rm eV}\,,\\
\nonumber&T_{c,\tau}\sim 6.09\times 10^{52}\,{\rm eV},\\
\nonumber&T_{c,\gamma}\sim 3.31\times 10^{8}\,{\rm eV}\,.
\end{align}

All three critical temperatures are comfortably larger than the Planck mass $M_P=1.22\times 10^{19}\,$GeV such that throughout the Universe's expansion history and modulo depercolation, which generates a nonthermal halo of particles correlated on the de Broglie wave length around a condensate core, the Bose-condensed state of e-, $\mu$-, and $\tau$-axions is guaranteed and consistent with $\xi\ll 1$, compare with Eq. (<ref>).

We now turn back to question (i). Explicit lump masses can be obtained from Eqs. (<ref>) based on the typical mass $M_e=6.3\times 10^{10}\,M_{\odot}$ of an e-lump. One has

\begin{align}
\label{lumpmassemutau}
&M_\mu=1.5\times 10^{6}\,M_\odot\,,\nonumber\\
&M_\tau=5.2\times 10^{3}\,M_\odot\,,\nonumber\\
&M_\gamma=1.0\times 10^{26}\,M_\odot\,.
\end{align}

A summary of the here obtained masses can be seen in Fig. <ref>.

An overview of the axion and lump masses proposed here: By definition is the mass of an e-lump associated with a typical galaxy mass, $\mu$-lumps are almost as heavy as galaxy centers and $\tau$-lump masses are similar to the mass of globular clusters.

For the computation of the respective gravitational Bohr radii according to Eq. (<ref>) both quantities, axion mass and lump mass, are required. To judge the gravitational stability of a given lump throughout its entire evolution a comparison between the typical Bohr radius $r_B$ and the typical Schwarzschild radius $r_{\rm SD}$, defined as

r_SD≡2M/M_p^2 ,
is in order. Using $M_e=1.86\times 10^{10}\,M_{\odot}$, Fig. <ref> indicates the implied values of the Bohr radii
$r_{B,e}$, $r_{B,\mu}$, and $r_{B,\tau}$ by dots on the curves of all possible Bohr radii as functions of their lump masses when keeping the axion masses fixed. Notice that for all three cases, e-lumps, $\mu$-lumps, and $\tau$-lumps, typical Bohr radii are considerably larger than their Schwarzschild radii. Indeed,
from Eqs. (<ref>), (<ref>), and (<ref>) it follows that

r_B/r_SD=1/2 κ^2 .
With $\kappa=314$ we have $r_B/r_{\rm SD}=4.92\times 10^4$. Arguing on an adiabatic basis by following the solid lines in Fig. <ref> down to their intersections with the dashed line, a slow increase in mass by a factor $\sim 2435.73$ of a given lump to reach the critical mass for black-hole formation is unlikely to occur by growth of an axion condensate through collisions with its peers. Moreover, since the mean mass density of a lump scales with the fourth power of the Yang-Mills scale it is excluded even for moderate hierarchies in Yang-Mills scales, such as $\Lambda_\tau/\Lambda_\mu\sim 17$ or $\Lambda_\mu/\Lambda_e\sim 200$, that a lump with a larger scale embedded into a lump with a lower scale can grow substantially by accreting the much less dense condensate of its surroundings.
What is conceivable though is that lumps of a larger Yang-Mills scale, embedded into a lump of a lower scale, catalyse the latter's gravitational compaction to the point of collapse, see discussion in Sec. 6.1.

Schwarzschild radius (dashed line) and gravitational Bohr radii (solid lines; dark blue: e-lump, turquoise: $\mu$-lump, black: $\tau$-lump and in orange the unpercolated $\gamma$-lump) as functions of lump mass in units of solar mass $M_\odot$. The dots indicate lump masses which derive from the typical e-lump mass $M_e=6.3\times 10^{10}\,M_{\odot}$ suggested by the analysis of the RCs of low-surface-brightness galaxies performed in Sec. <ref>. Note that because the axion condensate which is in association to the photon scale $\sim 10^{-4}$ eV, the $\gamma$-lump has a typical size which is by far larger than the current observable Universe. It hat no chance to depercolate, that means form lumps / get ripped into lumps. This is why it is spatially homogeneous and we interpret it as dark energy.