Axial Anomaly in Galaxies and the Dark Universe

Janning Meinert and Ralf Hofmann, Axial Anomaly in Galaxies and the Dark Universe, Universe 7, issue 6, article 198, 2021. Reproduced under the Creative Commons Attribution 4.0 International licence stated in the article; the published version is at doi.org/10.3390/universe7060198. Equations are described in words and identified by the number they carry in the source; the reference-number markers have been removed, and the four fit tables and the rotation-curve figure panels are described rather than transcribed (sections of tabulated per-galaxy parameters omitted for length; the complete tables and figures are at the source).

(On this site, the companion papers in Ralf Hofmann’s SU(2) programme are the thermal ground state and its nonthermal probes at /library/stm-9a9190b65f, the centre-vortex loops that carry the lepton identification at /library/stm-e1f7b72444, and the lepton spectra from evaporating hot spots at /library/stm-dcaf382795. The other routes to the dark sector without a new particle are Verlinde’s emergent gravity at /library/stm-601c8315b2, dark-matter superfluidity at /library/stm-395f34ea17 and /library/stm-25005f2061, and superfluid vacuum theory at /library/stm-4a4bcf1126. The measurement the dark-energy question now turns on is DESI, at /library/stm-15541611e8, and the size of the problem is set out at /library/stm-e0ebfdf437.)

Abstract

Motivated by the SU(2) modification of the cosmological model Lambda-CDM for the CMB, we consider isolated fuzzy-dark-matter lumps, made of ultralight axion particles whose masses arise due to distinct SU(2) Yang-Mills scales and the Planck mass. In contrast to the CMB theory, these Yang-Mills theories are in confining phases at zero temperature throughout most of the Universe’s history and associate with the three lepton flavours of the Standard Model of particle physics. As the Universe expands, axionic fuzzy dark matter comprises a three-component fluid which undergoes certain depercolation transitions when dark energy, a global axion condensate, is converted into dark matter. We extract the lightest axion mass of 0.675 times ten to the minus twenty-third electronvolts from well motivated model fits to observed rotation curves in low-surface-brightness galaxies from the SPARC catalogue. Since the virial mass of an isolated lump solely depends on the Planck mass and the associated Yang-Mills scale, the properties of an electron-family lump predict those of muon- and tau-family lumps. As a result, a typical electron-lump virial mass of about 6.3 times ten to the tenth solar masses suggests that massive compact objects in galactic centers, such as Sagittarius A star in the Milky Way, are merged muon- and tau-lumps. In addition, tau-lumps may constitute globular clusters. The CMB theory is always thermalised, and its axion condensate never has depercolated. If the axial anomaly indeed would link leptons with dark matter and the CMB with dark energy then this would demystify the dark Universe through a firmly established feature of particle physics.

Keywords: galaxy rotation curves; low surface brightness; dark matter; dark energy; ultralight axion particles; cores; halos; mass-density; profiles; pure Yang-Mills theory.

1. Introduction

Dark matter was introduced as an explanation for the anomalous, kinematic behavior of luminous test matter in comparison with the gravity exerted by its luminous surroundings, for example virialised stars within a galaxy or a virialised galaxy within a cluster of galaxies. That luminous matter can be segregated from dark matter is evidenced by the bullet cluster, in observing hot intergalactic plasma in X-rays in between localised dark-mass distributions seen by gravitational lensing.

The present Standard Model of Cosmology, Lambda-CDM, posits a spatially flat Universe with about 70 percent dark energy, inducing late-time acceleration. This model requires a substantial contribution of about 26 percent cold dark matter to the critical density and allows for a contribution of baryons of roughly 4 percent.

To determine all parameters of Lambda-CDM at a high accuracy, cosmological distance scales can be calibrated by high-redshift data — the inverse distance ladder, global cosmology — coming from precision observations of the Cosmic Microwave Background or from large-scale structure surveys probing Baryon Acoustic Oscillations. Alternatively, low-redshift data — the direct distance ladder, local cosmology — can be used by appeal to standard or standardisable candles such as cepheids, tip-of-the-red-giant-branch stars, and supernovae of types Ia and II. Recently, a comparison between global and local cosmology has revealed tensions in some of the cosmological parameter values, notably the Hubble constant and the amplitude-matter-density combination.

These interesting discrepancies motivate modifications of Lambda-CDM. A cosmological model aiming to resolve these tensions should target high-redshift radiation and the dark sector. In particular, models which are in principle falsifiable by terrestrial experiments and which pass such tests could lead to a demystification of the dark Universe. However, searches for weakly interacting, massive and stable particles, whose potential existence is suggested by certain extensions of the Standard Model of Particle Physics, so far have not produced any detection.

An attractive possibility to explain the feebleness of a potential interaction between the dark sector of cosmology and Standard Model matter, in terms of the large hierarchy between particle-physics scales and the Planck mass, is the theoretically and experimentally solidly anchored occurrence of an axial anomaly, which is induced by topological charge densities in the ground states of pure Yang-Mills theories. The axial anomaly acts on top of a dynamical chiral symmetry breaking mediated by a force of hierarchically large mass scale compared to the scales of the Yang-Mills theories. To enable the axial anomaly throughout the Universe’s entire history, chiral fermions — which acquire mass through gravitational torsion and which can be integrated out in a Planck-scale de Sitter background — need to be fundamentally charged under certain gauge groups. In such a scenario gravity itself, a strong force at the Planck scale, would induce the dynamical chiral symmetry breaking. The anomaly then generates an axion mass for particles that a priori are chiral Nambu-Goldstone bosons. Working in natural units where the speed of light, the reduced Planck constant and Boltzmann’s constant are one, equation 1 in the source states it: the axion mass is the square of the Yang-Mills scale divided by the Planck mass, the latter being 1.221 times ten to the twenty-eighth electronvolts.

The cold-dark-matter paradigm is successful in explaining large-scale structure in the Lambda-CDM context but exhibits problems at small scales, galactic and lower. While N-body simulations within Lambda-CDM reveal matter-density profiles of the galactic dark-matter halos that are characterised by a central cusp of the Navarro-Frenk-White type, falling as the reciprocal of the radial distance to the center of the galaxy, observations suggest a core or soliton profile subject to a constant central matter density. A model of fuzzy dark matter, according to the ground-state solution of the Schrodinger-Poisson system embedded into cosmological simulations, posits a condensate of free axion particles within the galactic core. For radii between the transition radius and the virial radius, and above three times the core radius, the associated central matter densities give way to a selfgravitating cloud of effective, nonrelativistic particles whose mass is the cubed de Broglie wavelength times the Navarro-Frenk-White density. Here the virial radius is defined by equation 3 in the source, that the Navarro-Frenk-White density there equals two hundred times three over eight pi, times the squared Planck mass, times the squared Hubble constant. Note that within the core region the correlation length in the condensate is given by the reduced Compton wavelength, the reciprocal of the axion mass. In what follows, we will refer to such a system — condensate core plus Navarro-Frenk-White tail — as a lump. In earlier work, fuzzy-dark-matter fits to the rotation curves of low-surface-brightness galaxies, which are plausibly assumed to be dominated by dark matter, produced an axion mass of 0.554 times ten to the minus twenty-third electronvolts. Note also that the cosmological simulation cited associates the axionic scalar field with dark-matter perturbations only, but not with the background dark-matter density, which is assumed to be conventional cold dark matter.

Another potential difficulty with Lambda-CDM, which fuzzy dark matter is capable of addressing, is the prediction of too many satellite galaxies around large hosts like the Milky Way or Andromeda. A recent match of observed satellite abundances with cosmological simulations within the fuzzy-dark-matter context yields a stringent bound on the axionic particle mass of more than 2.9 times ten to the minus twenty-first electronvolts. This bound is consistent with a Milky Way rotation-curve analysis giving 2.5, plus 3.6 and minus 2.0, times ten to the minus twenty-first electronvolts.

There is yet another indication that Lambda-CDM may face a problem in delaying the formation of large galaxies of about ten to the twelfth solar masses due to their hierarchical formation out of less massive ones. This seems to contradict the high-redshift observation of such galaxies and suggests that a component of active structure formation is at work.

Assuming axions to be a classical ideal gas of non-relativistic particles, the mass can be extracted from CMB simulations of the full Planck data subject to scalar adiabatic, isocurvature, and tensor-mode initial conditions — between ten to the minus twenty-fifth and ten to the minus twenty-fourth electronvolts, with a ten percent contribution to dark matter and a one percent contribution of isocurvature and tensor modes — and from a modelling of Lyman-alpha data with conservative assumptions on the thermal history of the intergalactic medium. For the XQ-100 and HIRES/MIKE quasar spectra samples one obtains, respectively, at least 7.12 times ten to the minus twenty-second and at least 1.43 times ten to the minus twenty-first electronvolts.

In our discussion in section 5 we conclude that three axion species of hierarchically different masses could determine the dark-matter physics of our Universe. When comparing the results of axion-mass extractions with fuzzy-dark-matter-based axion-mass constraints obtained in the literature it is important to observe that a single axion species always is assumed. For example, this is true of the combined axion-mass bound of more than 3.8 times ten to the minus twenty-first electronvolts, derived from modelling the Lyman-alpha flux power spectrum by hydrodynamical simulations, and it applies to the cosmological evolution of scalar-field based dark-matter perturbations yielding an axion mass of about 8 times ten to the minus twenty-third electronvolts.

In the present article we are interested in pursuing the consequences of fuzzy dark matter for the physics of dark matter on super-galactic and sub-galactic scales within a cosmological model which deviates from Lambda-CDM in three essential points. First, fuzzy dark matter is subject to three instead of one nonthermal axionic particle species, whose present cosmological mass densities are nearly equal. Second, axion lumps — condensate core plus halo of fluctuating density granules — cosmologically originate from depercolation transitions at distinct redshifts out of homogeneous condensates. Third, the usual, nearly scale invariant spectrum of adiabatic curvature fluctuations imprinted as an initial condition for cosmological cold-dark-matter evolution, presumably created by inflation, does not apply.

The first point derives from the match of axion species with the three lepton families of the Standard Model of particle physics. These leptons emerge in the confining phases of SU(2) Yang-Mills theories. According to the mass formula, axion masses are then determined by the universal Peccei-Quinn scale, the Planck mass, and the distinct Yang-Mills scales of the electron, muon and tau theories.

The second point is suggested by a cosmological model which is induced by the postulate that the CMB itself is described by an SU(2) gauge theory and which fits the CMB power spectra for temperature, temperature-polarisation cross-correlation and polarisation remarkably well except at low multipoles. The according overshoot in the temperature spectrum at large angular scales may be due to the neglect of the nontrivial, SU(2)-induced photon dispersion at low frequencies.

The third point relates to the fact that a condensate does not maintain density perturbations on cosmological scales and that the electron-lump depercolation redshift is about 53. As a consequence, constraints on axion masses from cosmological simulations by confrontation with the observed small-scale structure should be repeated based on that model. This, however, is beyond the scope of the present work.

To discuss the second point further, we refer to the work where a dark sector was introduced as a deformation of Lambda-CDM. This modification models a sudden transition from dark energy to dark matter at a redshift of 53. Such a transition is required phenomenologically to reconcile high-redshift cosmology — well below the Planckian regime but prior to and including recombination, where the dark-matter density is reduced compared to Lambda-CDM — with well-tested low-redshift cosmology. That a reduced dark-matter density is required at high redshift is a result of the temperature-redshift relation induced by the SU(2) description of the CMB. Depercolation of a formerly spatially homogeneous axion condensate, which introduces a change of the equation of state from an energy density equal to minus the pressure to zero pressure, is a result of the Hubble radius — the spatial scale of causal connectedness in a Friedmann-Lemaitre-Robertson-Walker Universe — exceeding by far the gravitational Bohr radius of an isolated, spherically symmetric system of selfgravitating axion particles. The value of the ratio of the two radii at depercolation so far is subject to phenomenological extraction, but should intrinsically be computable in the future by analysis of the Schrodinger-Poisson system in a thus linearly perturbed background cosmology whose dark sector is governed by axion fields subject to their potentials.

Roughly speaking, at depercolation from an equation of state where the energy density equals minus the pressure, the quantum correlations in the axionic system become insufficient to maintain the homogeneity of the formerly homogeneously Bose-condensed state. The latter therefore decays or depercolates into selfgravitating islands of axionic matter whose central regions continue to be spatially confined Bose condensates but whose peripheries are virialised, quantum correlated particle clouds of an energy density that decays rapidly with distance from the gravitational center, to approach the cosmological dark-sector density. On cosmological scales, each of these islands, or lumps, can be considered a massive nonrelativistic particle by itself, such that the equation of state of the associated ensemble becomes pressureless: the density of lumps then dilutes as the inverse cube of the cosmological scale factor.

For the entire dark sector, equation 4 in the source writes the dark-sector density parameter as a function of redshift: the cosmological-constant term, plus a primordial dark-matter term scaling as the cube of one plus the redshift, plus an emergent dark-matter term which scales the same way below the electron-lump depercolation redshift and freezes at its value there above it. Fits of this model to the CMB power spectra reveal that the emergent contribution is about half the primordial one. The primordial term denotes a contribution to the present dark-matter density parameter, while the emergent term refers to the emergence of dark matter due to the depercolation of a formerly homogeneous Bose-Einstein condensate into isolated lumps once their typical Bohr radius is well covered by the horizon radius. One may question that depercolation occurs suddenly at a single redshift, the only justification so far being the economy of the model. If a first-principle simulation of the Schrodinger-Poisson system plus background cosmology reveals that the transition from dark energy to dark matter during depercolation involves a finite redshift range, then this has to be included in the model.

After depercolation has occurred, a small dark-energy residual persists to become the dominant cosmological constant today. As we will argue in section 5, the primordial dark-matter density could originate from the stepwise depercolation of former dark energy in the form of super-horizon sized muon- and tau-lumps. Therefore, dark energy dominates the dark sector at sufficiently high redshift. However, due to radiation dominance dark energy then was a marginal contribution to the expansion rate. The model was shown to fit the CMB anisotropies with a low baryon density, the local value for the redshift of re-ionisation, and the local value of the Hubble constant from supernova distance-redshift extractions.

The purpose of the present work is to propose a scenario which accommodates the emergent dark matter, the primordial dark matter, and the cosmological constant. At the same time, we aim at explaining the two dark-matter parameters in terms of axial anomalies subject to a Planck-mass Peccei-Quinn scale and three SU(2) Yang-Mills theories associated with the three lepton families. In addition, an explanation of the cosmological-constant parameter is proposed which invokes the SU(2) Yang-Mills theory underlying the CMB. Hence, the explicit gauge-theory content of our model is the product of the electron, muon, tau and CMB SU(2) theories.

We start with the observation that ultralight bosons necessarily need to occur in the form of selfgravitating condensates in the cores of galaxies. Because these cores were separated in the course of nonthermal depercolation, halos of axion particles, correlated due to gravitational virialisation on the scale of their de Broglie wavelength, were formed around the condensates. Such a halo reaches out to a radius where its mass density starts to fall below two hundred times the critical cosmological energy density of the spatially flat Universe. A key concept in describing such a system, a lump, is the gravitational Bohr radius, defined in equation 5 in the source as the squared Planck mass divided by the lump mass and by the squared axion mass, where the lump mass should coincide with the virial mass. We use two fuzzy-dark-matter models of the galactic mass density to describe low-surface-brightness galaxies and to extract the axion mass: the Soliton-Navarro-Frenk-White model and the Burkert model.

Rather model independently, we extract a typical axion mass of about 0.7 times ten to the minus twenty-third electronvolts, which confirms the earlier value. With the mass formula this implies a Yang-Mills scale of about 287 electronvolts for the electron family. This is smaller than the value of 511 keV divided by 118.6, that is 4.31 keV, found where a link to an SU(2) Yang-Mills theory governing the first lepton family is made. Note that the larger value was extracted in the deconfining phase while the smaller value, obtained from the axion mass, relates to the confining phase. The suppression of the Yang-Mills scale is plausible because topological charges, which invoke the axial anomaly, are less resolved in the confining as compared to the deconfining phase. The gravitational Bohr radius associated with a typical electron-lump mass of about 6.3 times ten to the tenth solar masses turns out to be about 0.26 kiloparsecs.

Having fixed the scales of the CMB and electron theories and linked their lumps to dark energy and the dark-matter halos of low-surface-brightness galaxies respectively, we associate the lumps of the muon and tau theories with the primordial dark-matter parameter of the cosmological model. Within a galaxy, each individual muon- and tau-lump provides a mass fraction of about 2.3 times ten to the minus fifth and about 8.3 times ten to the minus eighth, respectively, of the mass of an electron-lump.

This paper is organised as follows. In section 2 we discuss features of lumps in terms of a universal ratio between reduced Compton wavelength and gravitational Bohr radius. As a result, a typical lump mass can be expressed solely in terms of Yang-Mills scale and Planck mass. The rotation curves of galaxies with low surface brightness, from the SPARC library, are analysed in section 3 using two models with spherically symmetric mass densities. Assuming that only one Planck-scale axion species dominates the dark halo of a low-surface-brightness galaxy in terms of an isolated, unmerged electron-lump, we extract the typical axion mass in section 3.2. In section 3.3 we demonstrate the consistency of axion-mass extraction between the two models: the gravitational Bohr radius, determined in the first, together with the lump mass obtained from the second, predicts an axion mass which is compatible with the axion mass extracted from the soliton-core density of the first. The typical value of the axion mass suggests an association with SU(2) Yang-Mills dynamics responsible for the emergence of the first lepton family. In section 4, this information is used to discuss the cosmological origin and role of lumps played in the dark Universe in association with the two other lepton families and the SU(2) gauge theory propounded to describe the CMB. As a result, on subgalactic scales the muon-lumps could explain the presence of massive compact objects in galactic centers such as Sagittarius A star in the Milky Way, while tau-lumps may relate to globular clusters. On super-galactic scales and below the electron depercolation redshift, however, lumps from all axion species act like cold dark matter. On the other hand, the CMB-lump’s extent always exceeds the Hubble radius by many orders of magnitude and therefore should associate with dark energy. Finally, in section 5 we discuss in more detail how certain dark structures of the Milky Way may have originated in terms of muon- and tau-lumps. We also provide a summary and an outlook on future work.

2. Gravitational Bohr radius and reduced Compton wavelength of a Planck-scale axion

We start by conveying some features of basic axion lumps, cosmologically originated by depercolation transitions, that we wish to study. Equation 6 in the source defines the reduced Compton wavelength of a species as the reciprocal of its axion mass, and equation 7 in the source defines the mean distance between axion particles within the spherically symmetric core of the lump as the cube root of the axion mass divided by the mean dark-matter mass density. Equation 8 in the source gives that mean density as the virial mass divided by four thirds pi times the cubed Bohr radius.

The energy densities of each of the three dark-energy-like homogeneous condensates of axionic particles prior to lump depercolation are assumed to arise due to Planckian physics. Therefore, each may only depend on the Planck mass and the axion mass of that species. Finite-extent, isolated, unmerged lumps selfconsistently are characterised by a fixed ratio between the reduced Compton wavelength — the correlation length in the condensate of free axion particles at zero temperature — and the Bohr radius.

Let us explain this. Causal lump segregation due to cosmological expansion, that is depercolation, which sets in when the Hubble radius becomes sufficiently larger than the Bohr radius, is adiabatically slow and generates a sharply peaked distribution of lump masses, and of Bohr radii, in producing typically sized condensate cores. These cores are surrounded by halos of axion particles that represent regions of the dissolved condensate and are nonthermally released by the mutual pull of cores during depercolation. In principle, we can state, as equation 9 in the source, that for an isolated, unmerged lump the ratio of Bohr radius to reduced Compton wavelength is a smooth dimensionless function of the ratio of axion mass to Planck mass, with a finite limit as that ratio goes to zero. This is because the typical mass of an isolated, unmerged lump, which enters the Bohr radius, is, due to adiabatically slow depercolation, by itself only a function of the two mass scales — axion mass and Planck mass — mediating the interplay between quantum and gravitational correlations that give rise to the formation of the lump. Since the mass ratio is much smaller than unity, we can treat the right-hand side as a universal constant. In practice, we will in section 3 derive the values of the electron-family Bohr radius and axion mass by matching dark-matter halos of low-surface-brightness galaxies with well motivated models of a lump’s mass density. As a result, we state a value of about 314 for that constant in section 4.

That relation, together with the mass formula, the Bohr-radius definition and the Compton-wavelength definition, implies equation 10 in the source for the mass of the isolated, unmerged lump: it is the cube of the Planck mass, divided by the universal constant and by the square of that species’ Yang-Mills scale.

This is important because it predicts that the ratios of lump masses solely are determined by the squares of the ratios of the respective Yang-Mills scales or, what is the same, by the ratios of the charged-lepton masses. Equation 11 in the source records them: the tau-to-muon lump mass ratio is the squared ratio of muon to tau mass, about 283 for muon over tau inverted; the muon-to-electron lump mass ratio is the squared ratio of electron to muon mass, about 2.3 times ten to the minus fifth; and the tau-to-electron lump mass ratio is the squared ratio of electron to tau mass, about 8.3 times ten to the minus eighth.

Moreover, equation 12 in the source fixes the ratio of the mean interparticle distance to the reduced Compton wavelength as the cube root of four thirds pi, times the four-thirds power of the universal constant times the Yang-Mills scale over the Planck mass. Since the Yang-Mills scales are far below the Planck mass, that ratio is far below one, and therefore a large number of axion particles are covered by one reduced Compton wavelength. This assures that the assumption of a condensate core is selfconsistent. A thermodynamical argument for the necessity of axion condensates throughout the Universe’s expansion history is given in section 4. The non-local and non-linear integro-differential Schrodinger equation, obtained from a linear Schrodinger equation and a Poisson equation for the gravitational potential governing the lump, has been analysed elsewhere, and an excitation of such a lump in terms of a wave function containing radial zeros was envisaged there. Here instead, we assume the isolated, unmerged lump to be in its ground state, parameterised by a phenomenological, positive mass density proportional to the squared modulus of the wave function, which represents the lump well.

Finally, equation 13 in the source gives the gravitational Bohr radius as the universal constant times the Planck mass divided by the squared Yang-Mills scale.

3. Analysis of rotation curves

In this section, we extract the axion mass from observed rotation curves of low-surface-brightness galaxies which fix the lump mass and a characterising length scale, the gravitational Bohr radius. This, in turn, determines the primary Yang-Mills scale associated with the lump. We analyse rotation curves from the SPARC library.

3.1. Fuzzy dark matter: Soliton-Navarro-Frenk-White versus Burkert model

To investigate, for a given galaxy and rotation curve, the underlying spherically symmetric mass density, it is useful to introduce the orbit-enclosed mass, equation 14 in the source: four pi times the integral of the squared radius times the density, out to the radius in question. Assuming virialisation, spherical symmetry, and Newtonian gravity, equation 15 in the source gives the orbital velocity of a test mass, a star, as the square root of Newton’s constant times the enclosed mass divided by the radius, with Newton’s constant being the inverse squared Planck mass. The lump mass is defined as the enclosed mass at the virial radius.

The mass-density profile of the Navarro-Frenk-White part of the combined model is given in equation 16 in the source: a central scale density divided by the radius over the scale radius, times the square of one plus that ratio. Note that this profile exhibits an infinite cusp as the radius goes to zero, and that the orbit-enclosed mass diverges logarithmically with the cutoff radius. In order to avoid the cuspy behavior at small radius, an axionic Bose-Einstein condensate, a soliton density profile, is assumed to describe the soliton region inside the transition radius. From the ground-state solution of the Schrodinger-Poisson system for a single axion species one obtains a good analytic description of the soliton density profile as equation 17 in the source: the core density divided by the eighth power of one plus 0.091 times the squared ratio of radius to core radius. On the whole, equation 18 in the source approximates the fuzzy-dark-matter profile by the soliton profile inside the transition radius and the Navarro-Frenk-White profile outside it.

For the Burkert model one assumes the mass-density profile of equation 19 in the source: a central mass density times the cubed scale radius, divided by the product of the radius plus the scale radius with the squared radius plus the squared scale radius.

3.2. Analysis of rotation curves in the combined soliton model

Using the enclosed-mass, orbital-velocity and profile relations, we obtain the orbital velocity of the combined model, which is fitted to observed rotation curves. This determines the transition radius, the scale radius and the core density. The scale density relates to these fit parameters by demanding continuity of the mass density at the transition radius, as set out in equation 20 in the source.

Examples of good fits, with reduced chi-squared below one, are shown in figure 1 of the source, which presents best fits to the rotation curves of seventeen SPARC galaxies with arrows marking, for each, the Bohr radius of the electron-lump, the core radius of the soliton, the transition radius from the soliton model to the Navarro-Frenk-White model, and the scale radius of the latter. The corresponding fit parameters are tabulated in tables 1 and 2 of the source; the galaxies are DDO170, F565-V2, F568-1, F571-V1, F574-1, F583-4, NGC3109, NGC3877, NGC4085, NGC6195, UGC00731, UGC00891, UGC06628, UGC07125, UGC07151, UGC11820 and UGC12632, with fitting constraints, heuristic and motivated by earlier results, of a virial radius below 200 kiloparsecs, transition and core radii below 6 kiloparsecs, and a ratio of transition to core radius above 0.1. (Per-galaxy parameter tables omitted for length; the complete tables are at the source.)

The derived axion mass is extracted from equation 21 in the source: the core density equals 1.9 times ten to the ninth, times the inverse square of the axion mass in units of ten to the minus twenty-third electronvolts, times the inverse fourth power of the core radius in kiloparsecs, in solar masses per cubic kiloparsec. The other derived quantities, the virial radius and virial mass, are obtained from the virial-radius definition and the enclosed mass. Figure 2 of the source shows the frequency distribution of the axion mass, based on a sample of seventeen best-fitting galaxies. The maximum of the smooth-kernel distribution, equation 22 in the source, is at 0.72 plus or minus 0.5 times ten to the minus twenty-third electronvolts.

3.3. Analysis of rotation curves in the Burkert model

Figure 6 of the source depicts the fits of the Burkert model to the same seventeen rotation curves used in the combined-model fits. Tables 3 and 4 indicate that three out of these seventeen are fitted with a reduced chi-squared above one. Therefore, we resort to a sample of eighty galaxies which fit with a reduced chi-squared below one.

Our strategy to demonstrate independence of the mean axion mass on the details of the two realistic models is to also determine it from the Bohr-radius definition. To do this, we use the value of the gravitational Bohr radius extracted in the previous section and the values of the virial mass extracted from rotation-curve fits within an ensemble of eighty SPARC galaxies to the Burkert model. Figure 5 of the source shows the resulting frequency distribution of virial masses, peaking at 2.9 plus or minus 4 times ten to the tenth solar masses; figure 7 plots the extracted virial masses against each galaxy’s central surface brightness; and figure 8 shows the resulting frequency distribution of eighty axion masses. Obviously, the maximum of that smooth-kernel distribution, 0.65 plus or minus 0.4 times ten to the minus twenty-third electronvolts, is compatible with the value from the combined model. Notice how the virial mass clusters around about 5 times ten to the tenth solar masses.

In our treatment of cosmological and astrophysical implications, we appeal to the mean of the two extractions, equation 24 in the source: an axion mass of 0.675 times ten to the minus twenty-third electronvolts.

4. Galactic central regions and the dark sector of the Universe

Interpreting the dark-matter structure of a typical low-surface-brightness galaxy as an electron-lump, we have a gravitational Bohr radius of 0.26 kiloparsecs from the combined model. Therefore, equation 25 in the source gives the value of the universal constant as 314. With the mean axion mass, the mass formula yields equation 26 in the source: a Yang-Mills scale of 287 electronvolts.

This is by only a factor of 15 smaller than the scale of the electron mass divided by 118.6 for an SU(2) Yang-Mills theory proposed to originate the electron’s mass in terms of a fuzzy ball of deconfining phase, where the 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 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 caloron centers subject to overlapping peripheries. The factor of 15 so far is a purely phenomenological result, which could be expected to be of order a hundred or higher, and which is plausible qualitatively because of the reduced topological charge density in the confining phase, where overlapping magnetic monopoles and antimonopoles, aligned within hardly resolved center vortices, are the topological charge carriers. The complex interplay between the would-be Goldstone nature of the axion, as prescribed by fermion interaction at the Planck scale, and the topological charge density of an SU(2) Yang-Mills theory deeply in its confining phase is anything but understood quantitatively so far. One may hope that simulations of the axion potential in a center-vortex model of the confining phase will yield more quantitative insights in the future. The authors add a footnote: 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 link between the masses of the three species of ultralight axions, whose fuzzy condensates form lumps of typical masses for the three families, 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, the electron-lump mass can be determined by mild modelling of direct observation, as done in section 3, while the muon- and tau-lump masses are predicted by an appeal to the mass-ratio relations. Such a scenario allows us to address two questions: first, the implication of a given lump’s selfgravity for its stability, and second, the cosmological origin of a given species of isolated lumps.

Before we discuss the first question 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 three Yang-Mills scales, related by the charged-lepton mass ratios, together with the mass formula and the extracted electron scale, yield the axion masses of equation 27 in the source: about 0.675 times ten to the minus twenty-third electronvolts for the electron family, about 2.89 times ten to the minus nineteenth for the muon family, and about 8.17 times ten to the minus seventeenth for the tau family.

Equation 28 in the source gives the critical temperature for the Bose-Einstein condensation of a quantum gas of free bosons of a given mass and mean number density as two pi divided by the mass, times the two-thirds power of the number density over the Riemann zeta function of three halves. Equation 29 in the source evaluates it: about 9.7 times ten to the thirtieth gigaelectronvolts for the electron family, 7.7 times ten to the thirty-ninth for the muon family, and 6.1 times ten to the forty-second for the tau family. All three critical temperatures are comfortably larger than the Planck mass of 1.22 times ten to the nineteenth gigaelectronvolts, such that throughout the Universe’s expansion history — modulo depercolation, which generates a nonthermal halo of particles correlated on the de Broglie wavelength around a condensate core — the Bose-condensed state of the three axion species is guaranteed and consistent with the interparticle-distance ratio being far below one.

We now turn back to the first question. Explicit lump masses can be obtained from the mass-ratio relations based on the typical electron-lump mass of 6.3 times ten to the tenth solar masses. Equation 30 in the source gives them: 1.5 times ten to the sixth solar masses for a muon-lump, and 5.2 times ten to the third for a tau-lump.

For the computation of the respective gravitational Bohr radii, both the axion mass and the lump mass are required. To judge the gravitational stability of a given isolated and unmerged lump throughout its evolution, a comparison between the typical Bohr radius and the typical Schwarzschild radius, defined in equation 31 in the source as twice the lump mass divided by the squared Planck mass, is in order. Figure 9 of the source indicates the implied Bohr radii for the three families by dots on the curves of all possible Bohr radii as functions of their lump masses when keeping the axion mass fixed, with the Schwarzschild radius drawn as a dashed line. Notice that for all three cases typical Bohr radii are considerably larger than their Schwarzschild radii. Indeed, equation 32 in the source shows that the ratio of the two is one half the squared universal constant. With that constant equal to 314 we have a ratio of 4.92 times ten to the fourth. An adiabatic pursuit of the solid lines in figure 9 down to their intersections with the dashed line reveals that an increase of lump mass by a factor of about 222 is required to reach the critical mass for black-hole formation. While this is unlikely to occur through mergers of electron-lumps within their peers, it is conceivable for merging muon- and tau-lumps.

The mean mass density of a lump scales with the fourth power of the Yang-Mills scale. With the hierarchies in Yang-Mills scales — about 17 between the tau and muon theories, about 200 between the muon and electron theories — it is conceivable that a sufficiently large number of lumps of a higher Yang-Mills scale, embedded into a lump of a lower scale, catalyse the latter’s gravitational compaction to the point of collapse.

With a muon-to-electron lump mass ratio of about 2.3 times ten to the minus fifth, a dark mass of the selfgravitating dark-matter disk of the Milky Way — exhibiting a radial scale of 7.5 to 8.85 kiloparsecs and a mass of 2 to 3 times ten to the eleventh solar masses — would contain a few previously isolated but now merged electron-lumps. This implies, by equation 33 in the source, a muon-lump mass of 4.7 to 7 times ten to the sixth solar masses.

The mass of the dark halo of the Milky Way, which is virialised out to about 350 kiloparsecs, is determined as 1.8 times ten to the twelfth solar masses. In addition to the halo and the disk, there is a ringlike dark-matter structure within 13 to 18.5 kiloparsecs of mass 2.2 to 2.8 times ten to the tenth solar masses. Since these structures probably are, judged within the here-discussed framework, due to contaminations of a seeding electron-lump by the accretion of tau- and muon-lumps, we ignore them in what follows. In any case, a virialised dark-matter halo of 350 kiloparsecs radial extent easily accommodates the dark mass ratio of about a tenth between the selfgravitating dark-matter disk and the dark halo in terms of accreted tau- and muon-lumps.

Interestingly, the lower mass bound of that muon-lump range is contained in the mass range of 4.5 plus or minus 0.4, or 4.31 plus or minus 0.36, times ten to the sixth solar masses for the central compact object extracted from orbit analysis of S-stars.

Next, we discuss the second question. Consider a situation where the gravitational Bohr radius exceeds the Hubble radius at some redshift. In such a situation, the lump acts like a homogeneous energy density — dark energy — within the causally connected region of the Universe roughly spanned by the Hubble radius. If the Bohr radius falls sizably below the Hubble radius then formerly homogeneous energy density may decay into isolated lumps. In order to predict at which redshift such a depercolation epoch has taken place we rely on the extraction of a redshift of 53 for the depercolation of electron-lumps. To extract the muon and tau depercolation redshifts we use the SU(2) cosmological model for the CMB with its published parameter values. Figure 10 of the source depicts the relative density parameters of that model as functions of redshift — dark energy, total matter both baryonic and dark, and radiation comprising three flavours of massless neutrinos and eight relativistic polarisations in a CMB subject to the SU(2) description — together with the Hubble radius as a dotted line, with the point of electron-lump depercolation marked by the cusps in dark energy and matter.

The strategy to extract the muon and tau depercolation redshifts out of information collected at the electron one is to determine the ratio, equation 34 in the source, of the Hubble radius of 16.4 megaparsecs at redshift 53 to the electron-lump Bohr radius of 0.26 kiloparsecs, which is 55,476. It is plausible that this can be promoted to a universal constant, independent of the Yang-Mills scale and temperature, again because of the large hierarchy between all Yang-Mills scales and the Planck mass. Moreover, the ratio of radiation temperature to the Planck mass remains very small within the regime of redshifts considered in typical CMB simulations. Using the same cosmological model, the Bohr-radius relation, and demanding that ratio to set the condition for muon- and tau-lump depercolation, equation 35 in the source yields depercolation redshifts of 40,000 for the muon family and 685,000 for the tau family. Those two are not modelled within the CMB cosmological model because the Universe then is radiation dominated.

Figure 11 of the source depicts a schematic evolution of the Universe’s dark sector under the four SU(2) Yang-Mills theories invoking Planck-scale induced axial anomalies, with the confining scales screened by the factor of 15 and the CMB scale of about ten to the minus fourth electronvolts in its deconfining phase. At the first epoch, gravity-induced chiral symmetry breaking at the Planck scale creates a would-be Goldstone boson which, due to the axial anomaly, gives rise to four ultralight axionic particle species whose gravitational Bohr radii are all much larger than the Hubble radius, so the associated energy densities should be interpreted as dark energy. As the radiation-dominated Universe expands, the smallest Bohr radius, that of the tau family, falls below the Hubble radius, and once the ratio is about 55,500 the tau-lumps depercolate at redshift 685,000. As the Universe expands further the muon Bohr radius falls below the Hubble radius and muon-lumps depercolate at redshift 40,000. The cosmological matter densities of tau- and muon-lumps are comparable; since the mass of an isolated tau-lump is smaller than that of a muon-lump by a factor of about 283, the number density of tau-lumps is larger by that factor. Upon continued expansion down to redshift 53 the electron-lumps depercolate, their number density smaller than that of muon-lumps by a factor of about 42,750. The CMB Bohr radius, 2.4 times ten to the tenth megaparsecs, is vastly larger than today’s Hubble radius of 4038 megaparsecs, so a depercolation of CMB-lumps up to the present is excluded — as a consequence, the condensate of CMB-axions is dark energy. The last panel shows a possible dark-matter configuration of a galaxy including tau-lumps and a single muon-lump inside an electron-lump.

Again, ignoring local gravitational binding effects, the dilution of tau- and muon-lump densities by cosmological expansion predicts that today we have 42,750 divided by the cube of one plus 53, that is 0.27, muon-lumps and 77 tau-lumps within one electron-lump. Local gravitational binding should correct these numbers to higher values, but the orders of magnitude — of order one for muon-lumps and of order a hundred for tau-lumps — should remain unaffected. It is conspicuous that the number of globular clusters within the Milky Way is in the hundreds, with typical masses between ten thousand and several hundred thousand solar masses. With a tau-lump mass of 5.2 times ten to the third solar masses it is plausible that the dark-mass portion of these clusters is constituted by a single or a small number of merged tau-lumps. In addition, in the Milky Way there is one central massive and dark object with about 4.5 plus or minus 0.4, or 4.31 plus or minus 0.36, times ten to the sixth solar masses. If, indeed, there is roughly one isolated muon-lump per isolated electron-lump today, then the mass range of the Milky Way’s dark-matter disk, interpreted as a merger of a few isolated electron-lumps, implies the muon-lump merger mass range given above. That range contains the mass of the central massive and dark object.

5. Discussion, summary, and outlook

5.1. Speculations on origins of the Milky Way’s structure

The results on mass ranges of tau-lumps, muon-lumps, and electron-lumps being compatible with typical masses of globular clusters, the mass of the central compact Galactic object, and the mass of the selfgravitating dark-matter disk of the Milky Way, respectively, is compelling. We expect that similar assignments can be made to according structures in other spiral galaxies.

Could the origin of the central compact object in the Milky Way be the result of tau- and muon-lump mergers? As figure 9 suggests, a merger of at least 222 isolated tau- or muon-lumps is required for black-hole formation. Since we know that the mass of the central compact object is about 4 times ten to the sixth solar masses, a merger of at least 222 muon-lumps is excluded for the Milky Way. Thus, only a merger of at least 222 tau-lumps, possibly catalysed by the consumption of a few muon-lumps, is a viable candidate for black-hole formation in our Galaxy. Such a process — merging of several hundred tau-lumps within the gravitational field of a few merging muon-lumps down to the point of gravitational collapse — would be consistent with the results of the S-star orbit analyses, which fit stellar orbits around the central massive object of the Milky Way extremely well to a single-point-mass potential. Indeed, the gravitational Bohr radius of a muon-lump is 7 times ten to the minus sixth kiloparsecs while the closest approach of an S2 star to the gravitational center of the central massive object is 17 light hours, or 5.8 times ten to the minus seventh kiloparsecs. Therefore, muon-lumps need to collapse in order to be consistent with a point-mass potential.

The Milky Way’s contamination with baryons, its comparably large dark-disk mass versus the mass of the low-surface-brightness galaxies analysed in section 3, and possibly tidal shear from the dark ring and the dark halo during its evolution introduce deviations from the simple structure of a typical low-surface-brightness galaxy. Simulations, which take all the here-discussed components into account, could indicate how typical such structures are, rather independently of primordial density perturbations.

Isolated tau-, muon-, and electron-lumps, which did not accrete sufficiently many baryons to be directly visible, comprise dark-matter galaxies that are interspersed in between visible galaxies. The discovery of such dark galaxies, pinning down their merger physics, and determinations of their substructure by gravitational microlensing and gravitational-wave astronomy could support the here-proposed scenario of active structure formation on sub-galactic scales.

5.2. Summary and outlook

In this paper, we propose that the dark Universe can be understood in terms of axial anomalies which are invoked by screened Yang-Mills scales in association with the leptonic mass spectrum. This produces three ultra-light axion species. Such pseudo Nambu-Goldstone bosons are assumed to owe their very existence to a gravitationally induced chiral symmetry breaking with a universal Peccei-Quinn scale of order the Planck mass, 1.22 times ten to the nineteenth gigaelectronvolts. We therefore refer to each of these particle species as Planck-scale axions. Because the axion mass is the squared Yang-Mills scale over the Planck mass, the screened Yang-Mills scale derives from knowledge of the axion mass. Empirically, the here-extracted screened scale of 287 electronvolts points to the first lepton family. This enables predictions of typical lump and axion masses in association with two additional SU(2) Yang-Mills theories associating with muon and tau leptons.

Even though the emergence of axion mass and the existence of lepton families are governed by the same SU(2) gauge principle, the interaction between these ultra-light pseudo scalars and visible leptonic matter is extremely feeble. Thus, the here-proposed relation between visible and dark matter could demystify the dark Universe. An important aspect of Planck-scale axions is their Bose-Einstein, yet non-thermal, condensed state. A selfgravitating, isolated fuzzy condensate, a lump, of a given axion species is chiefly characterised by the gravitational Bohr radius given in terms of the axion mass and the lump mass, the virial mass. As it turns out, for the electron family the information about the latter two parameters is contained in observable rotation curves of low-surface-brightness galaxies with similar extents. Realistic models for the dark-matter density profiles derive from ground-state solutions of the spherically symmetric Poisson-Schrodinger system at zero temperature and for a single axion species. These solutions describe selfgravitating fuzzy axion condensates. Two such models, the Soliton-Navarro-Frenk-White and the Burkert model, were employed in our present extractions under the assumption that the dark-matter density in a typical low-surface-brightness galaxy is dominated by a single axion species. Our result of 0.675 times ten to the minus twenty-third electronvolts is consistent with the earlier result of 0.554.

Interestingly, such an axion mass is close to the range between ten to the minus twenty-fifth and ten to the minus twenty-fourth electronvolts obtained by treating axions as a classical ideal gas of non-relativistic particles — in stark contrast to the Bose condensed state suggested by the condensation-temperature relation, or the gas surrounding it with intrinsic correlations governed by large de Broglie wavelengths. This value of the axion mass is considerably lower than typical lower bounds obtained in the literature: more than 2.9 times ten to the minus twenty-first electronvolts; 2.5, plus 3.6 and minus 2.0, times ten to the minus twenty-first; more than 3.8 times ten to the minus twenty-first; and about 8 times ten to the minus twenty-third. We propose that this discrepancy could be due to the omission of the other two axion species with the mass spectrum given above. For example, the dark-matter and thus baryonic density variations along the line of sight probed by a Lyman-alpha forest do not refer to gravitationally bound systems and therefore should be influenced by all three axion species.

Once axions and their lumps are categorised, questions about the cosmological origin of lumps and their role in the evolution of galactic structure can be asked. The first is addressed by consulting a cosmological model which requires the emergence of dark matter by lump depercolation at defined redshifts. Depercolation of electron-lumps at redshift 53 anchors the depercolations of the two other lump species; one obtains 40,000 and 685,000.

The critical temperature of the electron theory for the deconfining-preconfining phase transition, roughly equal to the temperature of the Hagedorn transition to the confining phase, is 9.49 kiloelectronvolts. A question arises whether this transition could affect observable small-scale angular features of the CMB. In the SU(2)-based cosmological model that temperature corresponds to a redshift of 6.4 times ten to the seventh; typically, CMB simulations are initialised at redshift ten to the ninth. Traversing the preconfining-deconfining phase transition at that redshift, an already strongly radiation dominated Universe receives additional radiation density and entropy. However, we expect that the horizon crossing of curvature perturbations at higher redshift, which may influence small-scale matter perturbations, will affect CMB anisotropies on angular scales above multipole 3000 only. Therefore, Silk damping would reduce the magnitudes of these multipoles to below the observational errors.

Up to the present, lump depercolation does not occur for the Planck-scale axion species associated with the CMB theory: here, the gravitational Bohr radius of the axion condensate always exceeds the Hubble radius by many orders of magnitude. As for the second question, the masses and Bohr radii of muon- and tau-lumps seem to be related with the central massive compact object of the Milky Way and with globular clusters, respectively. Within a given galaxy such active components of structure formation possibly originate compact stellar streams through tidal forces acting on tau-lumps. Whether this is supported by observation could be decided by a confrontation of N-body simulations of stars in the selfgravitating background of the externally deformed lump.

Apart from cosmological and astrophysical observation, which should increasingly be able to judge the viability of the here-proposed scenario, there are alternative terrestrial experiments which can check the predictions of the underlying SU(2) gauge-theory pattern. Let us quote two examples. First, there is a predicted low-frequency spectral black-body anomaly at low temperatures, around 5 kelvin, which could be searched for with a relatively low instrumental effort. Second, an experimental link to the electron-family theory would be the detection of the Hagedorn transition in a plasma at an electron temperature of 9.49 kiloelectronvolts and the stabilisation of a macroscopically large plasma ball at a temperature of 1.3 times 9.49 kiloelectronvolts. Such electron temperatures should be attainable by state-of-the-art nuclear-fusion experiments such as ITER or by fusion experiments with inertial plasma confinement.

Author contributions, funding and data

Both authors contributed to conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing of the original draft, review and editing, visualization, supervision, project administration and funding acquisition, and both have read and agreed to the published version of the manuscript. This research received no external funding. The SPARC library was analysed in support of this research; the processed data and program underlying this article will be shared on request to the corresponding author. The authors declare no conflict of interest.