An SU(2) gauge principle for the Cosmic Microwave Background: Perspectives on the Dark Sector of the Cosmological Model
AAn SU(2) gauge principle for the Cosmic Microwave Background:Perspectives on the Dark Sector of the Cosmological Model
Ralf Hofmann
Institut f¨ur Theoretische Physik, Universit¨at Heidelberg,Philosophenweg 16, D-69120 Heidelberg, Germany;[email protected]
Abstract
We review consequences for the radiation and dark sectors of the cosmologicalmodel arising from the postulate that the Cosmic Microwave Background (CMB) isgoverned by an SU(2) rather than a U(1) gauge principle. We also speculate on the pos-sibility of actively assisted structure formation due to the de-percolation of lump-likeconfigurations of condensed ultralight axions with a Peccei-Quinn scale comparable tothe Planck mass. The chiral-anomaly induced potential of the axion condensate re-ceives contributions from SU(2)/SU(3) Yang-Mills factors of hierarchically separatedscales which act in a screened (reduced) way in confining phases. a r X i v : . [ phy s i c s . g e n - ph ] A ug Introduction
Judged by decently accurate agreement of cosmological parameter values extracted from(i) large-scale structure observing campaigns on galaxy- and shear-correlation functions aswell as redshift-space distortions towards a redshift of unity by photometric/spectroscopicsurveys (sensitive to Baryonic Accoustic Oscillations (BAO), the evolution of Dark Energy,and non-linear structure growth), for recent, ongoing, and future projects see, e.g. [1, 2,3, 4, 5], (ii) fits of various CMB angular power spectra based on frequency-band optimisedintensity and polarisation data collected by satellite missions [6, 7, 8], and (iii) fits tocosmologically local luminosity-distance redshift data for Supernovae Ia (SNeIa) [9, 10]the spatially flat standard Lambda Cold Dark Matter (ΛCDM) cosmological model is agood, robust starting point to address the evolution of our Universe. About twenty yearsago, the success of this model has triggered a change in paradigm in accepting a presentstate of accelerated expansion induced by an essentially dark Universe made of 70 % DarkEnergy and 25 % Dark Matter.While (i) and (ii) are anchored on comparably large cosmological standard-ruler co-moving distance scales, the sound horizons r s, d and r s, ∗ , which emerge at baryon-velocityfreeze-out and recombination, respectively, during the epoch of CMB decoupling and there-fore refer to very-high-redshift physics ( z > . < z < .
3, which do not significantly de-pend on the assumed model for Dark Energy [11] and are robust against sample variance,local matter-density fluctuations, and directional bias [12, 13, 14, 15].Thanks to growing data quality and increased data-analysis sophistication to identifySNeIa [16] and SNeII [17] spectroscopically, to establish precise and independent geo-metric distance indicators [18, 19] (e.g., Milky Way parallaxes, Large Magellanic Cloud(LMC) detached eclipsing binaries, and masers in NGC 4258), tightly calibrated period-to-luminosity relations for LMC cepheids and cepheids in SNeIa host galaxies [19], the useof the Tip of the Red Giant Branch (TRGB) [20, 21] or the Asymptotic Giant Branch(AGB, Mira) [22] to connect to the distance ladder independently of cepheids, the highvalue of the basic cosmological parameter H (Hubble expansion rate today) has (see [21],however), over the last decade and within ΛCDM, developed a tension of up to ∼ . σ [19]in SNeIa distance-redshift fits using the LMC geometrically calibrated cepheid distanceladder compared to its low value extracted from statistically and systematically accuratemedium-to-high-l CMB angular power spectra [7, 8] and the BAO method [3]. Both, theCMB and BAO probe the evolution of small radiation and matter density fluctuations inglobal cosmology: Fluctuations are triggered by an initial, very-high-redshift (primordial)spectrum of scalar/tensor curvature perturbations which, upon horizon entry, evolve lin-early [23] up to late times when structure formation generates non-linear contributions inthe galaxy spectra [2, 3] and subjects the CMB to gravitational lensing [24].Recently, a cosmographic way of extracting H through time delays of strongly lensedhigh-redshift quasars, see e.g. [25, 26], almost matches the precision of the presentlymost accurate SNeIa distance-redshift fits [19] – H = 71 . +2 . − . km s − Mpc − vs. H =(74 . ± .
82) km s − Mpc − –, supporting a high local value of the Hubble constant andrendering the local-global tension even more significant [27]. The high local value of H = (72 · · ·
74) km s − Mpc − [19, 27] (compared to the global value of H = (67 . ± .
5) km s − Mpc − [8]) is stable to sources of systematic uncertainty such as line-of-sight1ffects, peculiar motion (stellar kinematics), and assumptions made in the lens model.There is good reason to expect that an improved localisation of sources for gravitational-wave emission without an electromagnetic counterpart and the increase of statistics ingravitational-wave events accompanied by photon bursts (standard sirenes) within specifichost galaxies will lead to luminosity-distance-high-redshift data producing errors in H comparable of those of [19], independently of any distance-ladder calibration [28], see also[29].Apart from the Hubble tension, there are smaller local-global but possibly also tensionsbetween BAO and the CMB in other cosmological parameters such as the amplitudeof the density fluctuation power spectrum ( σ ) and the matter content (Ω m ), see [30].Finally, there are persistent large-angle anomalies in the CMB, already seen by the CosmicBackground Explorer (COBE) and strengthened by the Wilkinson Microwave AnisotropyProbe (WMAP) and Planck satellite, whose ’atoms’ (a) lack of correlation in the TT two-point function, (b) a rather significant alignment of the low multipoles ( p -values of below0.1 %), and (c) a dipolar modulation, which is independent of the multipole alignment(b), indicate a breaking of statistical isotropy at low angular resolution, see [31] for acomprehensive and complete review.The present work intends to review and discuss a theoretical framework addressing apossibility to resolve the above-sketched situation. The starting point is to subject theCMB to the thermodynamics of an extended gauge principle in replacing the conventionalgroup U(1) by SU(2). Motivated [32] by explaining an excess in CMB line temperatureat radio frequencies, see [33] and references therein, this postulate implies a modifiedtemperature-redshift relation which places CMB recombination to a redshift of ∼ ∼ z , one requires a release of Dark Matter from early Dark Energywithin the dark ages at a redshift z p : lumps and vortices, formely tighly correlated withina condensate of ultralight axion particles, de-percolate into independent pressureless (andselfgravitating) solitons due to cosmological expansion, thereby contributing actively tonon-linear structure formation at low z . A fit to CMB angular power spectra of a cos-mological model, which incorporates these SU(2) features on the perfect-fluid level butneglects low- z radiative effects in SU(2) [34], over-estimates TT power on large angularscales but generates a precise fit to the data for l ≥
30. With z p ∼
53 a locally favouredvalue of H ∼
74 km s − Mpc − and a low baryon density ω b, ∼ .
017 are obtained.Moreover, the conventionally extracted near scale invariance of adiabatic, scalar curvatureperturbations comes out to be significantly broken by an infrared enhancement of theirpower spectrum. Finally, the fitted redshift ∼ z quasars [36] and distinctlydiffers from the high value of CMB fits to ΛCDM [7, 8].As of yet, there are loose ends to the SU(2) based scenario. Namely, the physics ofde-percolation requires extra initial conditions for matter density fluctuations at z p . In theabsence of a precise modelling of the ’microscopics’ of the associated soliton ensembles it is only a guess that these fluctuations instantaneously follow the density fluctuations ofprimordial Dark Matter as assumed in [34]. Moreover, it is necessary to investigate to whatextent profiles of the axion field (lumps of localised energy density) actively seed non-linear Lump sizes could well match those of galactic dark-matter halos, see Sec. 4. z [37], which are expected to contribute or even explainthe above-mentioned ’atoms’ (a), (b), and (c) of the CMB large-angle anomalies, see [38],and which could reduce the excess in low- l TT power of [34] to realistic levels, need tobe incorporated into the model. They require analyses in terms of Maxwell’s multipole-vector formalism [31] and/or other robust and intuitive statistics to characterise multipolealignment and the large-angle suppression of TT.This review-type paper is organised as follows. In Sec. 2 we briefly discuss the maindistinguishing features between the conventional ΛCDM model and cosmology based on aCMB which obeys deconfining SU(2) Yang-Mills thermodynamics. A presentation of ourrecent results on angular-power-spectra fits to Planck data is carried out Sec. 3, includinga discussion of the parameters H , n s , σ , z re , and ω b,0 . In Sec. 4 we interpret the roughcharacteristics of the Dark Sector employed in [34] to match ΛCDM at low z . In particular,we point out that the value of z p seems to be consistent with the typical dark-matterdensities in the Milky Way. Finally, in Sec. 5 we sketch what needs to be done to arrive ata solid, observationally well backed-up judgement of whether SU(2) CMB based cosmology(and its extension to SU(2) and SU(3) factors of hierarchically larger Yang-Mills scalesincluding their nonthermal phase transitions) may provide a future paradigm to connectlocal cosmology with the very early Universe. Throughout the article, super-natural units (cid:126) = k B = c = 1 are used. CMB vs. conventional CMB photon gas in Λ CDM
The introduction of an SU(2) gauge principle for the description of the CMB is motivatedtheoretically by the fact that the deconfining thermodynamics of such a Yang-Mills theoryexhibits a thermal ground state, composed of densely packed (anti)caloron [39] centerswith overlapping peripheries [40, 41], which breaks SU(2) to U(1) in terms of an adjointHiggs mechanism [42]. Therefore, the spectrum of excitations consists of one masslessgauge mode, which can be identified with the CMB photon, and two massive vectormodes of a temperature-dependent mass on tree level: thermal quasi-particle excitations.The interaction between these excitations is feeble [42]. This is exemplified by the one-loop polarisation tensor of the massless mode [43, 37, 44]. As a function of temperature,polarisation effects peak at about twice the critical temperature T c for the deconfining-preconfining transition. As a function of increasing photon momentum, there are regimesof radiative screening/antiscreening, the latter being subject to an exponential fall-off[37]. At the phase boundary ( T ∼ T c ) electric monopoles [42], which occur as isolated andunresolved defects deeply in the deconfining phase, become massless by virtue of screeningdue to transient dipoles [45] and therefore condense to endow the formely massless gaugemode with a quasiparticle Meissner mass m γ . This mass rises critically (with mean-field exponent) as T falls below T c [42]. Both, (i) radiative screening/antiscreening ofmassless modes and (ii) their Meissner effect are important handles in linking SU(2) cmb to the CMB: While (i) induces large-ange anomalies into the TT correlation [38] andcontributes dynamically to the CMB dipole [46, 47] (ii) gives rise to a nonthermal spectrum3f evanescent modes for frequencies ω < m γ once T falls below T c . This theoreticalanomaly of the blackbody spectrum in the Rayleigh-Jeans regime can be considered toexplain the excess in CMB radio power below 1 GHz, see [33] and references therein,thereby fixing T c = 2 .
725 K and, as a consequence of λ c = 13 .
87 = πT c Λ cmb [42], the Yang-Mills scale of SU(2) cmb to Λ cmb ∼ − eV [32].Having discussed the low-frequency deviations of SU(2) CMB from the conventionalRayleigh-Jeans spectrum, which fix the Yang-Mills scale and associate with large-angleanomalies, we would now like to review its implications for the cosmological model. Ofparamount importance for the set-up of such a model is the observtion that SU(2) cmb implies a modified temperature ( T )-redshift ( z ) relation for the CMB which is derived fromenergy conservation of the SU(2) cmb fluid in the deconfining phase in an FLRW universewith scale factor a normalised to unity today [48]. Denoting by T c = T = 2 .
725 K [32]the present CMB baseline temperature [6] and by ρ SU(2) cmb and P SU(2) cmb energy densityand pressure, respectively, of SU(2) cmb , one has a ≡ z + 1 = exp (cid:32) −
13 log (cid:32) s SU(2) cmb ( T ) s SU(2) cmb ( T ) (cid:33)(cid:33) , (1)where the entropy density s SU(2) cmb is defined by s SU(2) cmb ≡ ρ SU(2) cmb + P SU(2) cmb
T . (2)For T (cid:29) T , Eq. (1) simplifies to T = (cid:18) (cid:19) / T ( z + 1) ≈ . T ( z + 1) . (3)For arbitrary T ≥ T , a multiplicative deviation S ( z ) from linear scaling in z + 1 can beintroduced as S ( z ) = (cid:32) ρ SU(2) cmb ( z = 0) + P SU(2) cmb ( z = 0) ρ SU(2) cmb ( z ) + P SU(2) cmb ( z ) T ( z ) T (cid:33) / . (4)Therefore, T = S ( z ) T ( z + 1) . (5)Fig. 1 depicts function S ( z ). Amusingly, the asymptotic T - z relation of Eq. (3) also holdsfor the relation between ρ SU(2) cmb ( z ) and the conventional CMB energy density ρ γ ( z ) inΛCDM (the energy density of a thermal U(1) photon gas, using the T - z relation T = T ( z + 1)). Namely, ρ SU(2) cmb ( z ) = 4 (cid:18) (cid:19) / ρ γ ( z ) = (cid:18) (cid:19) / ρ γ ( z ) ( z (cid:29) . (6) That the deep Rayleigh-Jeans regime is indeed subject to classical wave propagation is assured by thefact that wavelengths then are greater than the spatial scale s ≡ πT | φ | − , separating a(n) (anti)caloroncenter from its periphery where its (anti)selfdual gauge field is that of a dipole [41]. The expression for s contains the modulus | φ | = (cid:112) Λ cmb / (2 πT ) of the emergent, adjoint Higgs field φ (Λ cmb ∼ − eV the Yang-Mills scale of SU(2) cmb ), associated with densely packed (anti)caloron centers, and, explicitely, temperature T . z . . . . . S ( z ) SU (2) CMB Λ CDM
Figure 1: The function S ( z ) of Eq. (4), indicating the (multiplicative) deviation from theasymptotic T - z relation in Eq. (3). Curvature in S ( z ) for low z arises from a breakingof scale invariance for T ∼ T = T c . There is a rapid approach towards the asymptotics (cid:0) (cid:1) / ≈ .
63 with increasing z . Figure adopted from [34].5herefore, the (gravitating) energy density of the CMB in SU(2) cmb is, at the same redshift z (cid:29)
1, by a factor of ∼ .
63 smaller than that of the ΛCDM model even though there areeight (two plus two times three) gauge-mode polarisations in SU(2) cmb and only two suchpolarisations in the U(1) photon gas.Not yet considering linear and next-to-linear perturbations in SU(2) cmb to shape typi-cal CMB large-angle anomalies in terms of late-time screening/antiscreening effects [38],Eq. (5) has implications for the Dark Sector if one wishes to maintain the successes ofthe standard cosmological model at low z where local cosmography and fits to distance-redshift curves produce a consistent framework within ΛCDM. As it was shown in [34],the assumption of matter domination at recombination, which is not unrealistic even forSU(2) cmb [34], implies that z ΛCDM , ∗ ∼ (cid:18) (cid:19) / z SU(2) cmb , ∗ (7)and, as a consequence, Ω ΛCDM ,m, ≈ SU(2) cmb ,m, , (8)where Ω m, denotes today’s density parameter for nonrelativistic matter (Dark Matterplus baryons), and z ∗ is the redshift of CMB photon decoupling in either model. Sincethe matter sector of the SU(2) cmb model, as roughly represented by Eq. (8), contradictsΛCDM at low z one needs to allow for a transition between the two somewhere in the darkages as z decreases. In [34] a simple model, where the transition is sudden at a redshift z p and maintains a small dark-energy residual, was introduced: a coherent axion field – adark-energy like condensate of ultralight axion particles, whose masses derive from U(1) A anomalies [49, 50, 51, 52, 53] invoked by the topological charges of (anti)caloron centersin the thermal ground state of SU(2) CMB and, in a screened way, SU(2)/SU(3) Yang-Millsfactors of higher scales – releases solitonic lumps by de-percolation due to the Universe’sexpansion. Accordingly, we haveΩ ds ( z ) = Ω Λ + Ω pdm , ( z + 1) + Ω edm , (cid:26) ( z + 1) , z < z p ( z p + 1) , z ≥ z p . (9)Here Ω Λ and Ω pdm , + Ω edm , ≡ Ω cdm , represent today’s density parameters for DarkEnergy and Dark Matter, respectively, Ω pdm , refers to primordial Dark Matter (thatis, dark matter that existed before the initial redshift of z i ∼ used in the CMBBoltzmann code) for all z and Ω edm , to emergent Dark Matter for z < z p . In [34] theinitial conditions for the evolution of density and velocity perturbations of the emergentDark-Matter portion at z p are, up to a rescaling of order unity, assumed to follow thoseof the primordial Dark Matter. cmb fit of cosmological parameters to Planck data In [34] a simulation of the new cosmological model subject to SU(2) cmb and the DarkSector of Eq. (9) was performed using a modified version of the Cosmic Linear AnisotropySolving System (CLASS) Boltzmann code [54]. Best fits to the 2015 Planck data [8] on theangular power spectra of the two-point correlation functions temperature – temperature6TT), electric-mode polarisation – electric-mode polarisation (EE), and temperature –electric-mode polarisation (TE), subject to typical likelihood functions used by the Planckcollaboration, were performed. Because temperature perturbations can only be coherentlypropagated by (low-frequency) massless modes in SU(2) cmb [41] the propagation of the(massive) vector modes was excluded in one version of the code (physically favoured).Also, entropy conservation in e + e − annihilation invokes a slightly different counting ofrelativistic degrees of freedom in SU(2) cmb for this epoch [55]. As a consequence, we havea z -dependent density parameter for (massless) neutrinos given as [34]Ω ν ( z ) = 78 N eff (cid:18) (cid:19) Ω SU(2) cmb ,γ ( z ) , (10)where N eff refers to the effective number of neutrino flavours (or any other extra relativistic,free-streaming, fermionic species), and Ω SU(2) cmb ,γ is the density parameter associated withthe massless mode only in the expression for ρ SU(2) cmb ( z ) of Eq. (6). The value N eff = 3 . ≤ l ≤
30, however. This excess could be attributable to the omission ofradiative effects in the low- z propagation of the massless mode. The consideration ofthe modified dispersion law into the CMB Boltzmann code presently is under way. Thefollowing table was obtained in [34]: 7
10 30 l . . . . . l ( l + ) π C TT l × − CMB
SU(2)
CMB + V ± Figure 2: Normalised power spectra of TT correlator for best-fit parameter values quotedin Tab. 1: Dashed, dotted, and solid lines represent ΛCDM, SU(2) cmb + V ± (not consideredin in the present work), and SU(2) cmb , respectively. For l ≤
29 the 2015 Planck datapoints are unbinned and carry error bars, for l ≥
30 grey points represent unbinnedspectral power. Figure adopted from [34]. 8
250 500 750 1000 1250 1500 1750 2000 l − − l ( l + ) π C T E l × − ΛCDMSU(2)
CMB
SU(2)
CMB + V ± Figure 3: Normalised power spectra of TE cross correlator for best-fit parameter valuesquoted in Tab. 1: Dashed, dotted, and solid lines represent ΛCDM, SU(2) cmb + V ± (notconsidered in in the present work), and SU(2) cmb , respectively. The 2015 Planck datapoints are either unbinned without error bars (grey) or binned with error bars (blue).Figure adopted from [34]. 9
250 500 750 1000 1250 1500 1750 2000 l − − − l ( l + ) π C EE l × − ΛCDMSU(2)
CMB
SU(2)
CMB + V ± Figure 4: Normalised power spectra of EE cross correlator for best-fit parameter valuesquoted in Tab. 1: Dashed, dotted, and solid lines represent ΛCDM, SU(2) cmb + V ± (notconsidered in in the present work), and SU(2) cmb , respectively. The 2015 Planck datapoints are either unbinned without error bars (grey) or binned with error bars (blue).Figure adopted from [34]. 10e would like to discuss the following parameters (The value of z p will be discussed inSec. 4.): (i) H , (ii) n s , (iii) σ , (iv) z re , and (v) ω b,0 . (i): The fitted value of H is in agree-ment with local-cosmology observation [19, 27] but discrepant at the 4.5 σ level with thevalue H ∼ (67 . ± .
5) km Mpc − s − in ΛCDM of global-cosmology fits as extracted bythe Planck collaboration [8], a galaxy-custering survey – the Baryonic Oscillation Spec-troscopy Survey (BOSS) [3] – , and a galaxy-custering-weak-lensing survey – the DarkEnergy Survey (DES) [2] – whose distances are derived from inverse distance ladders us-ing the sound horizons at CMB decoupling or baryon drag as references. Such anchorsassume the validy of ΛCDM (or a variant thereof with variable Dark Energy) at high z ,and the data analysis employs fiducial cosmologies that are close to those of CMB fits. Asa rule of thumb, the inclusion of extra relativistic degrees of freedom within reasonablebounds [3, 2] but also sample variance, local matter-density fluctuations, or a directionalbias in SNa Ia observations [12, 13, 14, 15] cannot explain the above-quoted tension in H .Note that the DES Y1 data on three two-point functions (cosmic shear auto correlation,galaxy angular auto correlation, and galaxy-shear cross correlation in up to five redshiftbins with z ≤ . m and σ (or S = σ / Ω m ), cannotusefully constrain H by itself. However, the combination of DES 1Y data with those ofPlanck (no lensing) yields an increases in H on the 1-2 σ level compared to the Planck(no lensing) data alone, and a similar tendency is seen when BAO data [3] are enrichedwith those of DES Y1, see Table II in [2]. Loosely speaking, one thus may state a mildincrease of H with an increasing portion of late-time (local-cosmology) information in thedata. (ii): The index of the initial spectrum of adiabatic, scalar curvature perturbations n s is unusually low, expressing an enhancement of infrared modes as compared to theultraviolet ones (violation of scale invariance). As discussed in [34], such a tilted spectrumis not consistent with single-field slow-roll inflation and implies that the Hubble parameterduring the inflationary epoch has changed appreciably. (iii): There is a low value of σ (initial amplitude of matter-density power spectrum at comoving wavelength 8 h − Mpcwith H ≡ h
100 km s − Mpc − ) compared to CMB fits [8] and BAO [3] although the DESY1 fit to cosmic shear data alone would allow for a value σ ∼ .
71 within the 1- σ margin[2]. This goes also for our high value of today’s matter density parameter Ω m,0 (the ratioof physical matter density to the critical density). (iv): The low value of the redshift to(instantaneous) re-ionisation z re (and the correspondingly low optical depth τ re ) comparedto values obtained in CMB fits [8] is consistent with the one extracted from the observationof the Gunn-Peterson trough in the spectra of high-redshift quasars [36]. (v): The lowvalue of ω b,0 – today’s physical baryon density – could provide a theoretical solution of themissing baryon problem [34] (missing compared to CMB fits to ΛCDM (CMB fits) and theprimordial D/H (Deuterium-to-Hydrogen) ratio in Big-Bang Nucleosynthesis (BBN), thelatter yielding ω b,0 = 0 . ± . σ lower than the Planck value of0 . ± . ω b,0 could be consolidated observationallythan either the idea that BBN is determined by isolated nuclear reaction cross sectionsonly or the observation of a truly primordial D/H in metal-poor environments or both willhave to be questioned in future ivestigations. There is a recent claim, however, that themissing 30-40 % of baryons were observed in highly ionized oxygen absorbers representingthe warm-hot intergalactic medium when illuminated by the X-ray spectrum of a quasarwith z > . cmb and the ΛCDM model as ob-tained in [34]. The best-fit parameters of ΛCDM together with their 68% confidenceintervals are taken from [8], employing the TT,TE,EE+lowP+lensing likelihoods. ForSU(2) cmb the HiLLiPOP+lowTEB+lensing likelihood (lowP and lowTEB are pixel-basedlikelihoods) was used, see [35]. The central values associate with χ = χ ll + χ hl (best fit)as quoted in the lower part of the Table.Parameter SU(2) cmb ΛCDM ω b,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± . . ± . ω pdm,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± . − ω edm,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± . − θ ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . ± . . ± . τ re . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± . . ± . A s ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . ± .
009 3 . ± . n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± . . ± . z p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 . ± . − β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± . − H / km s − Mpc − . . . . . . . . . . . . . . . . . . . . . . . 74 . ± .
46 67 . ± . z re . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . +0 . − . +1 . − . z ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715 . ± .
19 1090 . ± . z d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1640 . ± .
27 1059 . ± . ω cdm,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± . . ± . Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± .
006 0 . ± . m,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± .
006 0 . ± . σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ± .
020 0 . ± . / Gyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . ± .
10 13 . ± . χ ll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10640 10495 n dof ,ll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9207 9210 χ ll n dof ,ll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.156 1.140 χ hl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10552.6 9951.47 n dof ,hl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9547 9550 χ hl n dof ,hl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.105 1.04212sing the dispersion of localised radio bursts with z ≤ . ω b,0 is extracted from the ASKAP data in [59] to be consistent with CMB fitsand BBN albeit subject to a 50 % error which is expected to decrease with the advent ofmore powerful radio observatories such as SKA. The model for the Dark Sector in Eq. (9) is motivated by the possibility that a coherentcondensate of ultra-light particles – an axion field [53] – forms selfgravitating lumps orvortices in the course of nonthermal (Hagedorn) phase transitions due to SU(2)/SU(3)Yang-Mills factors, governing the radiation and matter content of the very early Universe,going confining. A portion of the thus created, large abundance of such solitons percolatesinto a dark-energy like state: the contributions Ω Λ and Ω edm , ( z p + 1) in Eq. (9) of whichthe latter may de-percolate at z = z p into a dark-matter like component Ω edm , ( z + 1) for z < z p , the expansion of the Universe increasing the average distance between thecenters of neighbouring solitons. By virtue of the U(1) A anomaly, mediated by topologicalcharge density [49, 50, 51, 52], residing in turn within the ground state of an SU(2)/SU(3)Yang-Mills theory [42], the mass m a of an axion particle due to a single such theory ofYang-Mills scale Λ is given as [53] m a = Λ M P , (11)where we have assumed that the so-called Peccei-Quinn scale, which associates with a dynamical chiral-symmetry breaking, was set equal to the Planck mass M P = 1 . × eV, see [60, 61] for motivations. Let (natural units: c = (cid:126) = 1) r c ≡ /m a (12)denote the Compton wavelength, r B ≡ M P M m a (13)the gravitational Bohr radius, where M ∼ M (cid:12) is the total dark mass of a typical(spiral) galaxy like the Milky Way, and d a ≡ (cid:18) m a ρ dm (cid:19) / (14)the interparticle distance where ρ dm indicates the typical mean energy density in DarkMatter of a spiral. Following [63], we assume ρ dm = (0 . · · · .
4) GeV cm − . (15)For the concept of a gravitational Bohr radius in a selfconsistent, non-relativistic potentialmodel to apply, the axion-particle velocity v a in the condensate should be much small thanunity. Appealing to the virial theorem at a distance to the gravitational center of r B , onehas v a ∼ (cid:16) Mm a M P (cid:17) [62]. In [62], where a selfgravitating axion condensate was treated non-relativistically by means of a non-linear and non-local Schr¨odinger equation to represent a13ypical galactic dark-matter halo, the following citeria on the validity of such an approachwere put forward (natural units: c = (cid:126) = 1): (i) v a (cid:28)
1. ii) d a (cid:28) r c is required forthe description of axion particles in terms of a coherent Bose condensate to be realistic.(iii) r B should be the typical extent of a galactic Dark-Matter halo: r B ∼ −
300 kpc.With Λ cmb ∼ − eV one obtains m a = 8 . × − eV, d a = (4 . · · · . × − pc, r c = 3 . × pc, and r B ∼ × kpc. While (i) and (ii) are extremely well satifiedwith v a ∼ − and d a r c ∼ − point (iii) is badly violated. ( r B is about 2 × thesize of the visible Universe).Therefore, the Yang-Mills scale responsible for the axion mass that associates withdark-matter halos of galaxies must be dramatically larger. Indeed, setting Λ = 10 − Λ e where Λ e = m e . is the Yang-Mills scale of an SU(2) theory that could associate with theemergence of the electron of mass m e = 511 keV [64, 65], one obtains m a = 1 . × − eV, d a = (2 . · · · . × − pc, r c = 1 . × pc, and r B ∼
232 kpc. In addition to (i)and (ii) with v a ∼ − and d a r c ∼ − also point (iii) is now well satisfied. If theexplicit Yang-Mills scale of an SU(2) theory, which is directly imprinted in the spectraof the excitations in the pre - and deconfining phases, acts only in a screened way in theconfining phase as far as the axial anomaly is concerned – reducing its value by a factor ofone hundred or so –, then the above axionic Dark-Sector scenario would link the theoryresponsible for the emergence of the electron with galactic dark-matter halos! In addition,the axions of SU(2) cmb would provide the Dark-Energy density Ω Λ of such a scenario.Finally, we wish to point out that the de-percolation mechanism of axionic solitons(lumps forming out of former Dark-Energy density) in the cosmological model based onSU(2) cmb , which may be considered to underly the transition in the Dark Sector at z p = 53described by Eq. (9), is consistent with the Dark-Matter density in the Milky Way. Namely,working with H = 74 km s − Mpc − , see Table 1 and [19], the total (critical) energy density ρ c, of our spatially flat Universe is at present ρ c, = 38 π M P H = 1 . × − eV . (16)The portion of cosmological Dark Matter ρ cdm , then is, see Table 1, ρ cdm , ∼ . ρ c, (17)which yields a cosmological energy scale E cdm , in association with Dark Matter of E cdm , ≡ ρ / , ∼ . . (18)On the other hand, we may imagine the percolate of axionic field profiles, which dissolvesat z p , to be associated with densely packed dark-matter halos typical of today’s galaxies.Namely, scaling the typical Dark-Matter energy density of the Milky Way ρ dm of Eq. (15)from z p (at z p +0 ρ dm yet behaves like a cosmological constant) down to z = 0 and allowingfor a factor of ( ω pdm,0 + ω edm,0 ) /ω edm,0 = 2 .
47, see Table 1, one extracts the energy scalefor cosmological Dark Matter E G , in association with galactic Dark-Matter halos andprimordial dark matter as E G , ≡ (cid:18) . ρ dm ( z p + 1) (cid:19) / = (0 . · · · . . (19)14 comparison of Eqs. (18) and (19) reveals that E cdm , is smaller but comparable to E G , . This could be due to the neglect of galactic-halo compactification through themissing pull by neighbouring profiles in the axionic percolate and because of the omissionof selfgravitation and baryonic-matter accretion/gravitation during the evolution from z p = 53 to present. (That is, the use of the values of Eq. (15) in Eq. (19) overestimates thehomogeneous energy density in the percolate.) The de-percolation of axionic solitons at z p , whose mean, selfgravitating energy density ρ dm in Dark Matter is nearly independentof cosmological expansion but subject to local gravitation, could therefore be linked tocosmological Dark Matter today within the Dark-Sector model of Eq. (9). The present article’s goal was to address some tensions between local and global cosmologyon the basis of the ΛCDM standard model. Cracks in this model could be identifiedduring the last few years thanks to independent tests resting on precise observational dataand their sophisticated analysis. To reconcile these results, a change of ΛCDM likely isrequired before the onset of the formation of non-linear, large-scale structure. Here, wehave reviewed a proposal made in [34], which assumes thermal photons to be governed byan SU(2) rather than a U(1) gauge principle, and we have discussed the SU(2) cmb -impliedchanges in cosmological parameters and the structure of the Dark Sector. Noticeably, thetensions in H , the baryonic density, and the redshift for re-ionisation are addressed infavour of local measurements. High- z inputs to CMB and BAO simulations, such as n s and σ , are sizeably reduced as compared to their fitted values in ΛCDM. The Dark Sectornow invokes a de-percolation of axionic field profiles at a redshift of z p ∼
53. This idea isroughly consistent with typical galactic Dark-Matter halos today, such as the one of theMilky Way, being released from the percolate. Axionic field profiles, in turn, appear to becompatible with Dark-Matter halos in typical galaxies if (confining) Yang-Mills dynamicssubject to much higher mass scales than that of SU(2) cmb is considered to produce theaxion mass.To consolidate such a scenario two immediate fields of investigation suggest themselves:(i) A deep understanding of possible selfgravitating profiles needs to be gained towardstheir role in actively assisted large-scale structure formation as well as in quasar emergence,strong lensing, cosmic shear, galaxy clustering, and galaxy phenomenology (Tully-Fisher,rotation curves, etc.), distinguishing spirals from ellipticals and satellites from hosts. (ii)More directly, the CMB large-angle anomalies require an addressation in terms of radiativeeffects in SU(2) cmb , playing out at low redshifts, which includes a re-investigation of theCMB dipole.
Funding:
None.
Conflicts of Interest:
There is no conflict of interest.
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