Cosmic Microwave Background as a Thermal Gas of SU(2) Photons: Implications for the High- Cosmological Model and the Value of

Presently, we are facing a tension in the most basic cosmological parameter, the Hubble constant . This tension arises when fitting the Lambda-cold-dark-matter model (CDM) to the high-precision temperature-temperature (TT) power spectrum of the Cosmic Microwave Background (CMB) and to local cosmological observations. We propose a resolution of this problem by postulating that the thermal photon gas of the CMB obeys an SU() rather than U() gauge principle, suggesting a high- cosmological model which is void of dark-matter. Observationally, we rely on precise low-frequency intensity measurements in the CMB spectrum and on a recent model independent (low-) extraction of the relation between the comoving sound horizon at the end of the baryon drag epoch and (). We point out that the commonly employed condition for baryon-velocity freeze-out is imprecise, judged by a careful inspection of the formal solution to the associated Euler equation. As a consequence, the above-mentioned tension actually transforms into a discrepancy. To make contact with successful low- CDM cosmology we propose an interpolation based on percolated/depercolated vortices of a Planck-scale axion condensate. For a first consistency test of such an all- model we compute the angular scale of the sound horizon at photon decoupling.

Concerning the number of massless neutrinos N ν , a conservative input is used: N ν = 3 [29]. This high-z model, composed of SU(2) CMB , baryonic matter, and massless neutrinos (N ν = 3), is sufficient to predict the sound horizon r s at the end of the baryon drag epoch which, in turn, can be confronted with the r s -H 0 relation, recently extracted from local cosmological observations [30], to determine the value of H 0 . The value of r s , as computed in a high-z model, rather sensitively depends on the definition of redshift z drag for baryon-velocity (v b ) freeze-out. Usually, z drag is identified with the maximum position of the so-called drag visibility function D drag [32,33]. However, inspecting the solution v b of the corresponding Euler equation, given as a functional of D drag , one concludes that this definition applies only in the limit of zero peak width. Realistic results for the ionization fraction χ e , obtained by numerical integration of the according Boltzmann hierarchy (recfast [34]), imply that the width of this peak extends over several hundred units of redshift in both cases ΛCDM and SU(2) CMB . As a consequence, a more precise definition of z drag is in order which associates with the left flank of D drag . Therefore, we will in the following refer to this corrected redshift for the freeze-out of v b as z lf, drag . Our value r s (z lf, drag ) ∼ 1660 -after intersection with the r s -H 0 relation of [30] -determines the value of H 0 in good agreement with the value obtained in [35]. Also, we would like to point out that, as a consequence of the corrected baryon-velocity freeze-out condition, the value of H 0 in ΛCDM, obtained by this method, is now at a 5σ discrepancy with the value published in [35].
To be able to compute the CMB power spectra, our consistent high-z SU(2) CMB cosmological model of Eq. (3) needs to be connected to the observationally well cross-checked ΛCDM low-z parametrization of the Universe's composition. To facilitate such an in-terpolation, a candidate real scalar field ϕ representing the dark sector is the so-called Planck-scale axion (PSA) condensate [36,37,38] which rests on chiral symmetry breaking within the Planckian epoch and the axial anomaly invoked by deconfining thermal ground states of Yang-Mills theories. Notice that the only Yang-Mills theory exhibiting the deconfining phase from today to well beyond recombination is SU(2) CMB . A model, where ϕ undergoes coherent and damped oscillations at late times such as to effectively represent ΛCDM, is falsified by the redshift z q , where the Universe's expansion starts to accelerate, being too high. This prompts the idea that interpolation between SU(2) CMB at high z and ΛCDM at low z is achieved by the U(1) topologically stabilized solitonic configurations (vortices) of the PSA condensate occurring in percolated form (due to a Berezinskii-Kosterlitz-Thouless phase transition following their very creation during a non-thermal phase transition at very high z) down to intermediate z where a depercolation transition partially liberates them to effectively represent a pressureless vortex gas. Whether or not the cores of depercolated PSA vortices properly serve as dark-matter halos in spiral galaxies to explain the observed flattening of rotation curves and the lensing signatures of bullet galaxies is an open question. Likewise, it is not yet guaranteed that this new cosmological model, which exhibits radiation domination and baryon freeze-out prior to photon decoupling, explains the observed angular power spectra of the CMB.
This work is organized as follows. In Sec. 2 we explain our high-z cosmological model SU(2) CMB , introduced in [28], and compare it with the conventional ΛCDM cosmology. The modification of decoupling conditions due to finite-widths visibility functions is discussed in Sec. 3. Based on this, we perform the computation of r s and confront it with the r s -H 0 relation of [30] to determine the value of H 0 . Subsequently, in Sec. 5 we investigate whether coherent and damped oscillations of the PSA field can realistically represent ΛCDM at low z -with a negative result. According to [28] we are thus led to propose an interpolation between high-z SU(2) CMB and low-z ΛCDM in terms of percolated PSA vortices which, at some intermediate redshift z p , partially undergo a depercolation transition. Such a model is demonstrated to be consistent with the extremely well observed angular scale of the sound horizon at photon decoupling [31]. Finally, we summarize our results and provide an outlook on how the new model can be tested further by confrontation with the power spectra of various CMB angular correlation functions.
2 Definition of cosmological model SU(2) CMB In a flat FLRW universe, a cosmological model is given in terms of the z-dependence of the Hubble parameter where H 0 is today's cosmological expansion rate and Ω i (z) = f i (z)Ω i,0 .
Here Ω i,0 is the fraction of the energy density ρ i,0 of fluid i to the critical density ρ c,0 today. The function f i (z) is determined by energy conservation subject to fluid i's equation of state. From now on we work in supernatural units (c = = k B = 1) where Newton's constant G has units of inverse mass. Table 1 lists the parameter values used subsequently.

The conventional ΛCDM model
In the conventional high-z ΛCDM model is H(z) given as Here non-relativistic matter decomposes into baryonic (b) and cold dark matter (CDM). The radiation component contains photons with two polarizations, two relativistic vector modes with three polarizations each, and N eff flavors of massless neutrinos with two polarizations each. Ω γ,0 is today's fraction of photonic to the critical energy density 1 .

Modifications of ΛCDM towards SU(2) CMB
In high-z SU(2) CMB the Hubble parameter is given as In this case, only baryonic matter is present. We reiterate that both models, Eq. (2) and Eq. (3), need to be supplemented by a dark sector to yield successful low-z ΛCDM cosmology, see Eq. (32). The radiation sector is modified due to a different number of relativistic degrees of freedom and due to the SU(2) CMB high-z temperature-redshift relation T (z), for details see [27,28].

The end of recombination
The comoving sound horizon r s at redshift z is defined as whereby c s is the sound velocity in the primordial baryon-electron-photon plasma, given as c s ≡ 1 The function R(z) is determined by 3/4 of the ratio of energy densities in baryons and photons. In ΛCDM we have R(z) ≡ 111.019 whereas in SU(2) CMB one obtains The values of η 10 can be read off Table 1.

Conventional freeze-out
The final stages of recombination can be characterized in a twofold way. One considers either (i) photon temperature freeze-out, which is relevant for the peak structure in the temperature-temperature (TT) angular power spectrum of the CMB, or (ii) baryon velocity freeze-out, which is detectable in the matter correlation function (galaxy counts). Concerning case (i), the conventional criterion, which fixes the redshift z * , reads where σ T denotes the total cross section for Thomson scattering, χ e is the ionization fraction (calculated with recfast), and n b e refers to the density of free electrons just before hydrogen recombination, given as Here Y P denotes the helium mass fraction in baryons (see Table 1). Concerning case (ii), the conventional criterion is defined as

Corrected freeze-out
We now show that conditions (8) and (10) are imprecise due to the finite widths of the respective visibility functions. To see this, we have to analyze the formal solution of the Boltzmann hierarchy for the temperature perturbation and of the Euler equation for v b [32,33,41]. Since the argument is similar for both cases we focus on the latter only. The Euler equation readsv where k is the comoving wave number (omitted as a subscript in the following), Θ 1 denotes the (relative) dipole of the temperature anisotropy [42], and Ψ represents the Newtonian gravitational potential. The overdot demands differentiation with respect to conformal time. Transforming the conformal time to a redshift dependence, the solution of Eq. (11) is Here τ drag is defined as and the visibility function D drag (z , z) is represented by .
In order to study freeze-out the function Θ 1 in Eq. (12) is considered slowly varying. Therefore, the variability integral solely depends on D drag within its peak region. In both cases ΛCDM and SU(2) CMB function D drag exhibits a broad peak in dependence of z whose shape and maxima does not depend on z, see Fig. 2. Note that Eq. (10) describes the maxima z max,drag of D drag (z , z). However, due to the finite width the integral in Eq. (12) is not saturated at z = z max,drag but rather ceases to vary for z < z lf,drag where lf denotes the maxima of the z derivative of D drag . Therefore, z lf,drag defines the freezeout point more realistically than z max,drag . According to Fig. 2 An analogous discussion applies to photon temperature freeze-out with the following results (see [28]):  [30]. Note the good agreement between the values of H 0 implied by r s (z lf,drag ) in SU(2) CMB and the extraction performed in [35]. On the other hand, r s (z drag ) reproduces the value of H 0 published in [39] which exhibits a 3σ tension compared to [35]. However, according to Fig. 3, the more realistic freeze-out value z lf,drag in ΛCDM entails Thus, in ΛCDM there actually is a 5σ discrepancy between the value of H 0 quoted in [35] and obtained by confrontation of r s with the r s -H 0 relation of [30].

Planck-scale-axion field and interpolation of high-z with low-z cosmology
Here we would like to analyze cosmological models which link low-z ΛCDM with high-z SU(2) CMB . We assume a dark sector which originates from a real, minimally coupled scalar field -a pseudo Nambu-Goldstone mode of dynamical chiral symmetry occurring at the Planck scale [36,37] -whose potential is due to the chiral U(1) A anomaly invoked by (anti)calorons of the deconfining, thermal ground state of Yang-Mills theories [1,2,3,6,43,44,45]. This prompts the name Planck-scale axion (PSA). The only Yang-Mills theory, which is deconfining well above recombination, is SU(2) CMB because otherwise there wouldn't be just one species of photons.  [30] in confrontation with the high-z predictions of r s (z lf,drag ) and r s (z drag ) in ΛCDM and SU(2) CMB (horizontal bands) of Eqs. (17). Vertical bands indicate the values of H 0 extracted in [39] (low) and in [35] (high). Note that there is a ∼3σ tension. However, a ∼7σ discrepancy exists between the The radiatively protected potential for the axion condensate ϕ, arising due to the thermal ground state of SU(2) CMB [44,45], reads as follows where Λ CMB ∼ 10 −4 eV, κ is a dimensionless factor of order unity, and the reduced Planck mass reads With a canonical kinetic term for ϕ the according equation of motion is where an overdot signals the derivative with respect to cosmological time. In a first attempt at a ΛCDM -SU(2) CMB interpolation we assume spatially homogeneous ϕ-field dynamics subject to ΛCDM constraints at low z. It turns out, however, that such a model predicts a value of z q , defined as the zero of the deacceleration parameter of about z q ∼ 3 which is much higher than the realistic value ∼ 0.7 obtained in ΛCDM. Therefore, as a second proposal we abolish the energy density arising from spatially homogeneous configurations of the field ϕ. Rather, we conceive the dark-matter sector in ΛCDM as a piece of energy density due to depercolated topological solitons (vortices) of the field ϕ which percolate instantaneously into a dark-energy like piece at some redshift z p such that z re z p z lf,drag . The origin of such a vortex percolate, with hierarchically ordered core sizes, could be due to Hagedorn transitions of Yang-Mills theories in the early universe which are accompanied by Berezinskii-Kosterlitz-Thouless transitions in the axionic sector. Today's value of Ω Λ would then be interpreted in terms of not-yet depercolated vortices. Indeed, in such an interpolation between ΛCDM and SU(2) CMB a value of z p ∼ 155 can be fitted to the angular size of the sound horizon at photon decoupling. At z lf,drag the extra contribution to dark-energy amounts to ∼ 0.65% of the baryonic energy density which is consistent with SU(2) CMB .

Spatially homogeneous, coherent oscillations
Here we discuss a cosmological model where the interpolation between ΛCDM and SU(2) CMB is attempted by a spatially homogeneous PSA field which undergoing damped and coherent oscillations at late times. This models a pressureless component (cold dark matter) and component with negative pressure (dark energy). Notice that these two components represent fluids that are not separately conserved.
The Hubble equation reads Here ρ r denotes radiation-like energy density including SU(2) CMB (for z ≤ 9 radiation energy density is severely suppressed in the cosmological model, for z > 9 the thermal ground state and the masses of the vector modes of SU(2) CMB can be neglected) and three flavours of massless neutrinos, ρ b is the energy density of baryons, in addition to the energy density 1 2φ 2 + V (ϕ) associated with the spatially homogeneous PSA field ϕ which evolves temporally. Eqs. (21) and (23) can be cast into fully dimensionless equations by rescaling with powers of m P in the following way In general, dimensionless quantities (after rescaling with the appropriate power of m P ) are indicated by the hat-symbol. After rescaling and in dependence of z Eqs. (21) and (23) transmute intô where a prime demands z-differentiation. In Eq. (26) we approximateρ r aŝ ρ r =ρ γ,0 · 0 (z < 9) 4(0.63) 3 1 + 7 32 16 23 With the initial conditionsφ and that Ω Λ coincides with typical fit value Ω Λ ∼ 0.7 obtained in ΛCDM cosmology [31]: Fig. 4 shows the deacceleration parameter q(z) for the model defined by Eqs. (25), (26), (29), and (30). Obviously, this model is falsified by a much too high value of the zero z q of q(z).

Percolated and unpercolated vortices
Here the basic idea invokes the fact that a PSA field ϕ, due to non-thermal phase transitions of the Hagedorn type (e.g., there should be an SU (2) , (29), and (30). Notice that the value of the zero z q of q(z) is z q ∼ 3. This is much higher than the realistic value z q ∼ 0.7 obtained in ΛCDM.
phase transition [46,47]. Effectively, this percolate represents homogeneous, constant energy density. As the universe expands the vortex percolate is increasingly stretched, and, at around some critical redshift z p z lf,drag it releases a part of its solitons characterized by some specific core size. The ensuing vortex gas acts cosmologically like pressureless matter. Vortices of larger core sizes remain trapped in the percolate. For this scenario to be a consistent interpolation of SU(2) CMB and ΛCDM we need to assure that z p z re ∼ 6 [48].
With the definition of Eq. (27) the cosmological model to be considered thus readŝ where ρ DS is the dark-sector energy density, defined aŝ whereρ Λ andρ CDM,0 are today's values of the dark-energy and cold-dark-matter densities associated with Eq. (30) and the value quoted in Table 1, respectively. In order to fix the value of z p we confront the model of Eqs. (31) and (32) with the observed angular scale θ * of the sound horizon at CMB photon decoupling, occurring at z lf, * . Theoretically, θ * is given as To match θ * = 0.597 • , as extracted in [39] from the TT power spectrum, we require z p = 155.4, see Fig. 5. This yields a percentage of vacuum energy at CMB photon decoupling of about Ω DM,0 Ω b,0 z p + 1 z lf, * + 1 The omission of vacuum energy in our SU(2) CMB high-z cosmological model of Eq. (3) thus is justified for the interpolating model defined in Eqs. (31) and (32).

Summary and outlook
In the present work we have analysed, based on a modified temperature-redshift relation for the CMB which, in turn, derives from the postulate that thermal photon gases are subject to an SU(2) rather than a U(1) gauge principle, a high-z cosmological model which is void of dark matter and considers three species of massless neutrinos. Such a model predicts (after a reconsideration of baryon-velocity freeze-out) a value of the sound horizon r s which, together with a model independent extraction of the r s -H 0 relation from cosmologically local observations in [30], yields good agreement with the value of H 0 determined by low-z observations in [35]. The same r s -H 0 relation predicts a low value of H 0 in standard ΛCDM cosmology which is at a 5σ discrepancy with the value given in [35]. Motivated by the above results, an interpolation between ΛCDM at low z and our new high-z model is called for. In a first attempt, we have investigated whether coherent and damped oscillations of a Planck-scale axion condensate can realistically accomplish thiswith a negative result. With [28] we were thus led to propose an interpolation in terms of percolated PSA vortices which, at some intermediate z p , partially undergo a depercolation transition. We have demonstrated this model to be consistent with the angular scale of the sound horizon at photon decoupling.