# Cosmic Microwave Background as a Thermal Gas of SU(2) Photons: Implications for the High-z Cosmological Model and the Value of H0

Authors: Steffen Hahn; Ralf Hofmann
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

Presently, we are facing a3σ tension in the most basic cosmological parameter, the Hubble constant H0. This tension arises when fitting the Lambda-cold-dark-matter model (ΛCDM) 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(2) rather than U(1) gauge principle, suggesting a high-z 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-z) extraction of the relation between the comoving sound horizon rs at the end of the baryon drag epoch and H0 (rsH0=const). 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 3σ tension actually transforms into a 5σ discrepancy. To make contact with successful low-z  ΛCDM 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-z model we compute the angular scale of the sound horizon at photon decoupling.


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## Body

## 1. Introduction

Since the pioneering work by Yang and Mills [1] on the definition of a local four-dimensional, classical, and minimal field theory, which is based on the nonabelian gauge group SU(2), much progress has been made in elucidating the role of topologically stabilized and (anti)-self-dual field configurations in building the nonperturbative ground state and influencing the properties of its excitations [2–8]. In particular, the deconfining phase is subject to a highly accurate thermal ground state estimate [9, 10], being composed of so-called Harrington-Shepard (anti)calorons [11]. This (cosmologically relevant) ground state invokes both an adjoint Higgs mechanism [12–15], rendering two out of three directions of the SU(2) algebra massive (free thermal quasiparticles), and a U(1)A chiral anomaly [2, 3, 5, 6], giving mass to the Goldstone mode induced by the associated dynamical breaking of this global symmetry. Radiative corrections to thermodynamical quantities, evaluated on the level of free thermal (quasi)particles, are minute and well under control [9, 10]. Note that this is in contrast to the large effects of radiative corrections attributed to the effective QCD action at zero temperature in [16, 17] which are exploited as potential inducers of vacuum energy in the cosmological context in [18–22]. However, it was argued in [23, 24] that QCD condensates, which contribute to the trace anomaly of the energy-momentum tensor (as implied by the effective action), do not act cosmologically.Postulating that thermal photon gases obey an SU(2) rather than a U(1) gauge principle, the SU(2) Yang-Mills scale can be inferred from low-(radio)frequency spectral intensity measurements, for example [25], of the Cosmic Microwave Background (CMB) [26], prompting the name SU(2)CMB. Below we will use the name SU(2)CMB synonymously for the implied cosmological model. To investigate the consequences of this postulate towards the equation of state radiative corrections are entirely negligible [9]. When subjecting local energy conservation in a Friedmann-Lemaître-Robertson-Walker (FLRW) universe to this equation of state the numerical temperature (T)-redshift (z) relation (T(z)) of the CMB follows; see Figure 1 [27, 28], where a comparison with the conventional U(1) photon gas is shown. The curvature of T/(T0(z+1)) (T0=2.725 K denoting today’s CMB temperature) at low z is due to the influence of the SU(2) Yang-Mills mass scale on the equation of state. In [28] an argument is given why recent observational “extractions” of T(z), which claim no deviations from the conventional behavior T(z)=T0(z+1), are circular. One has T/T0=0.63(z+1) at high z and therefore a lower slope compared to the conventional case. In an approximation, where recombination at z∗ is subjected to thermodynamics, the decoupling condition is ΓTh(T∗)=H(z∗) where ΓTh denotes the Thomson photon-electron scattering rate at the decoupling temperature T∗~3000 K. We have (Ω0,b+Ω0,DM)/Ω0,b~6.5≡Rm,1 where Ω0,b and Ω0,DM denote the respective ratios of today’s energy densities in baryons and cold dark matter to the critical density. Since z∗,SU(2)CMB/z∗,ΛCDM~1/0.63 this roughly matches (1/0.63)3~4≡Rm,2. If a strong matter domination can be assumed during recombination then Rm,1 should be equal to Rm,2 but, due to matter-radiation equality occurring at z~1080 in SU(2)CMB, this assumption is not quite met, explaining the mild discrepancy between Rm,1 and Rm,2. Still, we take this rough argument and the desired minimality of the cosmological model as motivations to omit cold dark matter in the high-z cosmological model which operates down to recombination and well beyond it.Figure 1
TheT-z scaling relation T/(T0(z+1)) in SU(2)CMB (solid). Note the emergence of T/T0=0.63(z+1) for z≳9 (dotted). The conventional U(1) theory for thermal photon gases associates with the dashed line. Data taken from [27] after slight and inessential correction.Concerning the number of massless neutrinosNν, a conservative input is used: Nν=3 [29]. This high-z model, composed of SU(2)CMB, baryonic matter, and massless neutrinos (Nν=3), is sufficient to predict the sound horizon rs at the end of the baryon drag epoch which, in turn, can be confronted with the rs-H0 relation, recently extracted from local cosmological observations [30], to determine the value of H0. The value of rs, as computed in a high-z model, rather sensitively depends on the definition of redshift zdrag for baryon-velocity (vb) freeze-out. Usually, zdrag is identified with the maximum position of the so-called drag visibility function Ddrag [31, 32]. However, inspecting the solution vb of the corresponding Euler equation, given as a functional of Ddrag, one concludes that this definition applies only in the limit of zero peak width. Realistic results for the ionization fraction χe, obtained by numerical integration of the according Boltzmann hierarchy (recfast [33]), 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 zdrag is in order which associates with the left flank of Ddrag. Therefore, we will in the following refer to this corrected redshift for the freeze-out of vb as zlf,drag. Our value rs(zlf,drag)~1660, after intersection with the rs-H0 relation of [30], determines the value of H0 in good agreement with the value obtained in [34]. Also, we would like to point out that, as a consequence of the corrected baryon-velocity freeze-out condition, the value of H0 in ΛCDM, obtained by this method, is now at a 5σ discrepancy with the value published in [34].To be able to compute the CMB power spectra, our consistent high-z  SU(2)CMB cosmological model of (3) needs to be connected to the observationally well cross-checked ΛCDM low-z parametrization of the universe’s composition. To facilitate such an interpolation, a candidate real scalar field φ representing the dark sector is the so-called Planck-scale axion (PSA) condensate [35–37] 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 zq, 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 nonthermal 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 Section2 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 Section 3. Based on this, we perform the computation of rs and confront it with the rs-H0 relation of [30] to determine the value of H0. Subsequently, in Section 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 zp, 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 [38]. 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 ModelSU2CMB

In a flat FLRW universe, a cosmological model is given in terms of thez-dependence of the Hubble parameter (1)Hz=H0∑iΩiz,where H0 is today’s cosmological expansion rate and Ωi(z)=fi(z)Ωi,0. Here Ωi,0 is the fraction of the energy density ρi,0 of fluid i to the critical density ρc,0 today. The function fi(z) is determined by energy conservation subject to fluid i’s equation of state. From now on we work in supernatural units (c=ħ=kB=1) where Newton’s constant G has units of inverse mass squared. Table 1 lists the parameter values used subsequently.Table 1
Cosmological parameter values employed in the computations and their sources, taken from [28].
Parameter
Value
Source
H
0 (SU(2)CMB)
(
73.24
±
1.74
) km s−1 Mpc−1
[34]
H
0 (ΛCDM)
(
67.31
±
0.96
)  km s−1 Mpc−1
TT + lowP, [38]
T
0
2.725 K
[39]
Ω
γ
,
0
h
2
2.46796
×
1
0
-
5.
Based onT0=2.725 K
Ω
b
,
0
h
2
0.02222±0.99923
TT + lowP[38]
Ω
C
D
M
,
0
h
2
0.1197
±
0.0022
TT + lowP, [38]
η
10
6.08232±0.06296
Based onΩγ,0h2, TT + low P[38]
Y
P
0.252±0.041
TT, [38]
N
e
f
f
3.15±0.23
Abstract, [38]


### 2.1. The ConventionalΛCDM Model

In the conventional high-z  ΛCDM model H(z) is given as (2)Hz=H0Ωb,0+ΩCDM,0z+13+1+784114/3NeffΩγ,0z+141/2.Here nonrelativistic 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 Neff flavours of massless neutrinos with two polarizations each. Ωγ,0 is today’s fraction of photonic energy density to critical energy density (for details see [38]).


### 2.2. Modifications ofΛCDM towards SU2CMB

In high-z  SU(2)CMB the Hubble parameter is given as (3)Hz=H0Ωb,0z+13+4·0.6341+73216234/3NνΩγ,0z+141/2.In this case, only baryonic matter is present. We reiterate that both models, (2) and (3), need to be supplemented by a dark sector to yield successful low-z  ΛCDM cosmology; see (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].


## 2.1. The ConventionalΛCDM Model

In the conventional high-z  ΛCDM model H(z) is given as (2)Hz=H0Ωb,0+ΩCDM,0z+13+1+784114/3NeffΩγ,0z+141/2.Here nonrelativistic 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 Neff flavours of massless neutrinos with two polarizations each. Ωγ,0 is today’s fraction of photonic energy density to critical energy density (for details see [38]).


## 2.2. Modifications ofΛCDM towards SU2CMB

In high-z  SU(2)CMB the Hubble parameter is given as (3)Hz=H0Ωb,0z+13+4·0.6341+73216234/3NνΩγ,0z+141/2.In this case, only baryonic matter is present. We reiterate that both models, (2) and (3), need to be supplemented by a dark sector to yield successful low-z  ΛCDM cosmology; see (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].


## 3. The End of Recombination

The comoving sound horizonrs at redshift z is defined as (4)rsz=∫z∞dz′csz′Hz′,whereby cs denotes the sound velocity in the primordial baryon-electron-photon plasma, given as (5)cs≡131+R.The function R(z) is determined by 3/4 of the ratio of energy densities in baryons and photons. In ΛCDM we have(6)Rz≡111.019η10z+1, whereas in SU(2)CMB one obtains (7)Rz≡111.019η100.634z+1.The values of η10 can be read off Table 1.


### 3.1. 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 redshiftz∗, reads (8)τz∗=σT∫0z∗dzχeznebzz+1Hz=1, where σT denotes the total cross section for Thomson scattering, χe is the ionization fraction (calculated with recfast), and neb refers to the density of free electrons just before hydrogen recombination, given as (9)nebz=410.48·10-10η101-YPz+13cm-3. Here YP denotes the helium mass fraction in baryons (see Table 1). Concerning case (ii), the conventional criterion is defined as (10)τdragz=σT∫0zdz′χez′nebz′z′+1Hz′Rz′=1.


### 3.2. 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 vb [31, 32, 40]. Since the argument is similar for both cases we focus on the latter only. The Euler equation reads (11)v˙b=z˙z+1vb+kΨ+τ˙dragΘ1-vb,where k is the comoving wave number (omitted as a subscript in the following), Θ1 denotes the (relative) dipole of the temperature anisotropy [41], 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 (11) is (12)vbzz+1=limZ↗∞∫zZdz′e-τdragz′,zHz′z′+1τ˙dragz′Θ1z′+kΨz′~limZ↗∞∫zZdz′Ddragz′,zΘ1z′.Here τdrag is defined as (13)τdragz′,z≡∫zz′dz′′τ˙dragz′′Hz′′,and the visibility function Ddrag(z′,z) is represented by (14)Ddragz′,z≡e-τdragz′,zτ˙dragz′Hz′z′+1.In order to study freeze-out the function Θ1 in (12) is considered slowly varying. Therefore, the variability of the integral solely depends on Ddrag within its peak region. In both cases ΛCDM and SU(2)CMB function Ddrag exhibits a broad peak in dependence of z′ whose shape and maxima do not depend on z; see Figure 2. Note that (10) describes the maxima zmax,drag′ of Ddrag(z′,z). However, due to the finite width the integral in (12) is not saturated at z=zmax,drag but rather ceases to vary for z<zlf,drag where lf denotes the maxima of the z′ derivative of Ddrag. Therefore, zlf,drag defines the freeze-out point more realistically than zmax,drag. According to Figure 2’s caption the values of zdrag,zlf,drag deviate substantially. Namely, (15)zdrag=1813,zmax,drag=1789,zlf,drag=1659SU2CMB,zdrag=1059,zmax,drag=1046,zlf,drag=973ΛCDM.An analogous discussion applies to photon temperature freeze-out with the following results (see [28]):(16)z∗=1694,zmax,∗=1694,zlf,∗=1555SU2CMB,z∗=1090,zmax,∗=1072,zlf,∗=988ΛCDM.Figure 2
Normalised functionDdrag(z′,z), defined in (14), if z≤zmax,drag for SU(2)CMB (a) and ΛCDM (b). Redshift zlf,drag is defined as the position of the maximum of dDdrag/dz′ (position of left flank of Ddrag) whereas zmax,drag denotes the position of the maximum of Ddrag. The value of zdrag, defined in (10), essentially coincides with zmax,drag: zdrag=1813~zmax,drag=1789 for SU(2)CMB and zdrag=1059~zmax,drag=1046 for ΛCDM. This should be contrasted with zlf,drag=1659 for SU(2)CMB and zlf,drag=973 for ΛCDM. The hatched area under the curve determines the freeze-out value of vb/(z+1).
(a)
(b)


## 3.1. 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 redshiftz∗, reads (8)τz∗=σT∫0z∗dzχeznebzz+1Hz=1, where σT denotes the total cross section for Thomson scattering, χe is the ionization fraction (calculated with recfast), and neb refers to the density of free electrons just before hydrogen recombination, given as (9)nebz=410.48·10-10η101-YPz+13cm-3. Here YP denotes the helium mass fraction in baryons (see Table 1). Concerning case (ii), the conventional criterion is defined as (10)τdragz=σT∫0zdz′χez′nebz′z′+1Hz′Rz′=1.


## 3.2. 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 vb [31, 32, 40]. Since the argument is similar for both cases we focus on the latter only. The Euler equation reads (11)v˙b=z˙z+1vb+kΨ+τ˙dragΘ1-vb,where k is the comoving wave number (omitted as a subscript in the following), Θ1 denotes the (relative) dipole of the temperature anisotropy [41], 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 (11) is (12)vbzz+1=limZ↗∞∫zZdz′e-τdragz′,zHz′z′+1τ˙dragz′Θ1z′+kΨz′~limZ↗∞∫zZdz′Ddragz′,zΘ1z′.Here τdrag is defined as (13)τdragz′,z≡∫zz′dz′′τ˙dragz′′Hz′′,and the visibility function Ddrag(z′,z) is represented by (14)Ddragz′,z≡e-τdragz′,zτ˙dragz′Hz′z′+1.In order to study freeze-out the function Θ1 in (12) is considered slowly varying. Therefore, the variability of the integral solely depends on Ddrag within its peak region. In both cases ΛCDM and SU(2)CMB function Ddrag exhibits a broad peak in dependence of z′ whose shape and maxima do not depend on z; see Figure 2. Note that (10) describes the maxima zmax,drag′ of Ddrag(z′,z). However, due to the finite width the integral in (12) is not saturated at z=zmax,drag but rather ceases to vary for z<zlf,drag where lf denotes the maxima of the z′ derivative of Ddrag. Therefore, zlf,drag defines the freeze-out point more realistically than zmax,drag. According to Figure 2’s caption the values of zdrag,zlf,drag deviate substantially. Namely, (15)zdrag=1813,zmax,drag=1789,zlf,drag=1659SU2CMB,zdrag=1059,zmax,drag=1046,zlf,drag=973ΛCDM.An analogous discussion applies to photon temperature freeze-out with the following results (see [28]):(16)z∗=1694,zmax,∗=1694,zlf,∗=1555SU2CMB,z∗=1090,zmax,∗=1072,zlf,∗=988ΛCDM.Figure 2
Normalised functionDdrag(z′,z), defined in (14), if z≤zmax,drag for SU(2)CMB (a) and ΛCDM (b). Redshift zlf,drag is defined as the position of the maximum of dDdrag/dz′ (position of left flank of Ddrag) whereas zmax,drag denotes the position of the maximum of Ddrag. The value of zdrag, defined in (10), essentially coincides with zmax,drag: zdrag=1813~zmax,drag=1789 for SU(2)CMB and zdrag=1059~zmax,drag=1046 for ΛCDM. This should be contrasted with zlf,drag=1659 for SU(2)CMB and zlf,drag=973 for ΛCDM. The hatched area under the curve determines the freeze-out value of vb/(z+1).
(a)
(b)


## 4. The Value ofH0

Subjecting the freeze-out redshifts of (15) to (4) under consideration of (2) and (3) yields (17)rszdrag=129.22±0.52MpcSU2CMB,rszlf,drag=137.19±0.45MpcSU2CMB,rszdrag=147.33±0.49MpcΛCDM,rszlf,drag=154.57±3.33MpcΛCDM.In Figure 3, these (H0 independent) values of the sound horizon are confronted with the rs-H0 relation of [30]. Note the good agreement between the values of H0 implied by rs(zlf,drag) in SU(2)CMB and the extraction performed in [34]. On the other hand, rs(zdrag) reproduces the value of H0 published in [38] which exhibits a 3σ tension compared to [34]. However, according to Figure 3, the more realistic freeze-out value zlf,drag in ΛCDM entails (18)H0=64.5±1kms-1Mpc-1.Thus, in ΛCDM there actually is a 5σ discrepancy between the value of H0 quoted in [34] and obtained by confrontation of rs with the rs-H0 relation of [30].Figure 3
Thers(zlf,drag)-H0 relation (curved band) of [30] in confrontation with the high-z predictions of rs(zlf,drag) and rs(zdrag) in ΛCDM and SU(2)CMB (horizontal bands) of (17). Vertical bands indicate the values of H0 extracted in [38] (low) and in [34] (high). Note that there is a ~3σ tension. However, a ~7σ discrepancy exists between the H0 values of (64.3±1.1) km s−1 Mpc−1 and (72.9±1.2)  km s−1 Mpc−1 associated with the intersections of rs(zlf,drag) in ΛCDM and in SU(2)CMB, respectively, with the rs(zlf,drag)-H0 relation. Taking H0=(73.24±1.7) km s−1 Mpc−1 from [34] the discrepancy between this value and (64.3±1.1) km s−1 Mpc−1 is about 5σ.


## 5. 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 [35, 36], 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 [1–3, 6, 42–44]. 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 would not be just one species of photons.The radiatively protected potential for the axion condensateφ, arising due to the thermal ground state of SU(2)CMB [43, 44], reads as follows: (19)Vφ=κΛCMB4·1-cos⁡φmP,where ΛCMB~10-4 eV, κ is a dimensionless factor of order unity, and the reduced Planck mass reads (20)mP≡1.22×10198πGeV=8πG-1/2.With a canonical kinetic term for φ the according equation of motion is (21)φ¨+3Hφ˙+ddφVφ=0,where an overdot signals the derivative with respect to cosmological time.In a first attempt at aΛCDM - 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 zq, defined as the zero of the deacceleration parameter (22)qz≡z+12H^2H^2′-1,of about zq~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 fromspatially 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 zp such that zre≪zp≪zlf,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 ΩΛ 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 zp~155 can be fitted to the angular size of the sound horizon at photon decoupling. At zlf,drag the extra contribution to dark-energy amounts to ~0.65% of the baryonic energy density which is consistent with SU(2)CMB.


### 5.1. Spatially Homogeneous, Coherent Oscillations

Here we discuss a cosmological model where the interpolation betweenΛCDM and SU(2)CMB is attempted by a spatially homogeneous PSA field 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(23)H2=8πG312φ˙2+Vφ+ρb+ρr≡8πG3ρc.Here ρr 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; ρb is the energy density of baryons, in addition to the energy density 1/2φ˙2+Vφ 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 mP in the following way: (24)V=mP4V^,ρi=mP4ρ^ii=b,r,φ=mPφ^,H=mPH^. In general, dimensionless quantities (after rescaling with the appropriate power of mP) are indicated by the hat-symbol. After rescaling and in dependence of z (21) and (23) transmute into (25)φ^′′z+1H^2+φ^′12z+12H^2′-2z+1H^2+dVdφ^=0,(26)H^2=13V^+ρ^b,0z+13+ρ^r1-1/6z+12φ^′2,where a prime demands z-differentiation. In (26) we approximate ρ^r as (27)ρ^r=ρ^γ,0·0z<940.6331+73216234/33z+14z≥9.With the initial conditions (28)φ^z=zi=φ^i,φ^′z=zi=0 for sufficiently large zi (no roll; in practice one safely can choose zi~50) the solution to (25) subject to (26) is unique. To fix the values of κ in (19) and φ^i in (28) we demand (29)ρc,0=3H028πG=3mP4H^02and that ΩΛ coincides with typical fit value ΩΛ~0.7 obtained in ΛCDM cosmology [38]: (30)ΩΛ=mP4ρc,0limz↘0V^-12z+1H^φ^′2~0.7. Figure 4 shows the deacceleration parameter q(z) for the model defined by (25), (26), (29), and (30). Obviously, this model is falsified by a much too high value of the zero zq of q(z).Figure 4
The deacceleration parameterq(z) of (22) for the model defined by (25), (26), (29), and (30). Notice that the value of the zero zq of q(z) is zq~3. This is much higher than the realistic value zq~0.7 obtained in ΛCDM.


### 5.2. Percolated and Unpercolated Vortices

Here the basic idea invokes the fact that a PSA fieldφ, due to nonthermal phase transitions of the Hagedorn type (e.g., there should be an SU(2)e Yang-Mills theory of scale Λe≫ΛSU2CMB going confining at T~Λe) is subject to U(1)A winding and in this way creation of a density of percolated topological solitons (vortex percolate) with a hierarchical ordering of core sizes. Percolation could be understood as a Berezinskii-Kosterlitz-Thouless phase transition [45, 46]. Effectively, this percolate represents homogeneous, constant energy density. As the universe expands the vortex percolate is increasingly stretched, and at around some critical redshift zp≪zlf,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 zp≫zre~6 [47, 48].With the definition of (27) the cosmological model to be considered thus reads (31)H^2=13ρ^b+ρ^DS+ρ^r,where ρ^DS is the dark sector energy density, defined as (32)ρ^DS=ρ^Λ+ρ^CDM,0·z+13z<zpzp+13z≥zp,where ρ^Λ and ρ^CDM,0 are today’s values of the dark-energy and cold-dark-matter densities associated with (30) and the value quoted in Table 1, respectively.In order to fix the value ofzp we confront the model of (31) and (32) with the observed angular scale θ∗ of the sound horizon at CMB photon decoupling, occurring at zlf,∗. Theoretically, θ∗ is given as (33)θ∗=rszlf,∗∫0zlf,∗dz/Hz.To match θ∗=0.597∘, as extracted in [38] from the TT power spectrum, we require zp=155.4; see Figure 5. This yields a percentage of vacuum energy at CMB photon decoupling of about (34)ΩDM,0Ωb,0zp+1zlf,∗+13~0.65%. The omission of vacuum energy in our SU(2)CMB high-z cosmological model of (3) thus is justified for the interpolating model defined in (31) and (32).Figure 5
Functionθ∗(zp) for ΩΛ=0.7, ΩDM,0=0.26, Ωb,0=0.04, Ωγ,0=4.6×10-5, and H0=73.24 km s−1 Mpc−1 for the high-z  SU(2)CMB and low-z   ΛCDM interpolating cosmological model considered. Also indicated is the value θ∗=0.597∘ (dashed line), fitted to the CMB TT power spectrum.


## 5.1. Spatially Homogeneous, Coherent Oscillations

Here we discuss a cosmological model where the interpolation betweenΛCDM and SU(2)CMB is attempted by a spatially homogeneous PSA field 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(23)H2=8πG312φ˙2+Vφ+ρb+ρr≡8πG3ρc.Here ρr 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; ρb is the energy density of baryons, in addition to the energy density 1/2φ˙2+Vφ 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 mP in the following way: (24)V=mP4V^,ρi=mP4ρ^ii=b,r,φ=mPφ^,H=mPH^. In general, dimensionless quantities (after rescaling with the appropriate power of mP) are indicated by the hat-symbol. After rescaling and in dependence of z (21) and (23) transmute into (25)φ^′′z+1H^2+φ^′12z+12H^2′-2z+1H^2+dVdφ^=0,(26)H^2=13V^+ρ^b,0z+13+ρ^r1-1/6z+12φ^′2,where a prime demands z-differentiation. In (26) we approximate ρ^r as (27)ρ^r=ρ^γ,0·0z<940.6331+73216234/33z+14z≥9.With the initial conditions (28)φ^z=zi=φ^i,φ^′z=zi=0 for sufficiently large zi (no roll; in practice one safely can choose zi~50) the solution to (25) subject to (26) is unique. To fix the values of κ in (19) and φ^i in (28) we demand (29)ρc,0=3H028πG=3mP4H^02and that ΩΛ coincides with typical fit value ΩΛ~0.7 obtained in ΛCDM cosmology [38]: (30)ΩΛ=mP4ρc,0limz↘0V^-12z+1H^φ^′2~0.7. Figure 4 shows the deacceleration parameter q(z) for the model defined by (25), (26), (29), and (30). Obviously, this model is falsified by a much too high value of the zero zq of q(z).Figure 4
The deacceleration parameterq(z) of (22) for the model defined by (25), (26), (29), and (30). Notice that the value of the zero zq of q(z) is zq~3. This is much higher than the realistic value zq~0.7 obtained in ΛCDM.


## 5.2. Percolated and Unpercolated Vortices

Here the basic idea invokes the fact that a PSA fieldφ, due to nonthermal phase transitions of the Hagedorn type (e.g., there should be an SU(2)e Yang-Mills theory of scale Λe≫ΛSU2CMB going confining at T~Λe) is subject to U(1)A winding and in this way creation of a density of percolated topological solitons (vortex percolate) with a hierarchical ordering of core sizes. Percolation could be understood as a Berezinskii-Kosterlitz-Thouless phase transition [45, 46]. Effectively, this percolate represents homogeneous, constant energy density. As the universe expands the vortex percolate is increasingly stretched, and at around some critical redshift zp≪zlf,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 zp≫zre~6 [47, 48].With the definition of (27) the cosmological model to be considered thus reads (31)H^2=13ρ^b+ρ^DS+ρ^r,where ρ^DS is the dark sector energy density, defined as (32)ρ^DS=ρ^Λ+ρ^CDM,0·z+13z<zpzp+13z≥zp,where ρ^Λ and ρ^CDM,0 are today’s values of the dark-energy and cold-dark-matter densities associated with (30) and the value quoted in Table 1, respectively.In order to fix the value ofzp we confront the model of (31) and (32) with the observed angular scale θ∗ of the sound horizon at CMB photon decoupling, occurring at zlf,∗. Theoretically, θ∗ is given as (33)θ∗=rszlf,∗∫0zlf,∗dz/Hz.To match θ∗=0.597∘, as extracted in [38] from the TT power spectrum, we require zp=155.4; see Figure 5. This yields a percentage of vacuum energy at CMB photon decoupling of about (34)ΩDM,0Ωb,0zp+1zlf,∗+13~0.65%. The omission of vacuum energy in our SU(2)CMB high-z cosmological model of (3) thus is justified for the interpolating model defined in (31) and (32).Figure 5
Functionθ∗(zp) for ΩΛ=0.7, ΩDM,0=0.26, Ωb,0=0.04, Ωγ,0=4.6×10-5, and H0=73.24 km s−1 Mpc−1 for the high-z  SU(2)CMB and low-z   ΛCDM interpolating cosmological model considered. Also indicated is the value θ∗=0.597∘ (dashed line), fitted to the CMB TT power spectrum.


## 6. Summary and Outlook

In the present work we have analyzed, 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 rs which, together with a model independent extraction of the rs-H0 relation from cosmologically local observations in [30], yields good agreement with the value of H0 determined by low-z observations in [34]. The same rs-H0 relation predicts a low value of H0 in standard ΛCDM cosmology which is at a 5σ discrepancy with the value given in [34].Motivated by the above results, an interpolation betweenΛCDM 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 this, with a negative result. With [28] we were thus led to propose an interpolation in terms of percolated PSA vortices which, at some intermediate zp, partially undergo a depercolation transition. We have demonstrated this model to be consistent with the angular scale of the sound horizon at photon decoupling.The new model needs to be tested against the various CMB angular spectra. Our hope is that radiative corrections in SU(2) Yang-Mills thermodynamics, which play out at low z, are capable of explaining the large-angle anomalies of the CMB [49].



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*Source: 7525121-2017-10-16.xml*