Frequency-redshift relation of the Cosmic Microwave Background

We point out that a modified temperature-redshift relation ($T$-$z$ relation) of the cosmic microwave background (CMB) can not be deduced by any observational method that appeals to an a priori thermalisation to the CMB temperature $T$ of the excited states in a probe environment of independently determined redshift $z$. For example, this applies to quasar-light absorption by a damped Lyman-alpha system due to atomic as well as ionic fine-splitting transitions or molecular rotational bands. Similarly, the thermal Sunyaev-Zel'dovich (thSZ) effect cannot be used to extract the CMB's $T$-$z$ relation. This is because the relative line strengths between ground and excited states in the former and the CMB spectral distortion in the latter case both depend, apart from environment specific normalisations, solely on the dimensionless spectral variable $x=\frac{h\nu}{k_B T}$. Since literature on extractions of the CMB's $T$-$z$ relation always assumes (i) $\nu(z)=(1+z)\nu(z=0)$ where $\nu(z=0)$ is the observed frequency in the heliocentric rest frame, the finding (ii) $T(z)=(1+z)T(z=0)$ just confirms the expected blackbody nature of the interacting CMB at $z>0$. In contrast to emission of isolated, directed radiation, whose frequency-redshift relation ($\nu$-$z$ relation) is subject to (i), a non-conventional $\nu$-$z$ relation $\nu(z)=f(z)\nu(z=0)$ of pure, isotropic blackbody radiation, subject to adiabatically slow cosmic expansion, necessarily has to follow that of the $T$-$z$ relation $T(z)=f(z)T(z=0)$ and vice versa. In general, the function $f(z)$ is determined by energy conservation of the CMB fluid in a Friedmann-Lemaitre-Robertson-Walker universe. If the pure CMB is subject to an SU(2) rather than a U(1) gauge principle, then $f(z)= \left({1/4}\right)^{1/3}(1+z)$ for $z\gg 1$, and $f(z)$ is nonlinear for $z\sim 1$.


INTRODUCTION
Angular correlations between directionally dependent temperature and polarisation fluctuations of the Cosmic Microwave Background (CMB) radiation (Mather et al. 1994) are important probes for the extraction of cosmological parameters (Aghanim et al. 2020).Since the observed angular correlations are mainly influenced by curvature-induced dark-matter potentials, which in turn cause acoustic oscillations of the baryon-electron-photon plasma prior to recombination, these parameters depend on high- physics when extracted from CMB data.Therefore, they are very sensitive to the temperature redshift relation (- relation) which is assumed in expressing the CMB's energy density () in terms of .
The thermodynamics underlying the CMB and the thermodynamics of a dense gas of absorber-emitter particles may be richer than they appear, such that the two situations need to be distinguished.
★ E-mail: R.Hofmann@ThPhys.Uni-Heidelberg.de† E-mail: J.Meinert@ThPhys.Uni-Heidelberg.de While the CMB can be represented by a photon gas within its bulk, absorber-emitter particles thermalise via electromagnetic waves whose emissions and absorptions are enabled by electronic transitions.Therefore, the conventional - relation may not hold universally but, depending on how the above two extreme situations are mixed, is modified as  ()/ ( = 0) =  (), where the function  () is specific to the generalising theory.Thermodynamics then immediately implies that also the - relation is also of the form ()/( = 0) =  ().Such is the case, for example, if the thermal photons (and low-frequency waves) of the CMB are identified with the Cartan modes of a single thermal SU(2) Yang-Mills theory, SU(2) CMB , in the deconfining phase (Hofmann 2016b;Hahn & Hofmann 2018).These modes interact only feebly within a small range of low temperatures and frequencies with the two off-Cartan quasiparticle vector modes.The fact that all gauge modes, massless and massive, are excitations of one and the same thermal groundstate adds additional  dependent energy density to that of thermal fluctuations: the ground-state energy density rises linearly in  in contrast to the rapidly attained Stefan-Boltzmann law ∝  4 associated with thermal fluctuations (Hofmann 2016b).
The purpose of the present paper is to point out that past observational extractions of the CMB's - relation from background-light absorbing systems, which are assumed to thermalise with the CMB in a conventional way, are bound to extract the standard U(1) - relation if participating frequencies (observed absorption lines) are blueshifted accordingly.This is also true of the observation of spectral CMB distortions inflicted by its photons scattering off hot electrons belonging to X-ray clusters along the line of sight, i.e., the thermal Sunyaev-Zel'dovich effect (thSZ).In Section 2, we discuss these two observational approaches in more detail.First, we analyse the extraction of  () from absorption lines within the continuous spectrum of a background source caused by a cloud in its line of sight, which is assumed to be thermalised with the CMB.Second, we discuss the distortions of the CMB spectrum according to the thSZ effect.In the former case, the frequency of the absorption line () = ( + 1)( = 0), which is assumed to coincide with the exciting CMB frequency, is used to extract a temperature  ().Note that in this case  () coincides with the present CMB's temperature  ( = 0) only if it is redshifted as  ()/( + 1) =  ( = 0).
The thermalisation within a photon gas far away from any charges is different from the thermalisation within absorber clouds.This is because the degrees of freedom invoked are not the same.
As a result,  * () = ( + 1) ( = 0) is interpreted as the CMB's - relation, while it is actually  () =  () ( = 0).In exploiting the thSZ effect for the CMB's - relation extractions, we observe a similar situation.In Section 3, we review Hofmann (2016b); Hahn & Hofmann (2018) how a modified - relation (and then the - relation) arises if the CMB is subject to deconfining SU(2) rather than U(1) quantum thermodynamics and how the Yang-Mills scale of such an SU(2) model, in the following referred to as SU(2) CMB , is fixed by observation.To archive this, the CMB radio excess in line temperature, see, e.g., Fixsen et al. (2011); Dowell & Taylor (2018a), is interpreted as an effect due to the transition between deconfining and preconfining SU(2) Yang-Mills thermodynamics.
We also discuss a number of alternative explanations of this effect.
Moreover, we discuss how another SU(2) model, SU(2) e (Hofmann 2017;Hofmann & Grandou 2022;Hofmann 2016a), whose two stable solitonic excitations in the confining phase represent the first-family lepton doublet, mixes with SU(2) CMB .Such a mixing depends, up to temperatures of ∼ 7.99 keV, on the degree of thermalisation prevailing in a local environment of electromagnetically interacting electronic charges within a certain range of frequencies and charge densities.The aforementioned upward shift in the CMB frequency () to absorption line frequency  * (), accompanied by a shift in the CMB temperature  () to cloud temperature  * (), would then be a consequence of an incoherent mixture of the Cartan modes of SU(2) CMB (thermal photonic fluctuation) with those of SU(2) e (thermalised electromagnetic waves) when moving from empty space to the interior of the cloud.Finally, in Section 4, we summarise the results of this paper; mention an observational signature which is sensitive to the CMB's - relation, the spectrum of ultra-high energy cosmic rays (UHECRs); and briefly discuss implications for Big Bang nucleosynthesis.
From now on, we work in natural units  =   = ℏ = 1, where  denotes the speed of light in vacuum,   is Boltzmann's constant, and ℏ refers to the reduced quantum of action.

OBSERVATIONAL 𝑇-𝑧 RELATION EXTRACTIONS FROM A PRESCRIBED 𝜈-𝑧 RELATION
In this section, we discuss two principle probes used in the literature to extract the redshift dependence of the CMB temperature  () up to  ∼ 6.34, see Riechers et al. (2022) for a useful compilation, and how these extractions are prejudiced by an assumed - relation of CMB frequencies.
The first class of probes comprises absorbing clouds of known redshifts, e.g., parts of damped Lyman- systems, in the line of sight of a distant quasar or a bright galaxy.Here, the assumed thermalisation with the CMB populates the fine-structure levels of the ground-states of certain atoms or ions (Songaila et al. 1994;Ge et al. 1997;Srianand et al. 2000;Molaro et al. 2002;Riechers et al. 2022) or excites rotational levels of certain molecules (Srianand et al. 2000;Noterdaeme et al. 2011Noterdaeme et al. , 2017) ) whose population ratios can be obtained from the respective absorption-line profiles within the broad background spectra.
Modelling environmentally dependent contributions to level populations, such as particle collisions or pumping by UV radiation, the relative level populations yield upper-limit estimates of  () at the redshift of the cloud.The limitations of this method are discussed in Maeder (2017).Note that in Muller et al. (2013), a solution of the rotational excitations of various molecular species could be provided directly from their spectra.
The second class of probes refers to the observation of the CMB spectrum within certain frequency bands along the lines of sight of X-ray clusters of known redshifts.A characteristic spectral distortion, known as the thermal Sunyaev-Zel'dovich effect (thSZ) (Zeldovich & Sunyaev 1969;Sunyaev & Zeldovich 1972) and caused by inverse Compton scattering of CMB photons off free, thermal electrons of these clusters is exploited to estimate  ().

Absorber clouds in the line of sight of a quasar or a bright galaxy
Estimates of  () using the relative populations due to the excitation of atomic (ionic) fine-structure transitions and molecular rotation levels by the CMB have a long history, see Bahcall & Wolf (1968) for the theoretical basis and (Noterdaeme et al. 2011(Noterdaeme et al. , 2017;;Songaila et al. 1994;Srianand et al. 2000;Muller et al. 2013;Riechers et al. 2022) for applications.Since sources of level excitations other than the CMB (e.g., collisions, UV pumping) have to be modelled for a given absorber, the extracted  () is usually seen as an upper bound on the true CMB temperature.
If the CMB is assumed to be the sole source of level population then the extraction of  () is facilitated in terms of the column density of the absorber species, depending on the measured line strength in the continuous spectrum of the background source, the transition frequencies, the temperature at which levels are thermalised, and the integrated opacity of the line.Apparently, this method is validated by the measurement in Roth et al. (1993) of the CMB temperature  ( = 0) = (2.726+0.023 −0.031 ) K in our Galaxy, analysing CN rotational transitions in five diffuse interstellar clouds.The value extracted in this way,  ( = 0), agrees well with the spectral CMB fit by COBE (Mather et al. 1994) In Muller et al. (2013) a molecular rich cloud within a spiral galaxy at  = 0.89 was observed towards the radio-loud, gravitationally lensed blazar PKS 1830-211 at redshift  = 2.5.Within the cloud, the rotational temperature  rot is defined via where   (  ) and   (  ) are the populations (degeneracies) of the upper and lower level, respectively, and  * () denotes the transition frequency in the cloud's rest frame.For the rotational excitations of ten molecules,  rot was interpreted to universally represent  ( = 0.89) because the molecular gas was estimated to be subthermally excited (rotational levels solely radiatively coupled to the CMB, negligible impact of collisions and the local radiation field).
Observations were performed within 3 wavelength bands at around  = 2, 3, 7 mm using two different instruments.In one (simplified) approach, the extraction of  rot from the two transitions in each molecular species was performed by pinning down the intersection of the two column densities  LTE depending on  rot .For a given transition,  LTE is defined as where   is the energy of the lower level, ( rot ) the partition function including all rotational excitations,  the dipole moment,   the observed line strength, and ∫  the integrated (observed) opacity of the line.Across the absorption lines of all molecular species considered, this approach produces values of  rot , which are quite consistent with the expectation  ( = 0.89) = (1 + 0.89) ( = 0) = 5.14 K.
Setting  rot =  * () in Eq. ( 2) and using  * () = (1 + )( = 0) is only consistent with the participating CMB photons being distributed according to a blackbody spectrum if  * () also redshifts as  * () = (1 + ) ( = 0); ( = 0) denotes the observed frequency of the transition in the heliocentric restframe.Therefore, this is an in-built feature of the model even though the CMB may, in reality, exhibit a different - relation (and then also - relation) 1 .
The situation is similar for the observation of atomic / ionic fine-structure transitions in absorbers at  > 0. Also, here, the very assumption of these excitations thermalising with the CMB ties the extracted - relation to the - relation used in converting observed (heliocentric) frequencies to transition frequencies in an absorber's restframe: The proper use of  () = 1 +  for absorption lines produces a higher cloud temperature  * () than CMB temperature  () if the latter is assumed to be described by an unmixed SU(2) model, see Section 3.
In Section 3.2, we will discuss in more detail a degree-ofthermalisation dependent mixing of Cartan excitations in two SU(2) gauge groups explaining why directed radiation, as issued by the background source and observed in a spectrally resolved way after having passed the absorber, obeys a conventional - relation while 1 One may think of the true CMB temperature (which would be lower in SU(2) CMB ) and participating CMB frequency being elevated by the same factor to  * () and  * () of a rotational excitation, respectively, by an incoherent mixing of a Cartan mode in SU(2) CMB and a Cartan mode of SU(2) e as the observer moves from empty space outside the cloud towards its interior.
the - relation of CMB photons necessarily follows that of the - relation, which may well be unconventional (Hahn & Hofmann 2018).

The thermal Sunyaev-Zel'dovich effect
The thermal Sunyaev-Zel'dovich effect (thSZ) is a distortion of the blackbody shape of the CMB spectrum that is induced by inverse Compton scattering of CMB photons off thermalised electrons in the X-ray plasmas of a given cluster of galaxies (Zeldovich & Sunyaev 1969;Sunyaev & Zeldovich 1972).Neglecting contributions from the weakly relativistic high-end part of the electrons' velocity distribution, the thSZ effect can be expressed in terms of a frequency dependent (line-temperature) shift Δ with respect to CMB baseline temperature  at the cluster's redshift  as (Hurier et al. 2014) Here   and   refer to the mass of the electron and the Thomson cross-section, respectively.Both the electron temperature   and the electron number density   depend on the proper distance parameter  along the direction ì  of the line of sight under which CMB photons interacting with a given X-ray cluster are observed.The dimensionless variable  is defined as  ≡ 2   .As Eq. (3) indicates, the thSZ effect factorises into an environmental part, determined by the thermodynamics of the X-ray cluster at redshift  and dubbed thSZ flux, and into a part which solely depends on .We note that the zero  0 of the second factor is where  ( = 0) = 2.726 K (Mather et al. 1994) denotes the CMB temperature today.As a consequence, the thSZ effect predicts  0 ∼ 217 GHz.The  dependence of  already can be extracted by focusing on the frequency  0 at which Δ  (, ì ) vanishes.By virtue of Eq. (4) a blueshift of  0 according to the - relation then yields the - relation Therefore, whatever the assumption on  () in the - relation of Eq. ( 5) is, this assumption necessarily transfers to the - relation of Eq. ( 6) if the intensity of the unperturbed CMB at any redshift  > 0 is to possess a blackbody frequency distribution 2 .
To suppress the statistical error in extractions of  (), a set of frequency bands, centered at {  }, usually is invoked in fitting the modelled thSZ emission law to the observations with respect to X-ray clusters within a given redshift bin .For example, in Hurier et al. (2014) the Planck frequency bands at 100, 143, 217, 353, and 545 GHz were used in multiple redshift bins, stacking of patches in a given redshift bin  and frequency band centered at   was performed, and the thSZ emission law was modelled by integrating Eq. ( 3) over bandpasses and by normalising it with the bandpass averaged calibrator emission law.Relevant fit parameters turned out to be the (stacked) thSZ flux   ,   , and the radiosource flux contamination   rad which subsequently were estimated by a profile likelihood analysis.The crucial point here is that, in the modelling of the thSZ emission law within redshift bin , a blueshifting of observation frequency   to  *  =  ()  needs to be applied, implying again the - relation where   now is the solution to and    denotes the stacked, observed thSZ flux within redshift bin .In Hurier et al. (2014) the use of  *  = (1 + )  thus necessarily leads to the conventional - relation  () = (1 + ) ( = 0).To the best of the authors' knowledge, the use of  () = 1 +  in the - relation is, however, common to all extractions of  () that appeal to the thSZ effect.
When the CMB gauge field represents the Cartan subalgebra of an SU(2) Yang-Mills theory, SU(2) CMB , it can be shown (Hahn & Hofmann 2018) that  () is different from  () = 1 +  in the - relation, see also Section 3.1, and therefore also in the - relation.This is because, in addition to thermal photons, the thermal ground-state in the deconfining phase of an SU(2) Yang-Mills theory is excited towards two vector modes subject to a temperature dependent mass.A feeble coupling of these two kinds of excitations leads to spectral distortions of CMB radiance deeply within the Rayleigh-Jeans regime (Hofmann et al. 2023) which is not targeted by Planck frequency bands.In Section 3 we review how this - relation arises, and we discuss why an according non-conventional - relation is to be expected from a thermalisation process which is dependent on the mixing angle between the Cartan modes of two SU(2) gauge models subject to disparate Yang-Mills scales.

𝑇-𝑧 RELATION AND 𝜈-𝑧 RELATION IN SU(2) CMB : THEORETICAL BASIS
In this section, we discuss in detail why a thermal gas of electromagnetic disturbances (far away from any emitting surface on the scale of a typical inverse frequency  −1 ) and with an isotropic and spatially homogeneous flux density of practically incoherent photons obeys a - relation and an associated - relation which are different from the - relation of directed, (partially) coherent radiation.

𝑇-𝑧 relation in SU(2) CMB
A pronounced distortion of the blackbody spectrum of radiance deeply within the Rayleigh-Jeans part was observed in Fixsen et al. (2011); Dowell & Taylor (2018b) and in references therein.To explain this highly isotropic CMB radio excess at frequencies below 1 GHz, we argued in Hofmann (2009) that the critical temperature  ( = 0) for the deconfining-preconfining phase transition of SU(2) Yang-Mills thermodynamics is very close to the present temperature of the CMB of  ( = 0) = 2.726 K (Mather et al. 1994).This may seem to be a somewhat fine-tuned situation.However, the difference with an ordinary tuning of parameters by hand is that, the dual gauge coupling thermodynamically rises rapidly as the temperature drops into the preconfining phase.Since the Cartan mode's extracted thermal quasiparticle mass is 100 MHz, which is around three orders of magnitude smaller than the critical temperature  c , it follows that  ( = 0) needs to be very close to  c .However, due to a presently incomplete understanding of supercooling of the deconfining into the preconfining phase and associated tunnelling there is a tolerance of ∼ 10 % in this dynamic tuning of the two temperatures, which corresponds to about 1 Gy of cosmic evolution (Giacosa & Hofmann 2007).The exact assignment  ( = 0) =  c , addressed further below, implies a Yang-Mills scale Λ CMB = 1.064 × 10 −4 eV, and thus it is justified to refer to the SU(2) Yang-Mills model, whose deconfining thermodynamics is assumed to describe the CMB, as SU(2) CMB .We quote below a number of alternative approaches to explaining the CMB radio excess.
Let us now review Hahn & Hofmann (2018), how the - relation of deconfining SU(2) CMB thermodynamics is derived from energy conservation in Friedmann-Lemaitre-Robertson-Walker (FLRW) universe of cosmological scale factor , normalized such that today ( where  and  denote energy density and pressure of deconfining SU(2) CMB thermodynamics, respectively.As usual, redshift  and scale factor  are related as  −1 =  + 1. Eq. ( 9) has the formal solution where the entropy density  is defined as By virtue of the Legendre transformation one has Substituting Eq. ( 13) into Eq.( 10) finally yields Here,  denotes an arbitrary mass scale.The formal solution ( 14) is valid for any thermal and conserved fluid subject to expansion in an FLRW universe.If the function () is known, then ( 14) can be solved for the - relation  ().Eqs. ( 11) and ( 14) exclude a ground-state dependence of the - relation, since the equation of state for ground-state pressure  gs and energy density  gs is  gs = − gs (Hofmann 2016a).
In deconfining SU(2) CMB thermodynamics, asymptotic freedom (Gross & Wilczek 1973;Politzer 1973) occurs nonperturbatively for  ≫  ( = 0) (Hofmann 2016a).The Stefan-Boltzmann limit is then well saturated, and therefore () is proportional to  3 .Moreover, at  ( = 0), due to a decoupling of massive vector modes, excitations represent a free photon gas.Therefore, ( ( = 0)) is proportional to  3 ( = 0).As a consequence, the ratio ()/( ( = 0)) in Eq. ( 14) reads where  refers to the number of relativistic degrees of freedom at the respective temperatures.We have () = 2 × 1 + 3 × 2 = 8 (two photon polarizations plus three polarizations for each of the two vector modes) and ( ( = 0)) = 2 × 1 (two photon polarizations).Substituting this into Eq.( 15), inserting the result into Eq.( 14), and solving for , we arrive at the high-temperature - relation Due to two vector modes of a finite,  dependent mass contributing to () at low temperatures, the - relation is modified as where the function S() is depicted in Fig. 1.In Section 3.1, we reviewed why  ( = 0) ≲  c .We now argue why  ( = 0) is excluded to be larger than  c : There is another contribution to the excess in line temperatures at low frequencies from the fact that the frequency of waves (populating the deep Rayleigh-Jeans spectrum up to const/ 2 ) and the frequency of photons (starting to represent the spectrum for frequencies larger than const/ 2 ) redshift differently.On the other hand, baseline temperature  () redshifts like the frequency of photons.That is, wavelike modes redshift as () = (1 + )( = 0) while temperature (photon frequency) redshifts weaker than that like  () = ()( + 1) ( = 0) with a numerically known function () < 1 of negative slope

𝑑𝑆 (𝑧) 𝑑𝑧
∼ −1 for  ≪ 1, see Fig. 1.This also contributes to an increase of line temperature at low frequencies compared to the Rayleigh-Jeans law since low frequencies are redshifted as usual but baseline temperature redshifts slower when lowering  in the vicinity of  = 0, see Fig. 2 and Eq. ( 18).Observationally (Fixsen  et al. 2011), the onset of this effect could be visible at  ∼ 1 GHz which implies the wavelengths of low-frequency waves to be larger than 30 cm, in turn, implying a critical temperature lower than 11.6 K (Hofmann 2016a).For the differential evolution of baseline temperature, we have Since the present CMB's line temperature rises steeply with a spectral index of −2.6 when lowering the frequency (Fixsen et al. 2011;Dowell & Taylor 2018b) the effect of Eq. ( 18), which, modulo a mild stacking of low frequencies, is frequency independent for  < const/ 2 , does not explain these large and variable line temperatures.Thus, we are again led to set  ( = 0) ∼  c to explain the observed steep rise in terms of wave evanescence (thermal Meissner mass).Therefore, the tuning  ( = 0) ∼  c is entirely explained by observation and does not require any ad hoc parameter coincidence.
Note that large and variable line temperatures cannot be explained in terms of the diffuse free-free emission facilitated by cosmological reionisation (Trombetti & Burigana 2013;Oh 1999).Interestingly, synchrotron radiation induced by weakly interacting massive particle (WIMPS) annihilations or decays in extra-galactic halos could match the low-frequency excess in CMB line temperature if a thermal annihilation cross section for light WIMPS is invoked (Fornengo et al. 2011).Galactic radio emission is excluded as an explanation by the isotropy of the signal (Seiffert et al. 2011).In Baiesi et al. (2020) stochastic frequency diffusion is used to explain the low-frequency excess in the present CMB (also dubbed 'space roar´) in terms of a primordial epoch of nonequilibrium conditions in the plasma.These conditions are modelled by a mild violation of the Einstein relation in the Kompaneets equation to allow for low-frequency localisation in the evolving photon distribution.The formation of the first generation of supermassive, cosmological black holes is speculated to explain the space roar in terms of synchrotron emission from the remnants (Biermann et al. 2014).
If the line-temperature excess can, indeed, be shown to persist to higher redshifts, including the onset of re-ionisation (cosmic dawn), then a potential explanation of the anomalously strong absorption of the redshifted 21-cm line by neutral hydrogen measured by the experiment to detect the global epoch of reionisation signature (EDGES) (Bowman et al. 2018) is enabled.This would falsify our proposal of the present space roar to solely be a very-low redshift phenomenon due to an admixture of Gaussian distributed evanescent waves to the conventional low-frequency Rayleigh-Jeans CMB spectrum3 .However, the strong absorption of the redshifted 21-cm line can also be explained by dark-matter induced cooling of the absorbing cosmic gas without having to invoke excess intensity of the CMB at low frequencies throughout cosmic dawn (Barkana 2018).

Anisotropic photon emission by electrons or isotropic and homogeneous thermalisation
In the framework of SU(2) Yang-Mills theory, why is it that spectral lines redshift according to the conventional - relation () = (1 + )( = 0) while the bulk of frequencies within the CMB follows a - relation which associates with the - relation of Section 3.1?
The electron and its neutrino are modelled by a onefoldselfintersecting, figure-eight like center-vortex loop and a single center-vortex loop, respectively, see Hofmann (2017) ;Hofmann &Grandou (2022) andHofmann (2016a).These excitations are immersed into the confining ground-state of SU(2) Yang-Mills theory.A mass formula can be derived for the electron which equates the frequency of a breathing monopole (Fodor & Rácz 2004;Forgács & Volkov 2004) (or the quantum selfenergy (De Broglie 1925, 1964)), contained within an extended ball-like blob associated with the region of vortex intersection, with the sum of the static monopole's rest mass and the energy content of deconfining SU(2) Yang-Mills thermodynamics of the blob (considering a mixing of two thermal gauge theories SU(2) CMB and SU(2) e at a temperature  =0 = 1.18  ,e where the pressure vanishes (Hofmann & Meinert 2023)).By invoking the value of the electron mass  e = 511 keV, this formula yields a value of the SU(2) Yang-Mills scale Λ e = 3.62 keV or a critical temperature  ,e = 7.99 keV for the deconfining-preconfining phase transition in SU(2) e (Hofmann & Meinert 2023).
In addition, one obtains a blob radius  0 ∼  0 , where  0 denotes the Bohr radius  0 = 0.592 Å.Also, the reduced Compton radius   , which roughly coincides with the core radius of the monopole   (Fodor & Rácz 2004;Forgács & Volkov 2004), turns out to be   ∼  0 where  ∼ 1/137 is the electromaganetic fine-structure constant.Modulo the electron's magnetic moment, carried by two closed vortex lines connecting to the blob, this matches with de Broglie's original interpretation of the electron (De Broglie 1925Broglie , 1964) ) and with the interpretation of the square of the wave function in wave mechanics (Schrödinger 1926) as a probability density for locating a point particle (Born 1926).Namely, in its restframe, the electron represents an extended (spatially homogeneous) vibration induced by a charged monopole whose core size is negligible on the scale of the blob size and whose rate of jump-like location changes within the blob volume matches the vibration frequency  0 ( 0 = ℎ 0 ,  0 the electron mass).
If the global temperature of a photon gas is smaller than  ,e then these photons must be thermalised with respect to SU(2) CMB with  ,CMB ∼ 10 −4 eV ≪ 7.99 keV ∼  ,e .On the other hand, a directedly propagating electromagnetic field (a wave) represents a nonthermalised mode and thus cannot be subject to the gauge group SU(2) CMB but rather is described by SU(2) Yang-Mills theories of much larger Yang-Mills scales (Hofmann 2016a).The process of converting these isolated waves, emitted by the charge-carriers that do not penetrate into the volume bounded by a closed, emitting spatial surface, into a thermal photon gas contained within this volume hence proceeds by chopping their coherent intensity distribution into grainy and short-lived energy-momentum packets by an increasing homogenisation and isotropisation of energy transport as more and more differently directed waves of varying oscillation frequencies superposition away from the emitting surface.This very process of thermalisation, producing a photon gas with the temperature of the emitting surface, can effectively be understood as a rotation of SU(2) modes of theories with large Yang-Mills scales into those of SU(2) CMB .

Thermalisation dependent mixing of two SU(2) gauge theories
For simplicity and due to its practical relevance 4 we consider the interplay of gauge groups SU(2) CMB and SU(2) e .The discussion in Section 3.2 can then be summarised as where ( ĀCMB  , Āe  ) refers to the rotated state reached from the initial state ( CMB ,  e ) for the effective gauge fields in the deconfining phases of SU(2) CMB and SU(2) e .The thermodynamically determined mixing angle   turns out to be close to the electroweak mixing angle (  = 30.84• ) if mixing within the interior of the blob -representing the center of the region of selfintersection of the center-vortex loop -is considered.Within this central domain the mixed deconfining-phase pressure is demanded to vanish (Hofmann & Meinert 2023).Note that in case of infinite-volume thermodynamics at high temperatures (high- cosmology) such a stability constraint on a finite-volume region is irrelevant, and one has   = 45 • .
In general, a change of the state of thermalisation induces a change of the rotation angle  =  (,  * , ).In particular, for the interaction of the CMB with absorber clouds  depends on the degree of thermalisation  invoked by the initial states of electromagnetically interacting electrons of temperature  * within the cloud and the temperature  of the CMB.Note that the thermalisation of these electronic states is influenced by these very lines dissipating directed background light.Via the degree of thermalisation , the mixing angle  also depends on the range of frequencies   ≤  ≤   of wavelike modes in SU(2) CMB and SU(2) e which mediate the interactions between the electrons.Due to the present CMB exhibiting the largest low-frequency interval of excitations being associated with waves throughout its cosmological history, see Section 3.1, and since the main frequency used for the extraction of  ( = 0) from background-source-absorber-cloud systems in the Milky Way (Roth et al. 1993) is 113.6 GHz, which is to the left of the peak frequency  = 160.4GHz of the present CMB's blackbody spectrum, it is qualitatively understandable that the same temperature as from the CMB blackbody spectral fit in (Mather et al. 1994) is observed for these systems.Due to the strong compression of the CMB wave spectrum with a factor ∝  −2 it is also plausible that temperatures extracted from a background-source-absorber-cloud system at earlier epochs differ from the associated CMB temperatures.
The quantitative computation of  in a situation where a system of interacting (bound) electrons of a given number density, invoking a range of frequencies   ≤  ≤   for these interactions, is immersed into the CMB, is a complex task which we hope to gain more insight about in the future.

SUMMARY AND OUTLOOK
In this paper, we discussed two main approaches to extract the CMB temperature at finite redshifts: the analysis of absorption line profiles originating from gases of atoms, ions, or molecules within the line of sight of a broad-spectrum and bright background source, and the thermal Sunyaev-Zel'dovich effect (thSZ).In the literature, the assumption of a conventional frequency-redshift relation (- relation) for CMB photons, considered to thermalise with relevant transitions in the cloud systems in the former case or to represent CMB spectral distortions in the latter situation, yields a conventional temperature-redshift relation (- relation) for the CMB.We argued, based on the blackbody spectrum at all redshifts, that this is a consequence of thermodynamics.Whatever the assumed - relation, observations necessarily produce the associated - relation and vice versa.If the CMB is subject to an SU(2) rather than a U(1) quantum gauge principle, we reviewed how the corresponding - relation changes.Consequently, the CMB's - relation is changed.Finally, on a qualitative level, we provided reasons for which the temperature of a cloud of known redshift may differ from the temperature of a pure photon gas representing the CMB far away from the cloud.This is because thermalisation in the cloud, in addition the interaction of bound electrons with wavelike CMB disturbances, also proceeds via emissions and absorptions of wavelike modes by bound electrons.These modes, however, are subject to another (confining-phase) SU(2) Yang-Mills theory of a much higher critical temperature: SU(2) e .
The existence of SU(2) e impacts Big Bang nucleosynthesis.Specifically, if the electron is subject to an SU(2) gauge-theory model, involving the two factors SU(2) CMB and SU(2) e (Hofmann 2017;Hofmann & Grandou 2022;Hofmann & Meinert 2023) (see also Meinert & Hofmann (2021)), then the Hagedorn temperature   = 6.66 keV of SU(2) e implies that the primordial Helium mass fraction of  = 1 4 ( = 2   1+   , where   denotes the neutronto-proton ratio at the onset of nucleosynthesis) is not induced by the nucleosynthesis of the light elements setting in at  = 65 keV, subject to   = 1 7 (the freeze-out value  = 1 5 at  ∼ 800 keV being reduced to   = 1 7 at  = 65 keV due to neutron decay).Rather, nucleosynthesis would start at  = 65 keV with   = 1, implying a Helium mass fraction of  = 1 prior to the Hagedorn transition.The value of  would subsequently be reduced to  ∼ 1 4 through collective Helium photo-disintegration by gamma quanta that are released across the Hagedorn transition.
Our conclusion regarding the extractions of the CMB temperature using the thSZ effect pursued in the literature is that they confirm the CMB blackbody spectrum at a finite redshift.However, under the assumed conventional - relation, the result of the - relation extraction is necessarily conventional as well.
The extraction of the CMB's - relation from an assumed thermalisation within absorber clouds, which also uses a conventional - relation for the relevant absorption lines, is questionable since the two systems, (i) a cloud immersed into the CMB and (ii) a pure CMB, exhibit different thermal degrees of freedom at a sufficiently high redshift: waves for (i) and photons for (ii).
One possibility for determining the effect of the - relation subject to SU(2) CMB is to study the flux of ultra-high-energy cosmic rays (UHECRs).In particular, there is a sensitive region below the ankle, that is, for 1 × 10 18 eV ≤  ≤ 6 × 10 18 eV.Due to the reduced CMB photon density at the same finite redshift, there is a higher flux of UHECRs under otherwise equal conditions for emission and propagation.Most prominently, the flux of protons is significantly increased in comparison to the use of the conventional - relation when fitted to UHECR data (Meinert et al. 2023).

Figure 1 :
Figure 1: Plot of function S() in Eq. (17).The curvature in S() at low  indicates the breaking of conformal invariance in the deconfining SU(2) Yang-Mills plasma for  ≳  ( = 0) with a rapid approach towards (1/4) 1/3 as  increases.The conventional - relation of the CMB, as used in the cosmological standard model ΛCDM, is associated with the dashed line S() ≡ 1. Figure adapted from Hahn et al. (2019).