On a possible nonequilibrium imprint in the cosmic background at low frequencies

The cosmic background radiation has been observed to deviate from the Planck law expected from a blackbody at $\sim$2.7 K at frequencies below $\sim$3 GHz. We discuss the abundance of the low-energy photons from the perspective of nonequilibrium statistical mechanics by specifying an evolution to a frequency distribution fitting the observed discrepancies. We mention possible physical mechanisms that enter the derivation of that dynamics, where a low-frequency localization is combined with photon cooling as result of e.g. induced Compton scattering. In that sense, the so called 'space roar' we observe today is interpreted as a nonequilibrium echo of the early universe.


I. INTRODUCTION
The cosmic microwave background (CMB) is a prime witness to the physics of the early universe. According to the standard model of physical cosmology, it carries information about an epoch before neutral atoms were formed, dating from some 10 5 years after the Big Bang (see e.g. [1, 2] for reviews). It is generally assumed then that matter and radiation were approximately in thermodynamic equilibrium, owing to the high efficiency of Compton scattering, bremsstrahlung and radiative Compton processes, with characteristic time-scales much shorter than the cosmic expansion time-scale. At the recombination era, when electrons and protons formed hydrogen atoms, light decoupled from matter and the photons started to move almost freely through the expanding universe, influenced by secondary effects only, such as the cosmological re-ionization associated to the formation of early stars and galaxies and the weak gravitational lensing induced by cosmic structures (see e.g. [3,4] for reviews and [5,6] for recent analyses). As a result, the distribution function of the CMB photons at later times t is supposed to follow the blackbody spectrum at an equilibrium temperature T (t). After a delicate subtraction of the intervening astrophysical emissions, the cosmic background appears today very close to a blackbody radiation [7] at a temperature of about 2.7 K, peaking at about 160 GHz, in very good agreement with the Planck spectrum from about 10 GHz up to about 600 GHz.
However, there is evidence of a systematic deviation from the Planck law of a blackbody at about 2.7 K at low frequencies, in the radio tail of the cosmic background. That aspect has been recently brought to attention by two independent types of observations: the CMB absolute temperature excess measured by ARCADE 2 [8] and the anomalously strong absorption of the redshifted 21 cm line from neutral hydrogen measured by EDGES [9]. After consideration of possible instrumental errors and after subtracting Galactic and extragalactic sources of low-frequency radiation, a strong residual emission remains in ARCADE 2 data [10], that is much larger than predicted by the standard theory of CMB spectral distortions. Consistently, the intriguing EDGES absorption profile amplitude, about 2-3 times larger than expected, could be explained by a much stronger background radiation with respect to standard predictions (see [11] for an alternative explanation assuming that the primordial hydrogen gas was much colder than expected). The scientific literature in fact abounds with experimental data from low-frequency radio surveys, some going back a long time: after subtraction for Galactic and extragalactic contributions, they all show an excess of soft (i.e. low-frequency) photons. A concise description of cosmic background spectrum data considered in this work is given in Section II.
We repeat that the CMB spectrum theory assumes (near-)equilibrium conditions, e.g. up to the time of recombination. The equilibrium distribution, the Planck law, is the quantum analogue of the Maxwell distribution for a classical ideal gas, and as emphasized already by the pioneers of statistical mechanics, it is the distribution to be typically expected as a consequence of counting with Bose statistics. For the kinetics and relaxation to the Planck distribution, we remind the reader in Section III about the Kompaneets equation [12], which is used in that context. It describes the evolution of the photon spectrum due to repeated Compton scattering off a thermal bath of non-relativistic electrons, possibly towards the equilibrium Planck distribution. In the Kompaneets equation, the Planck law entails zero current (in frequency space) as the result of a detailed balance between diffusion and drift.
That arises from the analogue of the Einstein relation or the second fluctuation-dissipation relation as it is called in the (classical) Fokker-Planck equation. Yet, kinetically there is localization at low frequencies, as implied by the smallness ∝ ν 2 of the kinetic coefficients.
The soft photons will not change their number (or density) so easily as the interaction with the plasma-environment is damped at low frequency. It is that kinetic aspect that is crucially important when (even slightly) violating the Einstein relation.
In this paper, we no longer assume that the universe at t 1 sec after the Big Bang was in thermodynamic equilibrium for the relevant degrees of freedom. In Section IV, we investigate how a nonequilibrium change in the Kompaneets equation allows one to reproduce the main features of the observational data, in particular the observed excess in the cosmic background at low frequency. The suggested mechanism is formally similar to the one for population selection in various nonequilibrium distributions, as has for example been discussed for population inversion in lasers [13], for kinetic proofreading in protein synthesis [14] and for suprathermal kappa-distributions in space plasma [15]. It may be theoretically summarized in the so called blowtorch theorem [16,17]. Violating the Einstein relation and adding a low-frequency source immediately leads to the abundance of soft photons. We illustrate such mechanism in detail using a modified Kompaneets equation to fit the relevant data. Good agreement is remarkably easy to obtain in that way. Notably, the imbalance between drift and diffusion, resulting in what is here an effective pumping towards low frequency, is thus understood as a nonequilibrium feature.
In other words, we put forward the hypothesis that the low-frequency excess in the CMB is a nonequilibrium imprint, originating at (or ultimately before) the time of the primordial plasma. We are also brought to investigate such an idea by the various analogies we see with the phenomenon of low frequency spectral power enhancement that has been observed in a number of different nonequilibrium systems, including disordered systems [18,19], fluids [20,21], driven macromolecules [22], and vibrating solids [23]. Also in those systems a low frequency spectral power enhancement shows the violation of the equilibrium statistics. This may be interpreted as a consequence of the fact that long wavelengths explore space scales in which the lack of equilibrium is more clearly manifested (although some specific theoretical models [24,25] allow the existence of low-frequency depletion instead of abundance). In the present paper we emphasize the role of low-frequency localization, as explicit in the physically standard Kompaneets equation, which combined with a nonequilibrium driving almost immediately produces the enhancement.
Finally, in Section V, we discuss a possible nonequilibrium mechanism, which would explain the violation of the Einstein relation in the Kompaneets equation describing the photon density updating in the earliest epochs. Nonequilibrium effects in the early universe have not been discussed extensively so far, and the ideas remain speculative. The present paper in Section V is also not providing a systematic discussion; important controversies remain concerning the turbulent nature and the low-entropy sources of the primordial plasma. We will discuss the issue in more detail in a follow-up paper. So far, the origin of nonequilibrium features can only be thought to reside ultimately with gravitational degrees of freedom that have influenced the nature of light and matter as may be expected in strongly non-Newtonian regimes of gravity.

II. OBSERVATIONAL FRAMEWORK
Measurements of the absolute temperature of the cosmic background are performed since the CMB discovery by [26] at 4.08 GHz. In this work we use the best data available for estimating the photon density. That is necessary for evaluating our theoretical model of Section IV. In particular we consider: 1. The data listed in Table 1 of the ARCADE 2 data interpretation paper [10], but not the FIRAS "condensed" data at 250 GHz). Table 1 in [27] devoted to the joint study of early and late CMB spectral distortions.

The data of
3. The measurements by the TRIS experiment together with the long wavelength compilation in Table 1 reported in [28]. 4. The extremely accurate measurements by FIRAS on board COBE [7,29]. We take from [7] the measurements at the five lowest FIRAS frequencies while the results in [29] are used above 68 GHz. A little rescaling is applied to the FIRAS data to account for the last absolute temperature calibration by [30] at T * = 2.72548 K. We do not include the data by the COBRA experiment and by the analysis of molecular lines, as they fall in the same range of the much more accurate FIRAS measurements.
5. The recent data between 0.04 GHz and 0.08 GHz by [31]. They refer to the extragalactic signal without any subtraction of the known contribution by extragalactic sources, that we perform as described below. Note that the value adopted by the authors for the extragalactic background temperature at 408 MHz is consistent with the one in Table 1 of [10], but not with the value in the subset of the older data in Table 1 of [27].
As is well known, excluding the ARCADE 2 measurements, the averaged temperature of the data at 1 GHz ν < ∼ 30 GHz is slightly below the FIRAS temperature determination at ν 30 GHz. On the other hand, the measurements below ∼ 1 GHz and the excess at 3.3 GHz claimed by ARCADE 2 indicate a remarkable temperature increase in the radio tail of the background radiation.
One should consider possible necessary corrections to the data, as other sources than CMB may have contributed. The relevance of the accurate subtraction clearly emerges in the ARCADE 2 data about the residual extragalactic emission presented in Table 1 of [10]. The authors derive the extragalactic signal after the subtraction of the Galactic emission. Their residual extragalactic emission assumes the model by [32] to describe the global contribution by unresolved extragalactic radio sources, expressed in terms of the antenna temperature T ant (ν) = c 2 /(2ν 2 k B ) Smax S min S N (ν, S) dS. Here c, k B , ν, S and N (ν, S) are the light speed, the Boltzmann constant, the photon frequency, the source flux density and the source differential number counts. On the other hand, recent studies [33][34][35][36] suggest an increase of N (ν, S) up to a factor ∼ 3 at ∼ 10µJy and of a factor ∼ 1.5 at ∼ 100µJy with respect to the differential number counts by [32], likely to be ascribed to faint star forming galaxies and radio-quiet AGNs. By simply rescaling at faint fluxes the differential number counts in [32] by such factors, we find a larger contribution of unresolved extragalactic radio sources (of about 30%, when expressed in terms of antenna temperature, since only a fraction of the global contribution by unresolved extragalactic radio sources comes from sources at faint flux densities), always to be subtracted from the signal to derive the residual extragalactic emission.
The data we adopt in this study for the cosmic background absolute temperature are listed in Appendix A (see Table I). We report the background temperatures according to quoted papers, e.g. assuming the signal treatment originally performed by authors (second column). For the data where the model by [32] was applied to subtract the global contribution by unresolved extragalactic radio sources, as e.g. in [10], we report also the background temperature derived applying the higher subtraction described above to account for possible higher differential number counts at faint flux densities (third column). In the case of the data by [31] we perform the subtraction using both the recipe by [32] and this higher model. See also Appendix A for further details. The data in Table I Table I), gives appreciable changes between 0.022 GHz and 0.08 GHz, for the two TRIS measurements around 0.7 GHz, and, but only weakly, for the two ARCADE 2 measurements around 3.3 GHz.
Many explanations have been tried to account for the residual low-frequency excess, and we cannot mention all attempts. For example, the diffuse free-free emission associated to cosmological re-ionization has been considered as one way to explain the ARCADE 2 and the radio background excess, but the signal spectral shape is steeper than that predicted for the free-free distortion [10]. Furthermore, the signal amplitude is much larger than those derived for a broad set of models (see e.g. [37,38]). Efforts have also been dedicated to explain the low-frequency background signal excess and the EDGES absorption profile in terms of astrophysical emissions, possibly in combination with particle physics phenomena (see e.g. [11,[39][40][41][42][43]). So far, there is no agreement in explaining the intriguing and still even questioned data (see e.g. [44][45][46]). The present study is aimed at taking a very different route than previous studies, to consider the background radiation excess in the radio tail as a true cosmological signal and to explain it in terms of nonequilibrium statistical mechanics.

III. KOMPANEETS EQUATION NEAR EQUILIBRIUM
The fundamental equation describing the kinetics of Compton scattering of photons and thermal electrons, which is relevant for the relaxation to the Planck distribution in the primordial plasma as well, has been introduced by Kompaneets [12] and by Weymann [47].
It is assumed that the energy exchanges are non-relativistic at electron temperature T e with k B T e m e c 2 and for photon energies hν m e c 2 . Then, the dimensionless occupation number n(t, ν) at time t and frequency ν, obtained from the spectral energy density E ν of where σ T is the Thomson cross section and N e is the electron density. We use a rescaled version of that Kompaneets equation, for the time-evolution of the photon occupation number n(τ, ν) with rescaled time τ = htσ T N e /(m e c), which is irrelevant for the stationary solution we are after. In terms of the photon density (per unit frequency) defined as ρ(τ, ν) := ν 2 n(τ, ν), with prefactor ν 2 being proportional to the density of states, the equation (2) reads We refer to the literature [48,49] for details of the derivation of (1). The starting point is a Boltzmann equation for photons interacting with a plasma where the main mechanism is elastic Compton scattering between electrons and photons. This is thought to be the primary mechanism for the (partial) thermalization of the CMB.
The stationary solution of (2) for which the expression between square brackets vanishes, k B T e ∂ ν n eq + h (1 + n eq )n eq = 0, is the equilibrium Bose-Einstein distribution which reduces to the Planck law for integration constant C = 0 (photon chemical potential).
Assuming thermal equilibrium between electrons and photons, with the usual, ∝ (1 + z), temperature scaling with redshift z because of cosmic expansion, we have T e (z) = T * (1 + z), which is the same scaling as for the photon frequency.
It is the nonlinear term ∼ n 2 in (2) that makes the "low-frequency" Rayleigh-Jeans for the Rayleigh-Jeans density corresponding to (3). Without that nonlinearity the stationary solution would be the Wien spectrum n Wien (ν) , which is a good approximation for high frequencies. We emphasize that in all events the Planck law solves (2) because it balances the diffusion term (second derivative) with the drift term (first derivative), independent of the prefactor ν 4 in front of the square bracket. That is the usual scenario for detailed balance (or reversible) dynamics [50], for which the stationary solution shows zero current in the frequency domain.
The next important observation is the emergence of a localization effect at low frequencies, realized by the power ν 4 in (2). Dynamically the escape rates away from low frequency are strongly damped, which implies for example slower relaxation for initial conditions peaking at low-frequencies. That frequency dependence can already be read off from the Klein-Nishina cross section (for Thomson to Compton scattering). Again, that kinetics is not visible in the equilibrium Planck distribution but it does play a role dynamically. In fact, the low-frequency localization is a typical wave phenomena: scattering is limited at low frequencies/large wavelengths.
To be complete we note that, in the above, we considered the Kompaneets equation including only Compton scattering. The evolution equation for the photon occupation number could be described by a "generalized" Kompaneets equation accounting also for other physical processes in the plasma and coupled to an evolution equation for the electron temperature [51]. Unavoidable photon production/absorption processes operating in cosmic plasma [2] include the double (or radiative) Compton scattering [52][53][54], the bremsstrahlung [55,56] and, in presence of primordial magnetic fields, the cyclotron process [57]. In (near-)equilibrium conditions their rates are derived assuming again detailed balance and, consequently, in combination with the Compton scattering, they tend to re-establish a Planckian spectrum, as the reversible (zero current) stationary solution. Other photon production/absorption processes are predicted in exotic models. Heating and cooling mechanisms not directly originating photon production/absorption can be also effectively added as source terms in the Kompaneets equation or in the evolution equation of the electron temperature, according to a variety of almost standard or exotic processes. The resulting spectra mainly depend, at high redshifts, on the process epoch, the global amount of injected photon energy and number density, the overall energy exchange, and, at low redshifts, also on the details of the considered mechanism (see e.g. [58]).
In the following section we neglect the effects of such additional mechanisms, focusing instead on the implication of "violating" the Einstein relation in the (simplest and most elementary version of the) Kompaneets equation.

IV. BREAKING THE EINSTEIN RELATION
The Kompaneets equation (2) The notation suggests to think of D(ν) as a frequency-dependent diffusion, and of γ(ν) as a frequency-dependent friction. We stick with this terminology for a moment, but we will clarify their meaning in Section V. The point here is that, again in analogy with the Fokker-Planck equation, the ratio of D and γ may be called an effective temperature.
In (2) with F (ν) function of the frequency.
Even though that stationary solution (7) again makes the square bracket in (6) vanish and solves D(ν)∂ ν n s = −γ(ν)(1+n s )n s , it does correspond to a physical breaking of the Einstein relation, as the effective temperature T (ν) is frequency-dependent. That temperature is just The experimental data (black dots with 1σ error bars) refer to the measurements discussed in Section II: panel (a) refers to smaller subtraction of extragalactic signal, panel (b) refers to the higher subtraction (see Table I). For each panel, the solid line is the best fit of the data with T (ν) obtained from (9). At 95% confidence level, and respectively for the cases of panel (a)  which can be seen as the background equivalent thermodynamic temperature, as immediate from the relation between n s (ν) and F (ν) in (7), and hints at the nonequilibrium nature of the phenomenon. From the data in Table I and as discussed in Section II, T (ν) results to be large at ν 1 GHz and to assume values around T * at ν 1 GHz: this trend needs to be reproduced by ν/F (ν).
We claim that (8) can easily reproduce the features of the cosmic background data in the whole frequency spectrum with simple assumptions on F (ν) in (7). We take the functional form From here onward, when referring to the comparison with observational data, we take T e equal to the present CMB temperature T * at high frequencies. Then, T (ν) scales as ν −α for ν ν 0 , while it conforms to the standard CMB temperature T * for ν ν 0 . We use (9) to fit the observations with free parameters α and ν 0 . The results of the fits are shown in Fig. 1(a) and (b) for the two data sets that we are considering, as detailed in Section II and listed in Table I of Appendix A. Overall there is a general agreement between the data and this model, with reduced χ 2 1.9 for both data sets (see also Appendix A). Only minimal differences in the retrieved best fit parameters are obtained for the two different radio background subtractions, without relevant changes of the whole picture. The retrieved values are ν 0 0.37 GHz and α 3.3, significantly larger than the one found using only the data in table 1 of [10] and likely difficult to explain in terms of synchrotron emitters.
As anticipated, an informed breaking of the equilibrium assumption suffices to reproduce qualitatively the low-frequency excess observed in the data. We should however not take (9) as the correct behavior at ultra-low frequencies, see the discussion in Appendix B.

V. LOW-FREQUENCY ABUNDANCE
A rewriting of (6) in the form suggests the nonequilibrium character of the modification. The first term on the right-hand side of (10) is the finite temperature contribution and the second part (purely diffusive) formally adds an infinite temperature contribution. It is this second diffusive term that violates the balance we discussed in the previous section. Comparing (10) with (6)  In other words, (7). When indeed the dimensionless factor B(ν)/ν 2 depends on ν, we cannot interpret (10) as a reversible Kompaneets equation (2) with a new (effectively global) temperature.
Numerical results are shown in Fig. 2(a). Hereafter we refer to the results from the fitting of the data set with the smaller subtraction, the numerical difference with respect to the other case being small however. The enhanced photon occupation in the low-frequency part of the spectrum is evident in Fig. 2(b) where we plot the function n s (ν), obtained by plugging in (7) the parameters α and ν 0 from the fits, and the equilibrium photon occupation from the Planck spectrum.
As mentioned above, the Kompaneets equation is not literally about a diffusion and drift in frequency space. The term in (10) containing B(ν) amounts to an additional frequencydependent activation mechanism, due to the electrons, that increases the photon intensity.
In general indeed, in (6), the term proportional to D(ν)∂ ν n is the transfer of (noisy) energy from the electrons (the medium) to the radiation in terms of increased intensity (number of photons). The γ(ν) relates to the ν-dependent loss of photons.
Alternatively, one may think of the last term in (10) as giving a noisy rate of increase of intensity with variance B(ν). That is similar to the phenomenon of "stochastic acceleration" for classical probes in a turbulent plasma, but here leading to the increase of low frequency photons. More formally, things get clearer by writing the modification of the Kompaneets equation (3) for the density ρ : We can now make a more rigorous analogy with the Fokker-Planck equation, as we truly deal with the photon density ρ per unit frequency: in the low-frequency approximation and, to be specific, assuming the above corresponding power law relations for F (ν) and D(ν) that imply B(ν) = [ν 2 /(α + 1)](ν/ν 0 ) −α , we have The (nonequilibrium) insertion of B(ν) ∝ ν 2−α increases the diffusion constant for small frequencies, but there is also negative friction for small frequencies via the term B(ν)/ν ∼ ν 1−α . The amplitude of the nonlinear term is unchanged of order one which reflects the essential localization as it derives from γ(ν) = ν 2 . The stationary solution of (13) is the modified Rayleigh-Jeans law (low-frequency regime in nonequilibrium), which is indeed what we got in the previous section, except that one should not take that solution all the (unphysical) way down to zero frequency (see also the discussion in Appendix B). It shows of course a drastic increase of the density with respect to the usual (low-frequency regime in equilibrium) Rayleigh-Jeans case (5) where ρ RJ (ν) = k B T e ν/h.
The abundance of soft photons can be seen as the result of the photon cooling due to the interaction with electrons. This should not be considered odd given the high effective temperature at low frequency, which is just a parameter related to the occupation statistics.
The question arises about the physical mechanism leading to this instability and nonequilibrium effect. The trigger of the additional photon intensity at low frequencies is arguably to be found in the original plasma, in the epoch from the quark to the hadron age of the universe. Under the low-entropy assumption for the very early universe [59], it is not so strange to consider that the primordial plasma needs not have started in global thermal equilibrium. A far-from-equilibrium initial plasma would have very large relaxation times for the low frequencies. Moreover, in space plasmas we see suprathermal tails in the electron velocity distribution, stemming from a high energy localization in the electronic degrees of freedom coupled to a turbulent electromagnetic field [15]. That may have contributed to the abundance of soft photons and together prevented thermalization before the radiation became free CMB, and a near-steady occupation (7) was installed. The low-frequency localization which is already present in the reversible Kompaneets equation is then the final complement to the high energy localization in the electron momenta-transfer.
Obviously, details of the mechanism will need to be added, and other scenarios may be imagined. Here we just notice that a proper quantum mechanical treatment of the interactions between photons and nonequilibrium collective plasma excitations leads in the semi-classical limit (see Eq. (A9) in [60]) to the dissipative Kompaneets equation (10). Also, we refer to [61] and to [62] for other examples and derivation of low-frequency distortions due to the induced Compton scattering. Another way to transfer energy from photons to electrons is to think of pair production from high-energy photons. Pair production in a rapidly expanding universe, such as under early inflation, will then create real long-lived high-energy electrons while depleting the high-frequency photon spectrum.

VI. CONCLUSIONS AND OUTLOOK
We have modified the Kompaneets equation within an effective nonequilibrium scenario by introducing a frequency-dependent diffusion. That leads to a violation of the Einstein relation and of the balance between diffusion and friction. The result is a clear enhancement of lower photon frequencies compatible with the best data available for the CMB spectrum.
One crucial ingredient is already present in the (reversible) Kompaneets equation: the lowfrequency localization. It is the combination of that low-frequency localization and the pumping towards larger wave vectors that creates a (new) stationary frequency distribution for the (nonequilibrium) Kompaneets equation.
We have tested our hypothesis by fitting the temperature of the cosmic background in a very wide frequency range, including frequencies where excess is observed. What seems mandatory for future explorations is an experimental effort devoted at more precise estimates of the cosmic background in the low-frequency tail, from about (10 − 20) GHz downward.
The frequency region between 0.1 GHz and 0.4 GHz is of particular relevance because no experimental data are available. Observations at frequencies even lower than so far performed seem to be important to test or to complement our picture, since they could reveal larger deviations from the blackbody radiation or the transition to regimes (expected towards zero frequency) different to the one explored in this work. On the other hand, the background temperature raising predicted by our model is already significant, having an amplitude comparable or larger than those produced by unavoidable mechanisms with typical parameters, at frequencies between a few GHz and (10 − 20) GHz, a region where foreground mitigation is likely less critical and extremely accurate observations with space missions are in principle feasible. Thus, verifications of our model could take advantage from the next generation of both radio facilities and CMB dedicated projects. It would also be interesting to study the isotropy of the low frequency excess, since the present approach neglects this issue. Finally, a more accurate comprehension of Galactic and extragalactic intervening astrophysical emissions is necessary.
While we have added preliminary thoughts about the physical mechanism behind the abundance of low-frequency photons, a better understanding of the physics mechanism that originates the nonequilibrium state is beyond the scope of this paper. We still conclude from the present analysis that low-frequency data may be evidence for important nonequilibrium features in the early universe, when quantum and gravitational effects were strongly influenced by special conditions (e.g. low entropy) at the time of the Big Bang. [1] L. Danese and G. de Zotti, Nuovo Cimento Rivista Serie 7, 277 (1977).
We report in Table I the data compilation described in Sect. II. As discussed in Sect.
II, for the data where the model by [32] was applied to subtract the global contribution by unresolved extragalactic radio sources, we considered also a higher subtraction to account for possible higher differential number counts at faint flux densities. These two somewhat different subtractions are also applied to the radio background data by [31]. For the other data sets we keep the original foreground treatments performed by the authors, the differences between the two above extragalactic foreground subtraction models being in any case much smaller than the quoted uncertainties.
We perform our fit first with a 2-dimensional grid in ν 0 and α, to explore the dependence of the χ 2 on parameters and to avoid a possible wrong convergence; to overcome the finite sampling of the grid method, we then use a nonlinear minimization tool, weighting data with their inverse squared error. The fit is achieved with a Levenberg-Marquardt method from initial values ν ini 0 = 0.5 GHz, α ini = 4. With significantly different initial values, the convergence of the algorithm is compromised and the final result may be easily discarded basing both on visual inspection and on the results of the grid method. Fit errors are extracted from the parameters confidence interval, with a default 95% confidence level.
So far, the reduced χ 2 1.9 we found for our model reflects the use of the (almost) complete available sets of data. This is not surprising, since different data sets are affected by different systematic effects and derived with different foreground treatments, and we are just assuming a simple model as in (9).
For an immediate comparison (and cross-check), if we consider only the data in Table   1 of [10], but using the full set of FIRAS data and not replacing it with the "condensed" FIRAS value at 250 GHz, we obtain a reduced χ 2 of 1.08 with best fit parameters ν 0 = 0.66±0.06 GHz and α = 2.55±0.10 (implying a power-law amplitude of 18.72 K at 0.31 GHz, formally in terms of equivalent thermodynamic temperature, see (9)), fully consistent within errors with those found in Table 2 of [10] for the power-law fit model, as expected.
Applying the higher extragalactic subtraction, we find a similar reduced χ 2 ( 1.07), ν 0 0.64 ± 0.06 GHz and, as expected, a slightly smaller value of α ( 2.52 ± 0.11). For both the two extragalactic subtraction models, replacing the full set of FIRAS data with the "condensed" FIRAS value at 250 GHz we find similar best fit values, but with a reduced χ 2 1.6 in agreement with the one found in [10].
This simple comparison between the results found using two different data set compilations underlines the relevance of a significant improvement of both background observations and foreground modeling.
Appendix B: Global photon energy and number density As anticipated in Section V, we expect the validity of the adopted function F (ν), see (7) and (9), to break toward zero frequency, depending on the value of α, because the low frequency divergence of F (ν) could formally imply infinite values of the global photon energy and number density. As observed in Section III, other than Compton scattering, photon production/absorption processes operating in cosmic plasma are expected to be relevant, particularly at low frequencies, in both near-equilibrium and nonequilibrium approaches. In this Appendix, we discuss the frequency range validity for the assumed F (ν) in the simple version of the Kompaneets equation considered here.
We can rewrite the (nonequilibrium stationary) photon occupation number n(ν) as where n P (ν) = 1/(e xe − 1) is the Planckian distribution, x e = hν/(k B T e ) and δn(ν) defines the departure of n(ν) from it. To calculate δn(ν) at low frequencies, where the excess is more relevant, we can rely on the Rayleigh-Jeans approximation simplifying the computation of the global photon energy and number density: Here E P = aT x α e,0 while for α = 2, we get for α sufficiently larger than 2.
In general, α > 3 (or α > 2) implies a divergence of E r (or of N r ) for x a → 0, or, more physically, that n(x e ) should have a substantial flattening at x e below a certain dimensionless frequency x a (or at ν below a present time frequency ν a ).
The relative difference of the global photon energy density with respect to the Planckian case, δE r /E P = (E r − E P )/E P f − 1, is less than a certain value ( 1) for x a > ∼ [(15/π)(x α e,0 / )/(α − 3)] 1/(α−3) if α > 3 (or x a > ∼ exp[(15/π 4 )x −3 e,0 ] if α = 3). Analogously, for α > 2, the relative difference of the global photon number density with respect to the Planckian case, δN r /N P = (N r − N P )/N P φ − 1, is less than for x a > ∼ [(1/I 2 )(x α e,0 / )/(α − 2)] 1/(α−2) . The requirement of a change in the redshift of matter-radiation equivalence less than ∼ 1%, comparable to the accuracy set by Planck [104], i.e. ∼ 10 −2 (a condition stronger than that set by standard cosmological nucleosynthesis), in the case of the best fit values of ν 0 and α found in Section IV implies x a > ∼ 2.3 · 10 −14 , corresponding to ν a > ∼ 1.3 · 10 −3 Hz, is certainly not stringent. For comparison, a much stronger condition δE r /E P < ∼ 10 −5 (not to be confused with the potential limits on spectral distortion parameters from analyses in the near-equilibrium approach usually performed at higher frequencies) requires x a > ∼ 2.3 · 10 −4 , corresponding to ν a > ∼ 0.013 GHz, a value approaching the minimum frequency of current cosmic background observations. In the case α = 3, for any significant value of , we find instead x a larger than a value always negligible in practice, as expected from continuity with the case α < 3.