Excess in the High-frequency Radio Background: Insights from Planck

We created cutout images for all four Planck bands at the location of NVSS sources. Each cutout is 25 × 25 pixels, with a pixel size that is 1/5 the FWHM of the Planck beam at the corresponding frequency. Accordingly, the size of each cutout is five times the FWHM of the Planck beam on a side. Our stacking analysis was carried out in a way to avoid multiple inclusion of sources in the image stacks. This is done by stacking on the location of NVSS sources with decreasing 1.4 GHz flux density, and removing any other NVSS sources that fall within a radius of three times the FWHM of the Planck beam at the current position for further consideration. The Planck beam FWHM values at 30, 44, 70, and 100 GHz are taken as 3265, 2700, 1301, and 994, respectively. Thus, any occurrence of double counting or inclusion of flux density from bright nearby sources is eliminated.

Once the cutouts are generated for each Planck band, we then stack all sources within a 1.4 GHz flux density bin having a width of 0.5 dex, except for the brightest bin, for which we allowed the size to be large enough to include ≳30 sources. The final 1.4 GHz flux density bins are given in Table 1. The all-sky Planck maps at these frequencies are provided in units of thermodynamic temperature (i.e., K_CMB), requiring a conversion to obtain integrated flux densities from the stacked Planck cutouts that are in the same units as the integrated 1.4 GHz flux densities in the NVSS catalog. We therefore first convert from units of K_CMB to Raleigh–Jeans brightness temperature using the following multiplicative factors of 0.979328, 0.95121302, 0.88140690, and 0.76581996 at 30, 44, 70, and 100 GHz, respectively (note, for 1.4 GHz this value is unity). These values were then converted into units of MJy sr⁻¹ using the multiplicative factors of 24.845597, 59.666236, 151.73238, and 306.81118 at 30, 44, 70, and 100 GHz, respectively.

Table 1.  Stacking Results

1.4 GHz Flux Density Bin	${S}_{30\,\mathrm{GHz}}^{\mathrm{stack}}$	${S}_{44\,\mathrm{GHz}}^{\mathrm{stack}}$	${S}_{70\,\mathrm{GHz}}^{\mathrm{stack}}$	${S}_{100\,\mathrm{GHz}}^{\mathrm{stack}}$
(mJy)	(mJy)	(mJy)	(mJy)	(mJy)	${\alpha }_{1.4\,\mathrm{GHz}}^{30\,\mathrm{GHz}}$	${\alpha }_{1.4\,\mathrm{GHz}}^{44\,\mathrm{GHz}}$	${\alpha }_{1.4\,\mathrm{GHz}}^{70\,\mathrm{GHz}}$	${\alpha }_{1.4\,\mathrm{GHz}}^{100\,\mathrm{GHz}}$
66 $\leqslant \,{S}_{1.4\mathrm{GHz}}\ \lt \$210	⋯	⋯	6 ± 2.4	4 ± 1.3	⋯	⋯	⋯	⋯
210  ≤  S1.4 GHz < 664	4 ± 11.4	6 ± 8.8	20 ± 1.8	18 ± 1.6	⋯	⋯	−0.74 ± 0.02	−0.69 ± 0.02
664 ≤ S1.4 GHz < 2100	87 ± 7.0	88 ± 7.6	86 ± 4.3	81 ± 3.4	−0.89 ± 0.03	−0.75 ± 0.03	−0.63 ± 0.01	−0.60 ± 0.01
2100  ≤  S1.4 GHz < 6642	394 ± 12.3	324 ± 16.2	319 ± 11.4	270 ± 10.6	−0.68 ± 0.01	−0.65 ± 0.02	−0.58 ± 0.01	−0.58 ± 0.01
6642 ≤ S1.4 GHz < 228260	2699 ± 58.0	2382 ± 67.4	2645 ± 55.2	2315 ± 41.0	−0.60 ± 0.01	−0.56 ± 0.01	−0.50 ± 0.01	−0.49 ± 0.01

Note. The measured central frequencies of the Planck data, which were used for the analysis, are 28.5, 44.1, 70.3, and 100 GHz. Spectral indices were calculated and used in the present analysis only when the significance of the stacked Planck flux densities were >5σ.

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Stacking is done by first removing sky values from the individual cutouts, taken as the median pixel value in an annulus defined by radii of one and two times the FWHM of the Planck beams. The sky subtracted cutouts are then combined by taking a mean of the pixels at each location in the stack. We take the mean stack, as opposed to the median, given that we are working with image cutouts that include detected sources (i.e., not residual maps, for which the latter would be more appropriate). The stacked images at 30, 44, 70, and 100 GHz are shown in Figure 1, for which we show images to just below the 1.4 GHz flux density bins that resulted in a statistically significant detection in the Planck bands. The integrated flux density of the stacked images, in units of mJy, is then taken by summing pixels within an aperture having a radius equal to the FWHM of the Planck beam, and then multiplying that number by an aperture correction factor of ≈1.07.

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Given the significantly larger size of the Planck beam compared to the 45'' NVSS beam, we account for multiple NVSS sources that fall within the integration aperture at each frequency by keeping track of their individual 1.4 GHz flux densities. This is done because each NVSS source may be contributing to the stacked Planck flux density at some level. We therefore mimic what Planck would see at 1.4 GHz, given the angular resolution at each frequency by convolving each NVSS source within a radius of 2.5 times the FWHM of the corresponding Planck beam using a Gaussian with the same FWHM. We then sum the contribution of 1.4 GHz emission from all sources within the integration aperture (i.e., with a radius equal to the FWHM of the Planck beam) as the total 1.4 GHz flux density associated with the Planck source. At a distance of 2.5 times the FWHM of the beam, a source contributes at the ≈0.1% level to the total flux density measured within our integration aperture.

The uncertainty of the stacked flux density is estimated through a Monte Carlo exercise. The above described stacking procedure was carried out 100 times on each Planck map, at a random position located at a distance that was between one and four times the FWHM of the corresponding Planck beam from the center of the NVSS position. The standard deviation of the Planck flux densities at the random positions was then taken as an estimate on the uncertainty for the stacked flux density in that 1.4 GHz flux density bin. The stacked flux densities, along with estimates on their uncertainties, are given in Table 1.

Using the results of the stacked flux densities, we estimate the corresponding spectral index between each of the stacked Planck flux densities and the corresponding mean 1.4 GHz flux density for that bin. Spectral indices are only calculated for 1.4 GHz flux density bins that yielded a statistically significant (i.e., >5σ) stacked flux density at the Planck frequencies. The spectral indices are given in Table 1 along with corresponding uncertainties and plotted in Figure 2. There is a clear trend of brighter 1.4 GHz sources having flatter spectral indices in all cases.

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Had we naively assumed that only the single, brightest NVSS source was associated with the location of our Planck cutouts, the total 1.4 GHz flux densities would be reduced resulting in significantly flatter stacked spectral indices. This is especially true at 30 GHz, where the beam area is ≈1900 times larger than that of NVSS. By including (weighted) contributions of nearby 1.4 GHz sources, the associated 1.4 GHz flux densities are larger by factors of ≈2.5, 2.1, 1.6, and 1.4 at 30, 44, 70, and 100 GHz, respectively, compared with what is measured when only associating the cutout with the single, brightest NVSS source. This results in corresponding, stacked spectral indices that are steeper, on average, by factors of ≈1.3, 1.2, 1.1, and 1.1 at 30, 44, 70, and 100 GHz, respectively; the differences are largest for the faintest flux density bins.

To illustrate that our stacking analysis is in fact resulting in realistic spectral indices, we overplot the 1.4–30 GHz spectral indices reported in Rodríguez-Gonzálvez et al. (2015). The spectral indices from our stacking analysis are clearly consistent with the mean and spread of values (asterisks) reported from direct detections of sources. Our values are also consistent with the results of Mason et al. (2009), who reported on a 31 GHz survey of 3165 known extragalactic radio sources included in NVSS using both the 100 m Robert C. Byrd Green Bank Telescope and the 40 m Owens Valley Radio Observatory telescope. These authors measure a mean 1.4–31 GHz special index of −0.917 with a standard deviation of 0.311 among their entire sample; this is also illustrated in Figure 2. This is significantly steeper than the average 1.4–31 GHz spectral index of ∼−0.7 reported by Muchovej et al. (2010), which was measured from a sample of 209 sources detected at 31 GHz with the Sunyaev–Zel'dovich Array and matched to 1.4 GHz counterparts in NVSS. It is worth pointing out that any spectral index distribution having the same mean as that obtained from our stacking analysis will yield the same results. However, evolution in the spectral index distribution as a function of flux density within each bin is important and not directly measured, which is a limitation of our study.

Having the spectral indices per each 1.4 GHz flux density bin, we use these to estimate the 30, 44, 70, and 100 GHz flux densities for each NVSS source through a linear interpolation over stacked bins with statistically significant detections. This is done by first applying the stacked spectral index to all 1.4 GHz sources that contributed to that specific stack to estimate a corresponding flux density for the Planck frequencies. For those NVSS sources having flux densities above or below where we were able to obtain a statistically significant detection through stacking, we use the spectral index from the highest and lowest 1.4 GHz flux density bins, respectively. Then, an ordinary-least-squares fit to the scatter plots per Planck frequency were used to estimate the corresponding flux density from all sources in the NVSS catalog.

Taking our estimated 30, 44, 70, and 100 GHz flux densities, we generate corresponding differential source counts shown in the left panel of Figure 3. In the right panel of Figure 3, we additionally plot the Euclidian normalized source counts at all frequencies. In both panels, we compare with other differential source count estimates at ∼30 GHz from the literature. The shaded region with the dotted–dashed line illustrates the predicted 30 GHz differential source counts over a flux density range of 0.7 < S_{31 GHz} < 15 mJy reported by Muchovej et al. (2010). The shaded region with the long-dashed line illustrates the predicted 30 GHz differential source counts from Mason et al. (2009) over a flux density range of 1 < S_{31 GHz} < 4 mJy. In both cases, to properly compare these measurements with ours at 28.5 GHz, we have scaled their results using their average 1.4–31 GHz spectral indices, which has a negligible effect.

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The differential source counts reported by Mason et al. (2009) are ≈16% larger than what we calculate over the same flux density range from our stacking analysis, and thus in very good agreement. Such a difference could easily arise from selection biases and calibration errors in their study. The agreement between our source counts and Mason et al. (2009) may not be too surprising, as both populations are NVSS-selected. However, both the 30 GHz differential source counts reported here and in Mason et al. (2009) are significantly lower than those reported by Muchovej et al. (2010). These authors state that the discrepancy can be explained by a small shift in the spectral index distribution for faint 1.4 GHz sources. However, even if we force a constant 1.4–30 GHz spectral index of −0.7 for all stacked flux density bins, we are still nowhere near able to increase our 30 GHz differential source counts to match theirs. Furthermore, models for number counts of galaxies at these frequencies (e.g., de Zotti et al. 2005; Tucci et al. 2011) appear to be much more consistent with what is reported here, and found by Mason et al. (2009). In both cases, however, our methodology enables us to push the 30 GHz source counts to an order of magnitude fainter in flux density, although we caution that our estimates at the faintest flux densities could be uncertain due to the absence of detections in the stacks at the faintest bins.

To extend the differential source counts to flux densities below the 2.1 mJy sensitivity limit of the NVSS catalog, we make use of recent work by Condon et al. (2012), who employed a P(D) analysis on a deep, confusion limited 3 GHz observation of the Lockman Hole to estimate differential source counts down to a corresponding 1.4 GHz flux density of ≈2 μJy. Following the conclusions of Condon et al. (2012), we adopt functional forms for the differential 1.4 GHz source counts of $n({S}_{1.4\mathrm{GHz}})=1.2\times {10}^{5}\,{S}_{1.4\,\mathrm{GHz}}^{-1.5}\,{\mathrm{Jy}}^{-1}\ {\mathrm{sr}}^{-1}$ (Condon 1984a) and $n({S}_{1.4\mathrm{GHz}})=57\ {S}_{1.4\,\mathrm{GHz}}^{-2.2}\ {\mathrm{Jy}}^{-1}\ {\mathrm{sr}}^{-1}$ (Mitchell & Condon 1985) for sources in the flux density ranges of 2 < S_{1.4 GHz} < 20 μJy and 20 < S_{1.4 GHz} < 2100 μJy, respectively. These are shown in right panel of Figure 3 as a dashed line.

As a comparison, we additionally make use of the Euclidean-normalized differential source counts reported by Owen & Morrison (2008), which were also carried out over the Lockman hole using deep 1.4 GHz observations [i.e., $n({S}_{1.4\mathrm{GHz}})=6\,{S}_{1.4\,\mathrm{GHz}}^{-2.5}\,{\mathrm{Jy}}^{-1}\ {\mathrm{sr}}^{-1}$]. This is illustrated by a dotted line in Figure 3. While the recent analysis of Condon et al. (2012) suggests that the source counts of Owen & Morrison (2008) are likely too large, arising from an overcorrection of source brightnesses to integrated flux densities near the brightness cutoff of their catalog, we include these values in our analysis as they likely provide a conservative estimate for the uncertainty in our measurements.

Finally, while not plotted, we additionally fit the cumulative source counts at each frequency with a fifth order polynomial such that

$$\begin{eqnarray}&&\mathrm{log}\left[\displaystyle \frac{N(\gt {S}_{\nu })}{{\mathrm{sr}}^{-1}}\right]=\displaystyle \sum _{i=0}^{5}{\xi }_{i}\mathrm{log}{\left(\displaystyle \frac{{S}_{\nu }}{\mu \mathrm{Jy}}\right)}^{i}.\end{eqnarray} \tag{ 1 }$$

The corresponding coefficients from the fits, along with the faintest flux density used in the fit (i.e., >S_ν), are given in Table 2.

Cumulative plots of the integrated extragalactic light from individual sources at each frequency, in units of brightness temperature, are show in Figure 4. Extrapolations to the sub-mJy 1.4 GHz population (i.e., $2\lt {S}_{1.4\mathrm{GHz}}\lt 2100$ μJy) based on the results of Condon et al. (2012) and Owen & Morrison (2008) are also shown, clearly illustrating that the latter introduces a sharp (divergent) increase in the contribution from faint sources to the total extragalactic background light at these frequencies. In Table 3, the integrated brightness temperatures at 1.4, 30, 44, 70, and 100 GHz for only NVSS sources (i.e., S_{1.4 GHz} > 2.1 mJy) are given. Also listed are the brightness temperature estimates for sources in the flux density range of $2\lt {S}_{1.4\mathrm{GHz}}\lt 2100$ μJy) based on the faint source count estimates from Condon et al. (2012) and Owen & Morrison (2008), where the latter results in a sky brightness that is nearly a factor of ∼2 larger than the former for S_{1.4 GHz} > 2 μJy.

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While we only integrate down to S_{1.4 GHz} ≈ 2 μJy, which is the minimum flux density that the Condon et al. (2012) P(D) analysis is sensitive to, continuing to integrate to lower flux densities results in the total brightness temperature from the Owen & Morrison (2008) counts to increase rapidly, and eventually exceed the ARCADE 2 total extragalactic sky brightness measurement. Specifically, by integrating down to 10 nJy, we find that the integrated brightness temperature assuming the Condon et al. (2012) fit to the faint source population converges near 111 mK, consistent within errors to what we obtain by only integrating down to where the P(D) analysis is no longer sensitive. However, the corresponding brightness temperature assuming the Owen & Morrison (2008) extrapolation to faint sources reaches a value of ≈63 K, which clearly violates the ARCADE 2 value, let alone the CMB temperature. This is, of course, due to the fact that to avoid Olbers' paradox, the differential source counts at faint flux densities must eventually achieve an index flatter than −2.0.

3.3.1. Comparison with other Work at 1.4 GHz

We note that the total contribution to the 1.4 GHz brightness temperature from extragalactic sources reported here (i.e., 111 ± 11 mK for S_{1.4 GHz} > 10 nJy) is slightly larger than that reported by Condon et al. (2012, i.e., 100 ± 10 mK for S_{1.4 GHz} > 10 nJy), even when using the same estimates for the contribution from faint sources below the detection limit of the NVSS. To determine the cause of this discrepancy, we plot the cumulative 1.4 GHz brightness temperature histograms using the data from Condon et al. (2012) and the NVSS data (i.e., this work) in Figure 5. The cumulative histograms appear to diverge at ≈175 mJy, where the 1.4 GHz brightness temperature using the NVSS data begins to increase more rapidly toward lower flux densities relative to the histogram from Condon et al. (2012). The location of the divergence is exactly where the data used in the Condon et al. (2012) study switched from single-dish, NRAO 91 m data (Machalski 1978) to various WSRT and VLA interferometric observations compiled by Condon (1984a).

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Unlike the NVSS maps, which obtain nearly uniform coverage, these older interferometric data required large correction factors for incompleteness caused by primary beam attenuation, as well as for partial source resolution by the smaller synthesized beams. Had there been some sharp feature near the 2.1 mJy limit of the NVSS survey data in Figure 5, it would likely indicate that the NVSS is incomplete or unreliable near that limit. Additionally, if the NVSS correction for CLEAN bias was incorrect, it would likely lead to a conspicuous feature at the faint end. Given that we observe a smooth transition between the two curves in Figure 5 likely suggests that the discrepancy is most likely arising from underestimated corrections to the older VLA and WSRT data compiled in Condon (1984a).

In the left panel of Figure 6 the integrated brightness temperatures from discrete sources at 1.4, 30, 44, 70, and 100 GHz are plotted against frequency using the faint source contribution estimates from both Condon et al. (2012) and Owen & Morrison (2008). These data are marginally well fit by a single power law having an index of ${\beta }_{1.4\,\mathrm{GHz}}^{100\,\mathrm{GHz}}\,=-2.69\pm 0.03$. If we instead only fit the data at the Planck frequencies, we obtain an index that is significantly flatter, being ${\beta }_{30\,\mathrm{GHz}}^{100\,\mathrm{GHz}}=-2.39\pm 0.12$. This is illustrated in the right panel of Figure 6, where the integrated brightness temperatures have been multiplied by ${\nu }^{-{\beta }_{1.4\mathrm{GHz}}^{30\mathrm{GHz}}}$. These values are consistent using either Condon et al. (2012) or Owen & Morrison (2008) for the faint source count estimates, being ${\beta }_{1.4\,\mathrm{GHz}}^{30\,\mathrm{GHz}}\,=-2.82\pm 0.06$. This is a statistically significant (i.e., 6.5σ) indication that the extragalactic sky brightness spectrum from discrete sources is flattening with increasing frequency.

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To ensure that this result is not driven by the extremely steep 1.4–30 GHz spectral index measured for the faintest 1.4 GHz flux density bin, we determine how much this value would have to change (i.e., flatten) to reduce the significance of the measured excess at 44, 70, and 100 GHz to <3σ. For this to occur, we find that the 1.4–30 GHz spectral index would need to flatten from −0.89 to a value that is ≳−0.77, a change of ≳4σ. This would suggest that our 1.4 GHz flux densities are being grossly overestimated for sources in that flux density bin, which seems unlikely. We additionally make sure that this result is not being significantly affected by potential Galactic CO emission that could be contaminating our 100 GHz flux densities. We therefore run the entire analysis after reducing the stacked 100 GHz flux densities by 4.5%, which is the median excess contribution of the measured 100 GHz flux densities of sources in the Planck Early Release Compact Source Catalog (ERCSC; Chen et al. 2016). We find that this has little effect, reducing the significance of the 44, 70, and 100 GHz excesses found in the right panel Figure 6 from 6.5 to 6.1σ.

In the top panel of Figure 7, thermodynamic temperatures from our integrated extragalactic brightness temperatures, added to a CMB temperature of 2.725 ± 0.001 K (Fixsen et al. 2011), are plotted against frequency again using the faint source contribution estimates from both Condon et al. (2012) and Owen & Morrison (2008). As recently pointed out by Condon et al. (2012), the integrated light from individual sources at GHz frequencies is substantially less than the fit to the ARCADE 2 extragalactic background measurements (Fixsen et al. 2011), which is given by the solid line $[{T}_{{\rm{b}}}\,={(24.1\pm 2.1{K}_{\mathrm{CMB}})(\nu /310\mathrm{MHz})}^{(-2.599\pm 0.036)}]$ and the points with error bars. We find this to be true even if we assume the Owen & Morrison (2008) sub-mJy differential source count results and only integrating down to a 1.4 GHz flux density of 2 μJy. Of course, as stated above, by integrating down to fainter flux densities and assuming that the shape of the differential sources counts reported by Owen & Morrison (2008) persists, results in integrated brightness temperates that greatly exceeds the limits imposed by ARCADE 2.

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In the bottom panel of Figure 7, the residuals between the fit to the ARCADE 2 data from Fixsen et al. (2011) and our integrated thermodynamic temperatures are plotted against frequency using the integrated source contribution estimates from Condon et al. (2012) and Owen & Morrison (2008). Each of these results in positive values. After removing the CMB temperature, the fit to the ARCADE 2 extragalactic sky brightness measurements is a factor of ≈4.5, 8.6, 6.6, 6.2, and 4.9 times brighter than what we measure from discrete sources at 1.4, 30, 44, 70, and 100 GHz, respectively, assuming the Condon et al. (2012) differential source counts at sub-mJy flux densities. If we instead assume the differential source counts of Owen & Morrison (2008), the ARCADE 2 total extragalactic sky brightness is still a factor of ≈2.3, 4.4, 3.4, 3.2, and 2.5 times brighter than what we measure from discrete sources at 1.4, 30, 44, 70, and 100 GHz. The above comparison between our measurements of the sky brightness from discrete sources at the Planck frequencies to what is estimated from the fit to the ARCADE 2 data may not be too enlightening, given that their fit was not constrained by data at frequencies ≳10 GHz. However, we note it here as it does yield values that are broadly consistent to the excess measured at ∼1 GHz. Furthermore, the spectrum of the residual is largely consistent with diffuse Galactic synchrotron emission.

Looking at a 10° × 10° patch of the S³ simulated extragalactic sky (Wilman et al. 2008),⁴ ≈75% of sources above the NVSS surface brightness limit are classified as FR-I radio galaxies (Fanaroff & Riley 1974), for which their brightness decreases from the central galaxy. The more luminous FR-II radio galaxies, which are identified by hot spots in their lobes at a large distance from the central galaxy core, make up ≈20% of sources above the NVSS surface brightness limit. Given the non-uniform magnetic fields and electron energy distributions in these geometrically complicated structures, along with the complexity of acceleration physics, their spectra exhibit a range in behavior and can deviate from a single, simple power law.

The observed spectrum, therefore, might simply be the result of combining different classes of radio galaxies. For instance, looking at detections in the band-merged Planck ERCSC (Chen et al. 2016), we can identify a handful of sources that have spectra similar to what we observe for the integrated extragalactic discrete-source spectrum (Figure 8). The fact that only a handful of sources have similar spectra may not be that surprising, given that the faintest 30 GHz flux density is ≈387 mJy, requiring a 1.4 GHz of ≈4.6 Jy (i.e., a negligible fraction of NVSS sources) to obtain a 1.4–30 GHz spectral spectral index of −0.82.

In Figure 9, we show probability distributions for spectral indices calculated at each of the frequencies being investigated here based on a simple two-component model presented in Condon (1984a). The observed spectral index distribution consists of a population of steep and flat power-law spectrum sources with a differential source count slope of −2 modeled by two Gaussians (see Appendix of Condon 1984a). At the fiducial frequency of 1.4 GHz, the steep component is defined by a Gaussian with amplitude 0.86, mean −0.93, and standard deviation 0.17 at z = 0, while the flat component is defined by a Gaussian with amplitude 0.14, mean −0.50, and standard deviation 0.38. Because the spectral index distributions have a finite width, as one selects sources at higher frequencies the observed spectral index will naturally flatten even if just considering the steep spectral index population alone. Furthermore, as one selects sources at higher frequencies, the contribution of the flat spectrum population begins to increase, leading to even flatter average spectral indices, with a mean shifting between −0.87 at 1.4 GHz and −0.12 at 100 GHz for this simple model.

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While the mean spectral index measured at the middle of the Planck frequencies is −0.26, ≈1σ flatter than what is measured here, this may be expected since flat spectrum sources selected at 1.4 and 5 GHz arising from synchrotron self-absorption will, in fact, steepen at higher frequencies as they become optically thin. Thus, the distribution of spectral indices arising from the combination of such sources and GPS sources could certainly lead to the observed discrete-source extragalactic spectrum, which may be the most plausible explanation.

Another explanation for the spectral flattening could be from free–free emission associated with star formation. If we instead fit the discrete-source extragalactic spectrum with a combination of a synchrotron component, having a spectral index equal to that measured between 1.4 and 30 GHz (i.e., α = −0.82, which would actually be an upper limit on the non-thermal spectral index), along with a thermal (free–free) emission component having a spectral index of α = −0.1, requires a thermal fraction at 30 GHz of ≳50%. This is consistent to what is found for typical star-forming galaxies in the universe (e.g., Condon & Yin 1990; Condon et al. 2002). However, over half of the extragalactic background contribution from discrete sources at 1.4 GHz is accounted for by sources stronger than S_{1.4 GHz} ≳ 2.1 mJy. At these flux density levels, sources are typically AGN (e.g., Condon 1984a; Condon et al. 2002; Wilman et al. 2008).

For instance, again looking at a 10° × 10° patch of the S³ simulated extragalactic sky, indicates that only ≈1% of sources above the NVSS surface brightness limit are star-forming galaxies. Consequently, we do not necessarily expect to see a spectral flattening at ≳30 GHz that one finds in star-forming galaxies arising from free–free emission, unless a significant fraction of such AGN also host ongoing star formation that is not captured in such simulations. However, spectra of canonical radio galaxies appear synchrotron-dominated through 100 GHz, rather than having an M82-like spectrum that becomes free–free dominated beyond ∼30 GHz. While difficult to assess, given the data in-hand, this explanation for the flattening of the discrete-source extragalactic sky brightness temperature by free–free emission associated with star formation seems unlikely.

It is worth seeing if the observed spectral flattening in the discrete-source extragalactic sky brightness temperature could be related to the excess of positrons reported by both the Payload for Antimatter-Matter Exploration and Light nuclei Astrophysics (PAMELA; Adriani et al. 2009) experiment and Alpha Magnetic Spectrometer on the International Space Station (AMS-02; Aguilar et al. 2013, 2014). If we are to assume a typical magnetic field strength of ∼5 μG in the host of radio galaxies, the energies for which cosmic-ray electrons emit at a critical frequency corresponding to the 4 Planck frequencies are 21, 26, 33, and 39 GeV. Integrating over these energies, PAMELA and AMS-02 report a fractional excess of positrons that corresponds to in an increase of ≈8% to the sky brightness temperature. This is well below the ≈30% excess in brightness temperature integrated over those same energies compared to what one would expect if the brightness temperature spectrum had a spectral index of β = −2.82 between 1.4 and 100 GHz, suggesting that the two are unrelated.

Another possible explanation for the observed spectral flattening in the discrete-source extragalactic sky brightness temperature could be related to additional emission from steep-spectrum (i.e., α < −1) radio halos associated with the massive clusters that many of the brightest radio sources that contribute significantly to the extragalactic sky brightness reside in. In this case, the assumption would be that the spectra from individual galaxies is, in fact, much flatter, similar to what is measured between 1.4 and 100 GHz for the ERCSC sources plotted in Figure 8 (i.e., β = −2.27 between 1.4 and 100 GHz), and there is a significant contribution of synchrotron emission from radio halos that is steepening the low-frequency end of the spectrum.

In Figure 10, we plot a model radio halo synchrotron spectrum having a spectral index of β = −3.5 (Feretti et al. 2012) scaled to the measured sky brightness temperature from discrete sources at 1.4 GHz. The 1.4 GHz flux density of the radio halo in Coma is S_{1.4 GHz} = 0.64 ± 0.035 Jy (Deiss et al. 1997). Assuming that the sky brightness temperature from discrete sources indeed has a flat spectrum consistent with what is measured across the 4 Planck bands (i.e., β = 2.39), the corresponding 1.4 GHz brightness temperature would be 29.21 mK, which is 76.42 mK less than what is measured. Consequently, it would take a contribution from ≈4.4 × 10⁵ such radio halos (i.e., ≈37% of all NVSS sources considered here) to account for this difference in the integrated sky brightness temperature, and thus is an unsatisfactory explanation.

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A final, and far more speculative, explanation for the observed spectral flattening in the discrete-source extragalactic sky brightness temperature could be related to the additional emission from dark matter (DM) annihilation that is expected to be associated with the massive clusters. Similar to a potential explanation by radio halos, this scenario assumes that the observed sky brightness temperature from galaxies is indeed quite flat, but there is some excess component that is steepening what is observed at the low-frequency end.

In Figure 10, we investigate such a scenario by looking at the possible contribution to the observed synchrotron spectrum from DM annihilation for a Coma-like cluster at z = 0.023 having a halo mass of 10¹⁵ M_☉ using the models of Storm et al. (2017). We consider DM annihilation to both $b\bar{b}$ and μ⁺μ⁻ integrated over a circular area having a radius of 300 kpc (20'), assuming a DM mass of m_χ = 100 GeV and cross section $\langle \sigma v\rangle =3\times {10}^{-26}$ cm³ s⁻¹. The spectrum from DM annihilation to μ⁺μ⁻ has a spectral index of β = −2.65 between 1.4 and 100 GHz, ≈3σ flatter than the spectral index of β = −2.82 ± 0.06 for the measured sky brightness temperature from discrete sources between 1.4 and 30 GHz, and thus may be excluded out as a candidate explanation. However, the spectrum from DM annihilation to $b\bar{b}$ has a spectral index of β = −3.26 between 1.4 and 100 GHz, suggesting that it could potentially contribute to any spectral steepening at the low-frequency end of our sky brightness temperature measurements. In Figure 10 we scale this DM annihilation spectrum to the measured sky brightness temperature at 1.4 GHz in Figure 10 as triangles.

Again, assuming that the sky brightness temperature from discrete sources indeed has a flat spectrum consistent with what is measured across the four Planck bands and that this difference is associated solely attributed to the contribution from DM annihilation for Coma-like halos based on the above assumptions and models, it would take ≈7.4 × 10⁶ such massive clusters, which is orders of magnitude beyond reality. If the assumed cross section is incorrect, and a value an order of magnitude smaller is more appropriate (i.e., $\langle \sigma v\rangle =3\times {10}^{-27}$ cm³ s⁻¹), the situation only becomes worse as it would in turn require an order or magnitude more clusters. Consequently, an explanation from DM annihilation seems highly unlikely.