INTERPRETATION OF THE ARCADE 2 ABSOLUTE SKY BRIGHTNESS MEASUREMENT

The sky brightness temperature contributed by discrete sources can be composed as the sum of two parts: the source population that has been characterized by existing surveys and the contribution of sources below the flux limit of existing surveys. We write this as

$$\begin{equation} T = T(S > S_{\mbox{\scriptsize limit}}) + \frac{\lambda ^2}{2 k_{\mbox{\scriptsize B}}} \int _{S_{\mbox{\scriptsize min}}}^{S_{\mbox{\scriptsize limit}}} \frac{\it dN}{\it dS} S dS, \end{equation} \tag{ 1 }$$

where T(S > S_limit) is contribution from sources with a flux S greater than the survey limit S_limit. The wavelength of observation is λ and k_B is the Boltzmann constant. We characterize sources below the survey limit with their differential number counts, dN/dS, and assume that there is a lower limit cutoff to the source population at a flux of S_min. Radio source count surveys reveal a faint-source population with differential number counts proportional to a power law

$$\begin{equation} \frac{\it dN}{\it dS} = \kappa S^{-\gamma }, \end{equation} \tag{ 2 }$$

where κ is the constant of proportionality and γ is the power-law index. An index γ of 2.5 corresponds to a static, Euclidean universe with uniform filling of sources, whereas faint radio surveys find γ in the range of 2.0–2.6 (Windhorst et al. 1993; Richards 2000; Fomalont et al. 2002, 2006; Kellerman et al. 2008; Owen & Morrison 2008).

Such source counts cannot extend to arbitrarily low fluxes, or the total contribution would diverge. A realistic distribution of sources, of course, would not have a sharp cutoff at S_min. In practice, we can characterize S_min as the flux below which the index γ falls below 2, as there will be negligible additional contribution to the integral below this limit.

Deep surveys of radio sources have been performed at a number of frequencies. Particularly useful are the surveys at 1.4 and 8.4 GHz with the Very Large Array (VLA). Fomalont et al. (2002) report the results of an 8.4 GHz survey, which we will examine as an example. This survey, with a limit of 7.5 μJy, finds a faint-end index to the number counts of γ = 2.11 ± 0.13. Windhorst et al. (1993) argue that the sources in the nJy flux range are dominated by ordinary spiral galaxies, which produces a natural lower limit of 30 nJy; below this limit there are insufficient galaxies.

Figure 2 shows a range of estimates for the contribution of discrete sources to the ARCADE 2 3.20 GHz measurement. We have calculated the expected contribution to the extragalactic sky temperature using the results of Fomalont et al. (2002) and varying the faint-end index, and plotting as a function of S_min. We have scaled the temperature from 8.4 to 3.2 GHz using a frequency spectral index of −2.75. This spectral index is characteristic of starburst and normal spiral galaxies with synchrotron emission, though at 8.4 GHz there could be some contribution from flat-spectrum sources such as active galactic nuclei. The sources fainter than 35 μJy described by Fomalont et al. (2002) have a spectral index distribution that peaks at −2.75. From this analysis, we conclude that the contribution is likely in the range of 5–10 mK at 3.20 GHz, which should be compared to the 62 ± 10 mK excess measured by ARCADE 2. We also note that the normalization of the differential number counts in the 8.4 GHz survey is sufficiently accurate to not contribute a significant source of error to this analysis.

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We have focused on the 8.4 GHz survey results as illustrative; these measurements are amenable to extrapolation to fainter fluxes and to the ARCADE 2 frequencies. As summarized by Kellerman et al. (2008), a variety of VLA measurements indicate a trend toward a differential number count index of γ = 2.5 in the faintest flux density bins, along with substantial scatter in the number density from different survey fields.

Our results, however, do not depend on measurements from a radio survey at any one frequency. Gervasi et al. (2008a) describe a more comprehensive analysis of the potential contribution of unresolved extragalactic radio sources by examining data from a wide variety of radio surveys, and fitting an empirical, two-population model to the survey data at each frequency. Their combined result for extragalactic radio source brightness versus frequency is described by a single power-law model, "Fit1," with amplitude 0.88 K at 0.61 GHz and a power-law index of −2.707. This model is also shown in Figure 1, indicating again that this contribution is insufficient for explaining the ARCADE 2 results. The brightness temperature values of this model, evaluated at the frequencies used in our analysis, appear in Table 1. Because the sample size and brightness limits of the radio surveys vary with frequency, the error in their estimate also varies with frequency. These frequencies are not identical to the frequencies measured by ARCADE 2. For Table 1, we adopt a fractional error estimate of 10% independent of frequency, which we believe is a conservative estimate of the error in the Gervasi et al. (2008a) modeling. These estimates for the contribution of extragalactic sources are consistent with the analysis described above.

It is also interesting to consider whether the existing surveys have missed significant flux from the sources. The low-frequency faint-source observations are primarily interferometric and have the possibility of overresolving the source and missing flux in extended low surface brightness emission. Henkel & Partridge (2005) consider the evidence for this and conclude that 20% may be an upper limit to this effect for mJy flux levels at 8.5 GHz. Fomalont et al. (2006) suggest that at 1.4 GHz only a few percent of sources are larger than 4 arcsec and that other reports of a larger figure in other surveys are actually confusion of multiple disparate sources. Garrett et al. (2000) compare their 1.4 GHz survey conducted using the Westerbork Synthesis Radio Telescope and its larger, 15 arcsec effective beam with previous VLA measurements of the same region with a similar noise level. Of a total of 85 sources in their survey, they find 22 not apparent in the previous VLA survey. Some of these 22 likely correspond to the combined flux of multiple sources that were resolved by the VLA measurements. At least two sources, however, appear to be relatively nearby discrete sources with emission from a large enough region to have been resolved out by the VLA measurements. We conclude from these studies that it seems unlikely that sufficient flux has been missed in surveys of known objects to explain our residual emission.

Another method to examine the possibility of extended low surface brightness emission in extragalactic sources is radio observations of the halos of nearby edge-on spirals. Irwin et al. (1999, 2000) report results of VLA surveys for radio emission from nearby edge-on spirals. These studies can elucidate the connection between the star formation processes that drive the far-IR background, and the supernova processes that drive radio emission, but do not provide evidence that large amounts of radio flux are missed in surveys of more distant sources.

The cosmic far-IR background has been detected at a level of approximately 10–20 nW m⁻² sr⁻¹ with FIRAS and DIRBE (Puget et al. 1996; Fixsen et al. 1998; Hauser et al. 1998). We can use the universal radio to far-IR correlation in star-forming galaxies (Condon 1992) to estimate the expected extragalactic radio background that can be attributed to galaxies contributing to the cosmic far-IR background. Haarsma et al. (2000) and Dwek & Barker (2002) specifically address this prospect. The conclusion of these studies is that the far-IR measurements are consistent with the existing surveys of radio galaxies described earlier. For example, the radio brightness temperature of 18 K at 178 MHz predicted by Dwek & Barker (2002) is within 1σ of the radio galaxy contribution modeled by Gervasi et al. (2008a). This emission is insufficient to account for the excess detected by ARCADE 2.

One can ask if there is a way around this limit by considering departures from the far-IR radio correlation associated with faintness or redshift; the physical processes of far-IR emission from dust heated by star formation and radio emission driven by supernovae are related but differ in timescale (see, e.g., Murphy et al. 2008). Garn & Alexander (2009) stack IR-selected galaxies and data from faint radio surveys and find that there is no evidence for a change in the far-IR to radio correlation with fainter galaxies. While some authors present the possibility that the far-IR correlation evolves with redshift (Vlahakis et al. 2007; Seymour et al. 2009), others see no evidence for this (Chapman et al. 2005; Frayer et al. 2006). In this paper, we follow Dwek & Barker (2002) and assume the measured correlation to hold in deriving our estimate for the contribution to the radio signal from star-forming galaxies.

Figure 1 shows that there is clear excess emission detected by ARCADE 2 in the 3 GHz channels compared to what is expected from the CMB plus the contribution of extragalactic radio sources. As noted above, the fit is performed on the combined data sets after subtraction of an estimate of the point-source contribution. The unexplained residual emission is consistent with a power law with amplitude 18.4 ± 2.1 K at 0.31 GHz, with a spectral index of β = −2.57 ± 0.05.

We have also experimented with inflating the assumed errors for the removal of extragalactic discrete sources. As noted in Section 3.1, we have assumed a fractional error of 10%, independent of frequency, for the discrete source contribution model. Inflating this error to 50% fractional error makes much less than a 1σ change in the values of our power-law fit parameters above, and only a small (∼10%) increase in the quoted errors in the fit parameters. This is because the error in the removal of the discrete sources is smaller than the other errors in the low-frequency measurements, as can be seen by inspection of Table 1.

Free–free distortions to the CMB spectrum can arise from energy released at lower redshifts (Bartlett & Stebbins 1991; Gnedin & Ostriker 1997; Oh 1999) and can be characterized by

$$\begin{equation} \Delta T_{\rm{ff}}(\nu)= T_0 \frac{Y_{\rm{ff}}}{x^2}, \end{equation} \tag{ 5 }$$

where Y_ff is the optical depth to free–free emission, T₀ is the undistorted CMB temperature, x is the dimensionless frequency hν/kT₀, and ΔT is apparent temperature distortion.

A lower limit can be placed on the optical depth to free–free emission from late time effects of Y_ff > 8 × 10⁻⁸ (Haiman & Loeb 1997). The current upper limit of Y_ff < 1.9 × 10⁻⁵ comes from combining data from FIRAS and previous ground-based CMB spectrum measurements (Bersanelli et al. 1994).

We have performed a four-component fit (Equation (3)) to the data to assess whether there is evidence to support a free–free spectral distortion to the CMB, compared to the three-parameter fit described in the previous section. The four-fit components consist of a constant CMB temperature, a power-law amplitude, a power-law index, and a free–free amplitude. The fit parameters are presented in Table 2. The addition of the free–free amplitude does not improve the reduced χ² of the fit and is therefore not justified by the data.

We have also examined a two-parameter fit consisting of a constant CMB component and free–free distortion component; the parameters are shown in Table 2. This fit produces a significantly worse reduced χ² and is therefore not consistent with the source of the unexplained emission.

The 2σ limits on the free–free amplitude at 1 GHz derived from our four-parameter fit are −0.49 K < ΔT_ff < 0.84 K. This corresponds to an upper limit on the free–free optical depth of

$$\begin{equation} |Y_{\rm{ff}}| < 1 \times 10^{-4}. \end{equation} \tag{ 6 }$$

This limit is less constraining than those reported by Bersanelli et al. (1994) and Gervasi et al. (2008b). This is the result of the additional degrees of freedom allowed by our four-parameter fit. As described above, the four-parameter fit is a better description of the data than the CMB plus free–free distortion fits performed by Bersanelli et al. (1994) and Gervasi et al. (2008b). We conclude that the tighter constraints offered by those studies are likely too optimistic.

It is interesting to consider how future measurements might improve the Y_ff limit. We have used the existing fits and asked how much tighter the limit becomes if an additional measurement of some fixed fractional accuracy is added to the data set. We have run this test as a function of frequency; the frequency range with the greatest effect is between 0.3 and 3.0 GHz, where a factor of several tighter constraints is potentially achievable. The results are shown in Figure 3, where we have plotted the size of the 2σ errors on Y_ff as a function of the frequency of the additional measurement. We have assumed that the measurement point falls precisely on the existing fit. Better fractional error results in tighter constraints on Y_ff. We have also considered the impact of repeating this test with additional measurements at 3, 5, 8, 10, 30, and 90 GHz with 0.002 K precision, as might be obtained with a future flight of ARCADE. Such a measurement would enhance the power of future lower frequency measurements to constrain Y_ff and would improve limits to free–free distortions better than any single new measurement at lower frequency.

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Energy release early in the universe can distort the spectrum of the CMB. Energy injection at redshift z ≲ 10⁷ no longer results in a Planck spectrum, but instead forms a spectrum with a Bose–Einstein photon occupation number (Sunyaev & Zeldovich 1970). An example of such an effect is the damping of primordial density perturbations before the epoch of recombination (Hu et al. 1994).

A series of papers (Danese & De Zotti 1980; Burigana et al. 1991a, 1991b, 1995) has investigated in detail the shape of the resulting spectral distortions after inclusion of free–free processes which act at the low-frequency end. The shape of the distortion depends on the range of redshift over which such energy injection takes place. We have used their analytic description of such distortions to the CMB to provide a functional form to fit to our combined data set. The occupation number is given by a Bose–Einstein like distribution with a frequency-dependent chemical potential:

$$\begin{equation} \eta _{\rm{{BE}}} = \frac{1}{e^{\left[x/\phi _{\rm{{BE}}} - \mu (x) \right]} -1}, \end{equation} \tag{ 7 }$$

where

$$\begin{equation} \mu (x) = \mu _0 \exp {\left[-x_c(z_1)/(x/\phi _{\rm{{BE}}})\right]}, \end{equation} \tag{ 8 }$$

μ₀ is the chemical potential, x ≡ hν/kT is the dimensionless frequency, and ϕ_BE is the ratio of electron to radiation temperature with

$$\begin{equation} \phi _{\rm{{BE}}} \simeq (1 - 1.1 \mu _0)^{-1/4}. \end{equation} \tag{ 9 }$$

The characteristic dimensionless frequency, x_c(z₁), is given by an analytical approximation in Burigana et al. (1991a) and depends on the Hubble parameter, the baryon density, the present CMB temperature, and the number of relativistic species, N_ν. For N_ν = 3, T₀ = 2.725 K, and Ω_bh² = 0.0227 (Hinshaw et al. 2009), we find

$$\begin{equation} x_c(z_1) \simeq 5.7 \times 10^{-3}, \end{equation} \tag{ 10 }$$

which corresponds to a frequency of approximately 0.33 GHz.

The above discussion neglects the contribution of additional photons at low energy through free–free emission. This effect can be included through the following modification of the occupation number (Danese & De Zotti 1980):

$$\begin{equation} \eta = \frac{ 1 - [1-\eta _{\rm{{BE}}}(e^{x/\phi _{\rm{{BE}}}} -1)] e^{-y_{\rm{{abs}}}(x)} }{[e^{x/\phi _{\rm{{BE}}}} -1 ]}. \end{equation} \tag{ 11 }$$

Here, y_abs(x) is the optical depth to free–free absorption and depends on baryon density and other cosmological parameters. An analytic approximation is given in Burigana et al. (1991a).

We can compare the size of the corrections relative to a constant μ₀ = 9 × 10⁻⁵ distortion, assuming the cosmological parameters described above. The correction to include a frequency-dependent chemical potential decreases the distortion amplitude by approximately 3 mK at 1 GHz. The correction to include the contribution of additional photons at low energy through free–free emission decreases the distortion amplitude by 7 mK at 1 GHz relative to a constant μ distortion. The corrections described above become more important at frequencies below 1 GHz. We have included these corrections in our fits.

For fitting the data, the distorted spectrum is related to the occupation number by

$$\begin{equation} T_{\rm{{eff}}}(\nu) = (h \nu /k) / \ln {(1 + \eta ^{-1})}. \end{equation} \tag{ 12 }$$

The shape of the distortion is shown in Figure 4.

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In addition to the μ distortion, we have included a power-law amplitude and index to the fit parameters. For this fit, we have used the "condensed FIRAS" data set. The addition of the μ distortion as a free parameter does not substantially improve the reduced χ². The 2σ upper limit on μ is

$$\begin{equation} \mu < 6 \times 10^{-4}. \end{equation} \tag{ 13 }$$

This limit is not an improvement on the previous limit of 9 × 10⁻⁵ set using FIRAS (Fixsen et al. 1996). The combination of the ARCADE 2 data and the "full FIRAS" data set (with proper inclusion of the FIRAS error covariance, including absolute calibration uncertainty, and a nuisance parameter times the Galactic dust model) does not result in a substantial improvement in the previous FIRAS limit.

We note that sensitivity to a chemical potential distortion from the data in Table 1 is dominated by the ARCADE data and the FIRAS data point, due to the large uncertainties in the lower frequency radio surveys. Over the range 3–90 GHz, the chemical potential distortion resembles a power law with a negative amplitude, making it degenerate with the power-law fit for the radio background. There is not enough curvature between 3 and 90 GHz to precisely distinguish a chemical potential distortion from a power law, thus leading our less precise limit on μ from the low-frequency data alone.