DETECTION OF THE COSMIC FAR-INFRARED BACKGROUND IN AKARI DEEP FIELD SOUTH

Pre-flight measurements of the absolute gain, including detector responsivity, were carried out in all wave bands of AKARI in the laboratory, using an external blackbody source at temperatures ranging from 17 K to 30 K, with attenuators of 20–30 dB. The measured gain factors showed good agreement with each other, to within ±5%, and we adopted a mean value as the final gain factor.

To compare measurements done at different times and different conditions either in orbit or in the laboratory, we corrected the absolute gain value by referring to internal calibrator signals. The reproducibility of the calibrator signals was approximately ±5%, and this was also the limiting factor of calibration accuracy. In the SW bands at 60 and 90 μm, we estimated the accuracy of the pre-flight calibration to within ±7% by quadratically combining the above uncertainties.

While the monolithic Ge:Ga detectors for the SW bands had fairly good uniformity of responsivity over the array and showed high stability in orbit, the stressed Ge:Ga detector arrays for the LW bands showed considerably larger responsivity differences among pixels and strong radiation effects in orbit. These properties made it difficult to correct for responsivity in the LW bands, even with the internal calibrator (c.f. flat-field error), and caused additional calibration errors of ±6% and ±9% at 140 and 160 μm, respectively.

To measure the absolute gain of the detectors in orbit, we made observations of diffuse emission whose brightness had been previously accurately measured with COBE, as described in the last section.

For the SW bands, we observed the zodiacal emission at low ecliptic latitudes. To derive the gain factor, we compared our observed data with the COBE/DIRBE model of zodiacal emission, which is about 98% reliable at low ecliptic latitudes (Kelsall et al. 1998). In order to use this method to avoid the contamination from Galactic and extragalactic emission, we differentiated the sky brightness measured at different ecliptic latitudes in low cirrus regions with similar H i column densities ($N_{\rm H\,\mathsc{i}}$ ∼ 2 × 10¹⁹ cm⁻²) and compared the differential data with the zodiacal light model. The accuracy of this method is consequently limited by the calibration uncertainty of DIRBE, which is ∼10% (Hauser et al. 1998).

For the LW bands, we observed Galactic cirrus emission and compared the measured brightness with the COBE/DIRBE annual average map (AAM). Sky brightness at low galactic latitudes in this wavelength range is dominated by Galactic cirrus emission; seasonal variation of zodiacal emission is smaller than 3%, and the AAM map is thus valid for this calibration. In addition to the calibration uncertainty of DIRBE, the low-frequency noise and instability of the detector during the observations increased the uncertainties to ±15% and ±19% at 140 and 160 μm, respectively.

For the SW bands, the absolute gain factor of AKARI from pre-flight measurements showed excellent agreement with that from the in-orbit calibration. The difference was only 5% at both 65 and 90 μm. It is noteworthy that this difference is smaller than both the uncertainties of the pre-flight and in-orbit calibrations. For the LW bands, the pre-flight measurement gave a slightly different gain from that of the in-orbit calibration; the difference was 13% and 25% at 140 and 160 μm, respectively, but they agreed with each other within the uncertainties.

In this work, we adopted our own gain factor from the pre-flight measurement to indicate the final result of the CIB brightness. For the foreground subtraction, however, we adopted the in-orbit calibration factor to compare our data directly with the DIRBE result. In summary, the absolute calibration uncertainties of our CIB measurements, taking all systematics into account, are ±7% at both 65 and 90 μm, ±9% at 140 μm, and ±13% at 160 μm. These numbers are sufficiently small for significant detection of the CIB. More details about the absolute calibration are described in a separate forthcoming paper (S. Matsuura et al. 2011, in preparation).

Although the seasonal variation of zodiacal brightness is sometimes an obstacle, it is rather useful to know the contribution of dust near Earth to zodiacal brightness. In fact, conventional models of the zodiacal cloud based on the IRAS and COBE observations have been constructed by relying on the local dust density and temperature derived from seasonal variation (Reach 1988; Kelsall et al. 1998). The amplitude of seasonal variation is not affected by any uncertainty from the isotropic component of zodiacal emission.

In Figure 2, the measured sky brightness at 65 μm at various locations in ADF-S is plotted as a function of days after launch. Figure 3 is the same plot but at 90 μm. The sky brightness values are only for those dates when the field was observed, rather than the entire length of the mission. Statistical errors of sky brightness from the rms fluctuation of the image are on the order of a few kJy sr⁻¹ (smaller than the size of the plot symbols). The sky brightness in the second season of winter, about a year after launch, was significantly lower than that in the first and third seasons of summer, as expected from a one-year period of the seasonal variation of zodiacal brightness.

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For comparison, the sky brightness toward the LH and NEP, which has been measured as supplemental observations in previously well-explored fields, is also shown in Figures 2 and 3. The LH for both PV and MP is brighter than ADF-S because of the field's higher zodiacal brightness, despite the similar density of the Galactic cirrus region. The NEP is brighter than ADF-S due to the larger contribution from the Galactic cirrus, even though it has the lowest zodiacal brightness.

To measure the absolute brightness of the CIB, we subtracted the zodiacal foreground from each observation by using the same COBE/DIRBE model as we used for the in-orbit calibration. The zodiacal brightness in the AKARI bands was calculated by interpolation from the DIRBE bands, with color correction assuming a blackbody spectrum. The zodi-subtracted data are composed of only the Galactic cirrus emission and CIB, assuming the zodi-model is correct.

In Figures 2 and 3, residuals after subtraction of the zodiacal foreground are compared with the raw data. The zodi-subtracted data for ADF-S taken in different seasons show good agreement with each other, not only in the mean levels but also in the detailed structure, mainly due to the Galactic cirrus. In addition, the residuals for ADF-S and the LH agree with each other except for the detailed structure, while the NEP is much brighter than the others, especially at 90 μm, due to the larger contribution from the Galactic cirrus, as described later. At 65 μm, the difference in NEP brightness compared to the other fields is not very clear, since the sky brightness is dominated by zodiacal emission. The results indicate that the zodi-model is correct to an accuracy better than 5%, which is a typical value of the model's uncertainty (Kelsall et al. 1998).

The upper image of Figure 4 shows the final diffuse map at 90 μm produced by mosaicing the zodi-subtracted data. To obtain this mosaic image, destriping was made by first calculating the mean sky levels of the image strips taken by multiple scans, after removing bright sources via a 3σ rejection method, and then smooth filtering the time domain to the mean sky levels of multiple scans to correct any slight responsivity difference that may not have been perfectly corrected by the calibration sequence. Finally, all images were co-added. The window size used for the time-domain filter corresponds to three scan strips in the space domain. A large contiguous area was successfully mapped, except for small lost-data areas caused by problems with the ground station, such as the inlets in the upper part of the map. As clearly seen in the map, a large-scale structure from Galactic cirrus emission exists; the central region is darker than the surrounding area. Most bright spots can be attributed to nearby galaxies.

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The power spectrum after noise subtraction and PSF correction is shown in Figure 7. There exist two major components in the power spectrum. One is a constant component dominating at wavenumbers larger than 0.1 arcmin⁻¹, and the other shows a steep rise toward the smallest wavenumber following a k³ power law. The former is regarded as the shot noise (Poisson noise) mainly due to bright galaxies in the field, and the latter can be identified as Galactic cirrus fluctuations, as discussed in the following section.

We carried out a power spectrum analysis for the 65-μm data in the same way we did for the 90-μm data. However, the obtained power spectrum was not significant compared to noise power, suffering from low sensitivity. It was also difficult to obtain a significant power spectrum at 140 and 160 μm because of the presence of low-frequency noise originating from the responsivity change induced by cosmic-ray hits.

The raw power spectrum in Figure 7 is obviously contaminated by shot noise, with a flat spectrum at larger wavenumbers originating from randomly distributed bright galaxies in the field. To investigate the fluctuations originating from underlying faint galaxies (the building blocks of the CIB), we derived the power spectrum by removing the point sources down to the detection limit of AKARI, ∼20 mJy.

Figure 12 shows the PSF-corrected power spectra before and after point-source removal. The shot noise power, in contrast to the cirrus component, decreases to 10% of the value before source removal. The increased power at ∼1 arcmin⁻¹ may be a signature of the small-scale correlation between galaxies and source confusion, or it may be due to the uncertainty of the PSF, as seen in Figure 11. Study of the power spectrum at these small angular scales is reserved for future work. At the other wavelengths, we only obtained upper limits for the shot noise, which are not as meaningful as lower limits.

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The resultant CIB brightness was measured as the zero-intercept of the correlation between the zodi-subtracted map and the SFD map, as shown in Figure 13, and summarized in Table 2 with statistical and calibration errors and an uncertainty in the zodiacal emission model of 5% (Kelsall et al. 1998). In Table 2, the mean brightnesses of zodiacal emission and Galactic cirrus emission over the entire ADF-S are also shown as components of the measured total sky brightness. The measured brightnesses in Table 2 are all color corrected, assuming a flat spectrum. Although the zero-point offset of the SFD map may cause a systematic uncertainty in the CIB brightness, such uncertainty has been estimated to be smaller than 0.1 MJy sr⁻¹ at 100 μm (Finkbeiner et al. 2000), which does not significantly alter the results. In conclusion, we can say that we significantly detected the CIB in the SW bands. We also tentatively detected the CIB at 140 and 160 μm by extrapolating the cirrus brightness from 90 μm to each wavelength, assuming a T = 18.2 K blackbody spectrum.

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To obtain a more robust result we adopted another zodi-model with a very powerful assumption about the celestial signal—called the "very-strong no-zodi" principle, i.e., no extrasolar emission at 25 μm exists at high ecliptic latitudes (Wright 1998; Wright 2001; Levenson et al. 2007)—in which the zodi-model brightness becomes ∼10% higher than what was assumed for our foreground subtraction (L. Levenson 2009, private communication). On this assumption, the residual background decreases to zero at 65 μm but still remains at 90 μm, at a significantly positive value of 0.50 MJy sr⁻¹, because of the bluer color of zodiacal emission. This lower limit of the CIB at 90 μm is shown in Figure 13. In the LW bands, the zodi-model difference changes the residual background by at most 10%, which is smaller than the statistical error.

In Table 3, we summarize the zodi-subtracted sky brightness in various fields as a mean value, with the error estimated from the brightness variance for a number of observations in each field. The error does not include the calibration error or the zodi-model uncertainty. The table also indicates the residual brightness after subtracting the cirrus brightness, which was estimated from the H i 20-cm line map of the Leiden–Argentine–Bonn (LAB) survey as a systematic uncertainty of 10¹⁹ cm⁻² (Kalberla et al. 2005; Bajaja et al. 2005). The cirrus brightness was obtained by assuming a conversion factor from the H i column density to the 100-μm brightness of 0.6 ± 0.1 MJy sr⁻¹/10²⁰ cm⁻², as the median value of various COBE-based results with their variation as an uncertainty (Boulanger et al. 1996; Arendt et al. 1998; Reach et al. 1998; Schlegel et al. 1998; Lagache et al. 2000). For the wavelength conversion from 100 μm to the AKARI bands, we used the correlation coefficients in Figures 5 and 6. Since the angular resolution of the H i data is 10 arcmin, which is much larger than that of our data, we used the mean brightnesses of our data and the H i data in each field. The residual brightnesses after subtracting the cirrus brightness in different fields agree with each other within the error bars, at both 65 and 90 μm including the systematic uncertainty of the H i column density.

Table 3. Field Variance of CIB Brightness

Field	$N_{\rm H\,\mathsc{i}}$a	I65b	Residualc	I90b	Residualc
	(1020 cm−2)	(MJy sr−1)		(MJy sr−1)
ADF-S	0.67 ± 0.10	0.329 ± 0.003	0.21 ± 0.04	0.835 ± 0.004	0.54 ± 0.08
LH-MP	0.54 ± 0.10	0.311 ± 0.057	0.22 ± 0.08	0.873 ± 0.021	0.64 ± 0.08
NEP-DT	3.73 ± 0.10	0.986 ± 0.023	0.33 ± 0.11	2.038 ± 0.058	0.42 ± 0.26
Average			0.22 ± 0.04		0.59 ± 0.07

Notes. ^aTaken from the LAB H i 20-cm survey with a systematic uncertainty (Kalberla et al. 2005; Bajaja et al. 2005). ^bThe mean brightness after subtracting the zodiacal foreground, with the errors estimated by the variance for a number of observations in each field. ^cObtained by subtracting the H i-correlated component, assuming the conversion factor at 100 μm is 0.6 ± 0.1 MJy sr⁻¹/10²⁰ cm⁻², and extrapolating it to each wavelength by the blackbody spectrum with T = 18.2 K and n = 2. The error includes all systematic uncertainties.

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The ratio of the H i column density to the far-infrared brightness assumed above may be an overestimate, because it tends to be determined by the data in the brighter cirrus regions. In low cirrus regions, the ratio could be smaller due to the contribution of uncorrelated components, such as dust associated with ionized gas. In fact, the mean residuals for the three fields in Table 3 are slightly smaller than the results in Table 2 from the correlation of the ADF-S and SFD maps, though they are consistent with each other within the total error, including the calibration error. The contribution of cirrus emission can be more accurately estimated in the LW bands, where cirrus emission is brighter. However, we did not make the same analysis as we did for the SW bands, because accurate measurement of the CIB was possible only in ADF-S, which has highly statistical samples.

As seen in Figure 13, our results in the four photometric bands are in agreement with previous CIB measurements from COBE/DIRBE at 60, 100, and 140 μm (Hauser et al. 1998; Wright 2004; Odegard et al. 2007) within their measurement accuracies. They are also consistent with an upper limit from IRAS at 60 μm (Miville-Deschênes et al. 2002) as well as an upper limit at 90 μm and the measured CIB level in the 150–180-μm range measured by ISO (Juvela et al. 2009).

The integrated galaxy counts (Dole et al. 2004; Matsuura et al. 2007; Shirahata et al. 2008) account for only a few percentage points of the CIB brightness. A stacking analysis of the Spitzer 24-μm sources (Dole et al. 2006) substantially extended the lower limits at 70 and 160 μm from the prediction of the galaxy evolution model presented in Lagache et al. (2004). Our measured CIB brightness at 90 μm is obviously higher than the Spitzer result, and our result implies the existence of many unknown sources, which may have been missed in the stacking process since they were too faint to be detected at 24 μm with Spitzer.

A hypothetical source with such an excessive brightness should have peculiar spectra, with a peak around 100 μm, that make only a small contribution to the CIB in the other wave bands. High levels of the CIB at 100 μm already have been seen with COBE (Hauser et al. 1998; Finkbeiner et al. 2000) but have been thought to be due to calibration errors and foreground uncertainties (Dole et al. 2006). The 140- and 240-μm bands of the DIRBE instrument on board COBE can be cross-calibrated with the FIRAS instrument, and the DIRBE result of the CIB brightness is reduced to 0.6 times if the DIRBE data are transformed to the FIRAS photometric system and corrected for the zero-point difference (Hauser et al. 1998; Odegard et al. 2007). It is remarkable that this work corroborates the uncorrected DIRBE result using a different instrument.

If the excess emission at 90 μm originates from an isotropic component of the Galactic cirrus emission, which may not be accounted for in the SFD map, it should be tightly constrained by the previously measured H i column density, unless the gas-to-dust ratio is extraordinary. If the excess brightness at 90 μm of ∼ 0.4 MJy sr⁻¹ is associated with neutral hydrogen, the corresponding H i column density must be $N_{\rm H\,\mathsc{i}}$∼1 × 10²⁰ cm⁻². This is much higher than the measured H i column density toward ADF-S (see Table 3) and also unlikely, judging from the uncertainty of the H i observation. Moreover, if the temperature of such an isotropic dust emission is ∼17 K, similar to the bulk of the Galactic cirrus emission, the emission should appear at longer wavelengths as a strong excess, but this result is not significant in our data. Therefore, it is difficult to explain the excess brightness by dust emission associated with neutral hydrogen.

Unexpectedly large amounts of ionized gas at high Galactic latitudes in the form of a warm ionized medium (WIM) comprise another possible candidate for the excess emission. In fact, dust emission from a WIM was detected by the WHAM H_α survey, though the observed areas were limited to relatively dense regions at low Galactic latitudes (Lagache et al. 2000). The DIRBE results at high Galactic latitude regions show that the WIM component contributes little to the total dust emission and does not change much of the residual CIB brightness (Arendt et al. 1998; Odegard et al. 2007), though the far-infrared emissivity per H⁺ atom shows very large field variance.

Large differences in the residual brightnesses of different fields listed in Table 3 may be caused by the contribution of dust associated with a WIM. In fact, ADF-S is darker than the LH in the far infrared, while the H i column density toward ADF-S is higher than that toward the LH. However, any WIM could be dust-poor compared to the Galactic plane, because grain destruction is expected to occur in low-density regions of the interstellar medium, while the ionized hydrogen column density, in order to account for all excess brightness, must be higher than that of neutral hydrogen. Just as for neutral hydrogen, the dust temperature must be high enough to have an emission peak around 100 μm while not producing a significant excess at longer wavelengths, but the existence of such hot dust at high Galactic latitudes has not been reported. Future measurements of the H_α emission and other interstellar lines in ADF-S would give a firm limit of the WIM's contribution to the excess.

Star-forming galaxies at high redshift are another possible cause of the CIB excess in the far infrared. In Table 2, the measured CIB brightness is compared with the range predicted from a selection of galaxy evolution models that take the contribution of a population of luminous infrared galaxies peaking at z ∼ 1 and ultra-luminous infrared galaxies (ULIRGs) peaking at z > 2 into consideration (Lagache et al. 2004; Pearson et al. 2007; Franceschini et al. 2008; Takeuchi & Ishii 2010). Most of these galaxy evolution models give a lower brightness than our measured CIB at 65 and 90 μm, but the models agree with our results at 140 and 160 μm, the stacking analysis of Spitzer data at 70 and 160 μm, and the COBE results at wavelengths other than 100 μm, within the data uncertainties. A model prediction from Takeuchi & Ishii (2010) in Figure 13 shows a fairly good agreement with the measured CIB at wavelengths shorter than 140 μm, but it gives too high a brightness at longer wavelengths, especially in the submillimeter range (Puget et al. 1996; Fixsen et al. 1998). It is obvious that our results, especially at 90 μm, provide a tight constraint on galaxy evolution models.

In contrast to the CIB measurement, the source counts at 90 μm with AKARI are much lower than those predicted by contemporary galaxy evolution models, while the galaxy counts at 65 and 140 μm are consistent with the models and the Spitzer results at 70 and 160 μm (Matsuura et al. 2007; Shirahata et al. 2008). Such low counts at 90 μm have also been found by ISO (Rodighiero et al. 2003; Héraudeau et al. 2004), and it has been reported that many galaxy evolution models fail to reproduce the 90-μm counts (Chary & Elbaz 2001; Serjeant et al. 2004). At 90 μm, the integrated galaxy counts down to the detection limit of AKARI, ∼ 20 mJy, can only account for less than 10% of the CIB brightness (Matsuura et al. 2007). To reproduce the CIB brightness, a strong increase in the galaxy counts below the detection limit is necessary. If we assume a single power law with a power index of −1.5 (Euclidian) for the cumulative galaxy counts, dN/dS ∼ S^−1.5, the power law needs to extend to 0.01 mJy. Such high counts require an extraordinary evolution scenario because conventional galaxy evolution models predict a steep drop in differential 90-μm counts toward the faint end below 1 mJy.

Recently, Berta et al. (2010) reported the result of far-infrared deep surveys to resolve the CIB with $\it {Herschel}$/PACS (the PEP survey); they claimed that about half the CIB at 100 and 160 μm was resolved into point sources by integrating the source counts down to 3 mJy, and that the fraction increased to more than half by stacking the 24-μm sources. Altieri et al. (2010) obtained source counts much deeper levels, ∼1 mJy at 100 μm, toward the cluster lens Abell 2218, but the resolved CIB fraction remained similar to that of the PEP survey. Therefore, a much deeper far-infrared survey is still required to resolve the CIB excess.

To explain the previous COBE result of high CIB levels at 60 and 100 μm, the possible contribution of a new population at high redshift (z = 2–3), with high dust temperatures (T_d ∼ 60 K) heated by AGNs, or accretion onto a black hole has already been discussed (Finkbeiner et al. 2000; Blain & Phillips 2002; Blain et al. 2004). If this is true, our finding of the CIB excess at 90 μm could also be explained by such high-z hot sources. Many results from the Spitzer 24-μm and X-ray surveys suggest that LIRGs and ULIRGs at z ∼ 1, rather than AGNs at high redshift, constitute the main bulk of the CIB, and the contribution of AGNs to the CIB is thought to be at most 10% (Treister et al. 2006; Devlin et al. 2009). Therefore, the CIB excess is more likely to be composed of high-z hot sources of new galaxy populations than known, obscured AGNs, which can be detected at 24 μm. If the hot source is so heavily obscured that dust extinction is strong even in the mid-infrared, and if the dust grains are so small that the emissivity index is at a maximum, λ⁻², at longer wavelengths, then they can contribute to the CIB selectively at short far-infrared wavelengths.

Another possibility to explain our finding that the CIB at 90 μm is very smooth is the contribution of unknown diffuse emission, such as radiative decay of relic particles of the early universe, e.g., the cosmic neutrino background or dark matter particles (Bond et al. 1986; Ressell & Turner 1990). If the particle decay time is longer than the age of the universe, the decay photon-energy spectrum can have a clear SW cutoff, which may result in a spectral peak in the CIB at the decay energy.

The total integrated energy of the CIB excess in the AKARI bands is estimated to be ∼10 nWm² sr⁻¹, which is comparable to the CIB energy associated with the 24-μm galaxies of 24 nWm² sr⁻¹ (Dole et al. 2006). If we assume a constant star formation rate (SFR) over cosmic history of z > 1, the excess energy corresponds to the SFR of ∼0.1 M_☉ yr⁻¹ Mpc⁻³, which is an order of magnitude higher than that in the local universe (Hauser & Dwek 2001). If the radiation energy is produced by a starburst at z, the total hydrogen mass fraction converted to metals in this epoch is estimated to be ΔX ∼ 0.004(1 + z). For a redshift of z > 1, more than 1% of the baryon mass density need to be converted to metals. A starburst at high z values as the origin of the CIB excess may cause overproduction of metals beyond the metal abundance in the local universe. Black hole accretion may play an important role in generating background radiation at high redshift (Finkbeiner et al. 2000).

In the near infrared, the CIB excess that cannot be explained by the integrated flux of known galaxy populations has been reported, and the origin of the excess has been posited as first stars, or proto-galaxies, at z ∼ 10 and discussed in terms of the spectral signature of the redshifted Ly-α in the CIB near 1 μm, as seen in Figure 1 (Kashlinsky 2005; Matsumoto et al. 2005). Findings of the CIB excess in both the near and far infrared lead us to consider the connection; the UV light from the first massive stars, which produce the near-infrared excess, might be partly absorbed by the self-produced dust, re-emitted in the mid-infrared, and redshifted into the far-infrared band at present. Accretion to the first black hole as a remnant of the first star explosion might also be the hot source at high redshift. Dust production by the first stars and successive black hole formation and accretion are little understood, and this research would be furthered by future observations of infrared galaxies and many details of the CIB.

In order to obtain information on the CIB fluctuations from the measured power spectrum, it is necessary to estimate the fluctuation power of the foreground, such as zodiacal emission and the Galactic cirrus. Zodiacal emission, the most dominant foreground component, has been known to have a very smooth distribution, and its fluctuations in the mid-infrared have been measured with ISO to be smaller than 1% of the mean brightness at angular scales of 1 arcmin (Abraham et al. 1998). This value is already much smaller than the shot noise level in unresolved galaxies. Although an unexpected time-varying residual of zodiacal emission, after zodi-model subtraction, may contribute to the power spectrum over a broad range of angular scales, it has already been subtracted as part of the noise power estimated from the difference of the two sub-maps, as described in the last section. Thus, the main foreground is the Galactic cirrus, especially at large angular scales.

A k³ power law of the fluctuation spectrum of the Galactic cirrus has been reported by many authors, e.g., Miville-Deschênes et al. (2007), but it is remarkable that this work confirmed the same power law at very low flux levels at 90 μm. The power spectrum of the cirrus fluctuations can be expressed as

$$\begin{equation} P_{{\rm cirrus}}(k)=P_0(k/k_0)^{-\gamma }, \end{equation} \tag{ 3 }$$

where k is the wavenumber in arcmin⁻¹ and P₀ is the power normalized at k₀ = 0.01 arcmin⁻¹. γ is the power-law index, ranging from 2.5 to 3.1, and is known to be close to 3 for the optically thin case at low cirrus regions (Schlegel et al. 1998; Jeong et al. 2005; Miville-Deschênes et al. 2007). As shown in Table 4, our result gives a cirrus power at 90 μm of P₀ = (5.5 ± 0.5) × 10⁴ Jy² sr⁻¹, and the power at 100 μm is derived to be P₀ = (1.1 ± 0.1) × 10⁵ Jy² sr⁻¹ from the correlation coefficient between the ADF-S and SFD maps.

At high values of k, the power-law index may not always be the same as the value derived at low k, where the cirrus component dominates the total fluctuation power. In previous work with the ISO observations, it has been proved that a single power law for the cirrus power spectrum is valid up to k ∼ 0.3 arcmin⁻¹, though this result is certain only in relatively bright regions. Since there is no strong reason to change the power-law index at higher k, we assume a single power law over the entire wavenumber range to estimate the cirrus power.

It has been reported that the fluctuation power of the cirrus emission scales with the mean brightness as a power law, with an index of 2–3 depending on the brightness. According to Miville-Deschênes et al. (2007), the cirrus fluctuation power at 100 μm in the low-brightness regime (B < 10 MJy sr⁻¹) is given by

$$\begin{equation} P_0=2.7\times 10^6 B^{2.0}\, \rm Jy^2\, \rm sr^{-1}, \end{equation} \tag{ 4 }$$

where B is the mean brightness at 100 μm. If this is the case, our measured power of the cirrus fluctuations corresponds to the mean brightness of 0.14 MJy sr⁻¹ at 90 μm, which is slightly lower than the value in Table 2. The relation between the fluctuation power and the mean brightness may not be valid under such low cirrus conditions. The result implies that it is dangerous to estimate the mean CIB brightness from the fluctuation power.

We have derived the shot noise of unresolved galaxies, from the source-removed power spectrum in Figure 12, as the mean power level over a range of 0.2–0.8 arcmin⁻¹, where the PSF is relatively accurate, to be P_shot = (3.6 ± 0.2) × 10² Jy² sr⁻¹ at 90 μm. In Table 4, we summarize our results for the foreground estimates at small angular scales, where the shot noise dominates. The fluctuations of zodiacal emission were estimated to be negligible, as discussed in the last section. The cirrus component following a k³ power law is negligible at such small angular scales (k > 0.2 arcmin⁻¹). Therefore, the remaining shot noise can be attributed to the fluctuations from unresolved galaxies.

In Table 4, we compare our results with the previous work at 90 μm using ISOPHOT (Matsuhara et al. 2000) and with a range of shot noise levels predicted by various galaxy evolution models. Since the detection limit of AKARI for point sources is better than that of ISO, the AKARI result shows lower fluctuation power than the ISO result. We estimated the shot noise level for each galaxy evolution model by simply integrating the squared flux, S², with the differential source counts, dN/dS, as

$$\begin{equation} P_{{\rm shot}}=\int {S^2(dN/dS)dS}, \end{equation} \tag{ 5 }$$

for fluxes below 20 mJy. The measured shot noise power is three to five times lower than any other model predictions, which is clearly due to the high source counts of the models near the detection limit. The result implies that the main bulk of the CIB at 90 μm consists of sources much fainter than the detection limit.

Since the shot noise is the squared flux integrated with the source counts, it is more sensitive to the number density of bright sources than the mean CIB level, which is a linear integration of the flux. As described in the last section, the mean CIB level provides constraints on the source counts at the very faint end below the point-source detection limit. In contrast, shot noise can provide constraints on the source counts near the detection limit.

The measured shot noise level can be reproduced by measured source counts that are extended down to 1 mJy with a Euclidian power law (a constant for the differential counts). In such a simple scenario, sources fainter than 1 mJy have little contribution to the fluctuations but become the main contributor to mean brightness. To resolve the bulk of the CIB into point sources, future infrared telescopes with detection limits much better than 1 mJy at ∼100 μm will be required.

Since ADF-S covers a large contiguous area of 12 deg², the field contains various regions of different sources and cirrus densities. The power spectrum shown in Figure 12 gives the average properties of the entire field, but it can be biased by strong features of specific regions on the map.

To check for cosmic variance, we carried out power spectrum analyses for sub-maps with an area of 1 × 1 deg² cut from various positions, as shown in Figure 14. The first is the central region, with the minimum cirrus level and source densities; the second region contains the rich cluster of galaxies called DC 0428-53 (Abell S0463) (Dressler 1980); and the third is a relatively high cirrus region with a moderate source density. As shown in Figure 15, the power spectrum obtained from the cluster region is about an order of magnitude higher than the others. The central region shows the minimum fluctuation power, as expected, and the high-cirrus region has slightly higher power than the central region, though the point source contribution seems to be more prominent than the cirrus fluctuations. This result implies that the power spectrum is severely affected by the point source contribution and biased by high source-density regions.

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To estimate the effect of point sources on the power spectrum in detail, we made the same comparison of the three sub-maps after removing point sources. The result is shown in Figure 16. All power spectra at different sky positions show excellent agreement with each other, at an accuracy better than 5% in the overall range of the wavenumber. This result implies that the power spectrum that resulted after removing point sources is not affected by cosmic variance, such as the clustering of nearby galaxies, and can be regarded as a universal function of the CIB fluctuations.

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In the measured power spectrum in Figure 12, a small hump of ∼10³ Jy² sr⁻¹ is seen in intermediate range of 0.03–0.1 arcmin⁻¹, and its k-dependence seems to be less steep than the k³ power law of the Galactic cirrus. A possible interpretation of such an excess fluctuation at an intermediate angular scale is the clustering of star-forming galaxies, e.g., Perrotta et al. (2003). Lagache et al. (2007) reported finding galaxy clustering in the power spectrum at 160 μm with Spitzer after removing point sources brighter than 200 mJy. In Figure 17, we plot the Spitzer result from Lagache et al. (2007), multiplied by a scaling factor of 0.01, with our power spectrum. They claimed their power spectrum shows an excess component in the 0.01–0.1 arcmin⁻¹ range that cannot be explained by the shot noise of individual galaxies and the power-law spectrum of the Galactic cirrus. They compared a galaxy clustering model (the dot-dashed line in Figure 17) with the measured spectrum and directly derived the bias factor for structure formation in a dark matter universe to be b ∼ 2.4.

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As a result of the comparison of our power spectrum with the Spitzer result, the two spectra in k < 0.1 arcmin⁻¹ agree with each other surprisingly well, not only in the overall shape of the power-law component at very low frequencies by the Galactic cirrus and the shape of the galaxy clustering component but also in details such as wiggles at k = 0.01–0.1 arcmin⁻¹ and a dip around k ∼ 0.02 arcmin⁻¹, which may show an oscillatory structure of matter distribution. Such concordance between the results of two independent observations with different instruments and in different fields leads us to conclude that the fluctuations in both the AKARI and Spitzer power spectra have the same origin.

It has been reported that galaxy clustering was clearly found by the BLAST experiment in the submillimeter range (Viero et al. 2009). Since the BLAST bands are on the Rayleigh–Jeans side of the dust emission in the rest-frame galaxy, BLAST bands can be more sensitive to high-redshift galaxies than those of AKARI and Spitzer, with less contamination by the Galactic cirrus. The power spectrum analysis for the BLAST map with an area of 6 deg² found a strong signature of galaxy clustering, with the power spectrum showing a k² power law over the range of k ∼ 0.03–0.1 arcmin⁻¹ in all BLAST bands. Their power spectra show a very similar shape to those measured by AKARI and Spitzer even though the limiting fluxes for the source removal for these three missions are different—AKARI: ∼ 20 mJy, $\it {Spitzer:}$ ∼ 200 mJy, and BLAST: ∼ 500 mJy.

To test if the hump structure seen in the AKARI power spectrum could be real and to compare the AKARI data with the results of other missions regarding the galaxy clustering component using the same method, we made a model fit to our power spectrum using a k⁻² power law for galaxy clustering, in addition to shot noise and cirrus fluctuations, i.e.,

$$\begin{equation} P_{2}(k)=P_{{\rm cirrus}}+P_{{\rm shot}}+P_{{\rm cluster}} \end{equation} \tag{ 6 }$$

and

$$\begin{equation} P_{{\rm cluster}}=P_{c}(k/k_{c})^{-2}, \end{equation} \tag{ 7 }$$

where k_c = 0.03 arcmin⁻¹. However, such a simple model cannot reproduce the complex structure seen in the measured spectrum. For the case shown by the thin solid line in Figure 17, we obtained the best-fit parameters: P_cirrus(0.01) = (5.0 ± 0.4) × 10⁴ Jy² sr⁻¹ for γ = 3.0 ± 0.1, P_shot = (3.7 ± 0.01) × 10² Jy² sr⁻¹, and P_c = (0.0 ± 1.9) × 10² Jy² sr⁻¹. Therefore, we conclude that the existence of galaxy clustering is not significant in the AKARI power spectrum. However, the AKARI data can provide a 3σ upper limit to the galaxy clustering power of P_c < 5.7 × 10² Jy² sr⁻¹.

The power index of the cirrus component, γ, is known to be highly uncertain and varies from place to place in the sky, and the uncertainty directly affects the determination of the clustering component. Therefore, the upper limit of the clustering component obtained above is not a very firm result. The most reliable constraint on the amplitude of the clustering component can be obtained by assuming there is no contribution from the cirrus fluctuations at k = 0.03 arcmin⁻¹. This limit of the clustering component is directly obtained from the measured power spectrum and is P_c < 3.7 × 10³ Jy² sr⁻¹.

The fluctuation amplitude, δI_ν, of the clustering component relative to the mean CIB level, 〈I_ν〉, can be estimated from the power spectrum at a given wavenumber to be (Kashlinsky 2005; Viero et al. 2009)

$$\begin{equation} \delta I_{\nu }=\sqrt{(}{2\pi }k^{2}P_{2}(k)). \end{equation} \tag{ 8 }$$

Using this relation, the 3σ upper limit of P_c when there is no cirrus contribution, as described above, is transformed to the upper limit of the CIB fluctuation, δI_ν < 1.6 × 10⁻² MJy sr⁻¹, and its relative amplitude, δI_ν/〈I〉 < 2.3 × 10⁻².

It is notable that the CIB distribution at 90 μm, after removing the point sources to a faint flux level, is unexpectedly smooth. It is also noteworthy that the AKARI data are of high enough quality to detect such very low fluctuation levels.

A study of the SED of the CIB fluctuations should give constraints on the redshift distribution and SED of the underlying galaxies and provide important information on their origins. Figure 18 summarizes the CIB fluctuations measured with AKARI, Spitzer (Lagache et al. 2007), and BLAST (Viero et al. 2009) for shot noise levels at k = 0.33 arcmin⁻¹ and galaxy clustering at k = 0.03 arcmin⁻¹, with the fluctuation amplitude δI_ν. The AKARI data point for the clustering component is the upper limit obtained by assuming there is no cirrus contribution to the measured power spectrum at k = 0.03 arcmin⁻¹, as described in the last section. The upper limits on the CIB fluctuations at k ∼ 0.02 arcmin⁻¹ from the DIRBE analysis (Kashlinsky & Odenwald 2000) are also indicated. For comparison, the measured mean CIB levels and the model CIB by Lagache et al. (2004) are also shown in Figure 18.

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If the CIB fluctuations measured by AKARI, Spitzer, and BLAST originate from the same underlying galaxies, the combined spectrum should be comparable to the SED of the most dominant galaxy population. In Figure 18, we compare the combined spectrum of the measured CIB fluctuations with the SED of Arp220 taken from Klaas et al. (2001), which is redshifted to z = 2 and scaled to fit the data. Although there is a non-negligible discrepancy between the two spectra, the overall colors are very similar. Therefore, the fluctuating component is likely to be formed by such red galaxies at redshifts of z > 1, which are all resolved at 24 μm and account for ∼100% of the CIB in the submillimeter range (Devlin et al. 2009; Pascale et al. 2009; Viero et al. 2009). This result is consistent in that the color of the CIB fluctuations is very similar to the color of the mean CIB flux in the Spitzer 160-μm band and the BLAST bands, and it is also consistent with the mean CIB levels in those bands, which agree well with the model CIB for the selected 24-μm galaxies.

A remarkable result on the SED of the CIB fluctuations is that the relative fluctuation amplitude at 90 μm, δI/〈I〉, is quite small compared with those at other wavelengths. This is caused by both the very red color of the CIB fluctuations and the high CIB level at 90 μm, which is exceeds the integrated flux of the selected 24-μm galaxies and can account for the mean CIB levels at longer wavelengths. According to this result, the CIB should have a spectrum peaking around 90 μm if it does not contribute much at 160 μm and cannot be detected at 24 μm. The underlying galaxies building the CIB should also be individually faint and numerous to keep the mean CIB levels high while keeping the CIB fluctuations low.

Although hot high-z sources with far-infrared luminosities on the order of 10¹³L_☉, discussed in Blain et al. (2004), can considerably contribute to the CIB at 90 μm, such very luminous and less numerous galaxies would largely contribute to the CIB fluctuations rather than the mean CIB flux. According to the measured CIB properties as discussed above, the bulk of the CIB may consist of hot high-z sources of lower luminosity but higher surface density. A simple way of reducing the CIB fluctuations without an SED change is to assume higher redshifts and higher temperatures by keeping T_d/(1 + z) a constant, and this could be the case of our result.

As described in the previous section, a more exotic source than galaxies, such as the radiative decay of the relic particles of the early universe, may also contribute to the CIB. To obtain a definite conclusion to the origin of the CIB excess in the AKARI band, further investigation is required. To prove the hot high-z source and/or decaying particle hypothesis, large-aperture space telescope missions, such as SPICA (Nakagawa 2008), are required to resolve the far-infrared CIB into point sources at very low flux levels (below 0.1 mJy; Swinyard & Nakagawa 2009). Spectral measurement of the continuous SED of the CIB over a wide infrared range would also be a powerful discriminator of the origin. Such observations can be simply made with a small-aperture telescope on a small satellite or sub-orbital missions, such as rocket experiments. Future space missions to obtain a direct measurement of the mid-infrared CIB while mitigating the contamination by zodiacal emission require out-of-ecliptic missions or deep space missions beyond the asteroid belt (as the source of zodiacal dust), which will be powerful enough to probe into the underlying galaxies or particles that compose the CIB excess (Matsuura 2002).