Annu. Rev. Astron. Astrophys. 2001. 39: 249-307
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3.6. Fluctuation Analysis

If the CIB originates at least in part from discrete sources, then fluctuations in the number of sources in an observer's field of view will produce fluctuations in the measured background brightness. Hence, measurement of fluctuations in the extragalactic background reveal information about the number and distribution of contributing sources. Fluctuations can be characterized by the two dimensional autocorrelation function, C(theta), of the sky brightness or by the corresponding two-dimensional power spectrum.

Fluctuation measurements do not directly yield the total background light. However, at least two approaches have been used to constrain the infrared background light using fluctuation measurements. For a given model of the source comoving luminosity density, three-dimensional correlation function, and spectral energy distribution, one can calculate both the autocorrelation function on angular scales of interest and the background. Measurements of the fluctuations can then be used to constrain the background light in the context of that model. An alternative approach is to use fluctuation measurements to constrain the behavior of source number counts below the detection threshold for individual sources, and then to integrate the counts to constrain the total light.

Measurement of extragalactic background fluctuations in the infrared is easier than direct measurement of the CIB because it is not necessary to determine the zero point of the photometric scale. However, measurement of CIB fluctuations otherwise has many of the same challenges as CIB measurement, since foreground sources and instrumental noise can be dominant contributors to the measured power spectrum or autocorrelation function of the sky brightness. For potential CIB fluctuation detections, it is also necessary to demonstrate isotropy, i.e., that the fluctuations are uniform over the sky. Table 2 summarizes results of studies of infrared background fluctuations, which we express as the rms value of measured fluctuations, delta (nu Inu), or as the source power spectrum, PS, depending on what was reported by the investigator. In either case, the reported values refer to the fluctuations in the extragalactic background. If the investigator related the fluctuations to the CIB brightness, we show that as well.

An early search for fluctuations in the CIB at 2.2 µm was conducted by Boughn et al. (1986), who used ground-based instruments and standard sky-chopping techniques to measure fluctuations. They reported upper limits on the brightness fluctuations on scales from 10" to 30" and from 60" to 300", and used these limits to constrain models of young galaxy formation without attempting to state limits on the intensity of the CIB.

Kashlinsky et al. (1996a) studied the fluctuations in the DIRBE maps and used this information to set constraints on the mean intensity of the CIB. They calculated the expected zero-lag correlation, C(0), from sources clustered like galaxies. They found that C(0)1/2 = delta(nu Inu) is of the order of 5%-10% of that part of the CIB arising from such sources, and that this result is insensitive to the details of their assumptions about the galaxy three-dimensional spatial correlation function and luminosity evolution. The variance in the DIRBE maps at 1.25, 2.2, and 3.5 µm was found to be dominated by the contribution from Galactic stars. Analyzing a few relatively dark fields at high Galactic and ecliptic latitude, they masked discrete sources and removed linear gradients. They found quite different values of C(0) in their different fields and interpreted the lowest value as an upper limit on the CIB fluctuations. In accord with their model analysis, they multiplied the limits on C(0)1/2 by a factor of 10 and reported the results as implied upper limits on nu Inu arising from sources clustered like galaxies, which they assumed to be the dominant sources of the CIB (Table 2).

Kashlinsky et al. (1996b) extended this type of analysis into the far infrared, including DIRBE wavelengths of 100 µm and shorter. They analyzed most of the sky (384 patches of 10° × 10° each), first removing the IPD contribution with an early version of the model of Kelsall et al. (1998). After masking discrete sources and removing the smoothly varying background in each patch, they found that the fluctuations in the residual maps varied systematically over the sky, indicating that these fluctuations arose largely from residual foreground sources. They reported the minimum value of the observed variance as an upper limit on the CIB fluctuations, finding the same upper limits at 1.25, 2.2, and 3.5 µm as those of Kashlinsky et al. (1996a). They reported new limits on the fluctuations in the DIRBE bands from 4.9 to 100 µm, and again multiplied these limits by 10 to obtain upper limits on the background arising from sources clustered like galaxies. This method of constraining the CIB is particularly appealing in this wavelength range, where the high brightness of the IPD makes direct measurement extremely difficult.

Kashlinsky & Odenwald (2000) further analyzed the fluctuations in the DIRBE maps at 1.25-4.9 µm. They found that the variation of C(0) could be fitted with a power law in csc(|b|) plus a positive constant term, with residuals showing no significant dependence on Galactic latitude or longitude in directions removed from strong Galactic signals. Their analysis indicated that instrument noise, data reduction methods, and the Galaxy were not likely sources of the spatially constant component of the fluctuations. They therefore suggested that these approximately isotropic fluctuations may arise from the CIB. Kashlinsky & Odenwald also reanalyzed the 12 to 100 µm data, obtaining upper limits on the residual fluctuations slightly lower than those reported by Kashlinsky et al. (1996b). Using their argument that the fluctuations are 5%-10% of the CIB, we have multiplied their fluctuation limits at these wavelengths by 15 to obtain the implied upper limits on the CIB shown in Table 2, results entirely consistent with those explicitly stated by Kashlinsky et al. (1996b).

Matsumoto et al. (2000) studied sky brightness fluctuations in the IRTS data. After removing the discernible stars and large-scale features, they found a fluctuation spectrum dominated by read-out noise at wavelengths longer than 2.6 µm. They did a Monte Carlo calculation of the expected fluctuations due to faint stars using the Galaxy model of Cohen (1997). After removing the readout noise and calculated fluctuations from faint stars, they found positive excess fluctuations from 1.4 to 2.6 µm similar in magnitude to those reported by Kashlinsky & Odenwald (2000) (Table 2). They also integrated their data from 1.4 to 2.1 µm and calculated the spatial power spectrum along their observational strip, finding an indication of structure on scales of 1°-2°. This structure in the power spectrum agrees roughly with expected CIB fluctuations from galaxy clustering (Jimenez & Kashlinsky 1999).

Somewhat contradictory evidence regarding CIB fluctuations in the near infrared was reported by Wright (2001b), who presented upper limits on the CIB fluctuations at 1.25 and 2.2 µm. Wright determined the standard deviation of the residuals after using 2MASS data to remove starlight from the DIRBE maps (Section 3.4.2). He interpreted the standard deviation in his four fields as an upper limit on CIB fluctuations because there were potential noncosmic contributions to the fluctuations that he ignored. Wright's 1.25 µm upper limit is lower than the 92% confidence interval for the detection of Kashlinsky & Odenwald (2000) and well below the data of Matsumoto et al. (2000) (Table 2). His 2.2 µm limit is below the detections reported by Matsumoto et al. and Kashlinsky & Odenwald but within the 92% confidence interval of Kashlinsky & Odenwald.

Burigana & Popa (1998) searched for evidence of the CIB at submillimeter wavelengths by looking for isotropically distributed fluctuations in the FIRAS maps. They found consistent values of C(0)1/2 in several fields at high galactic latitude and interpreted this as evidence for an extragalactic origin of the fluctuations (Table 2).

Lagache & Puget (2000) reported detection of CIB fluctuations based on analysis of deep 170 µm survey data in a 30' × 30' field (Marano 1 region) obtained with the ISOPHOT instrument on the Infrared Space Observatory (ISO) mission. The primary foreground contribution to fluctuations at this wavelength is expected to be Galactic cirrus, which has been shown to have a steep power spectrum P propto k-3 in previous studies (Gautier et al. 1992, Wright 1998) (k is the spatial frequency). After removing sources brighter than 100 mJy from their map, the observed power spectrum varied as k-3 for k < 0.2 arcmin-1 but decreased more slowly in the range 0.25 arcmin-1 < k < 0.6 arcmin-1. They showed that the excess power above that expected for cirrus was much larger than instrumental noise and attributed these fluctuations to unresolved extragalactic sources. Since this observation was carried out in a single small field, they did not demonstrate isotropy of the fluctuation signal. Their value is comparable to the upper limit found by Kashlinsky et al. (1996b), Kashlinsky & Odenwald (2000) at 100 µm. It is about 5%-10% of the CIB brightness measurements at 140 and 240 µm (Table 1), consistent with the model-based arguments of Kashlinsky and colleagues.

Matsuhara et al. (2000) analyzed fluctuations in ISOPHOT 90 and 170 µm maps in two 44' × 44' fields in the Lockman Hole. As a result of this choice of field, fluctuations due to cirrus did not contribute appreciably to the measured power spectrum. Their observed power spectra were flatter and brighter than expected for cirrus fluctuations. The 90 to 170 µm color of the fluctuations was also warmer than that of cirrus. They concluded that they had detected fluctuations from faint star-forming galaxies below their detection limits of 150 and 250 mJy at 90 and 170 µm, respectively. Adjusting their 170 µm result to the 100 mJy threshold used by Lagache & Puget (2000), they concluded that the 170 µm CIB fluctuations in the Lockman Hole field were consistent with those in the Marano 1 field. Although consistency in two fields does not constitute a strong isotropy test, it does provide further evidence that the reported fluctuations are indeed extragalactic in origin. From these measurements, Matsuhara et al. estimated limits on the behavior of source counts below their detection limits. Using the source count constraints, they estimated the CIB contributions from sources brighter than 35 mJy and 60 mJy at 90 and 170 µm, respectively. The integrated 170 µm source light accounts for about 10%-20% of the measured CIB (Table 1).

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