5 Discussion

5.1 CIB fluctuations at 60 and 100 $\mu$m

We have shown here that the power spectrum of high latitude fields at 60 and 100 $\mu$m is characterized by a break at small scales (near 0.02 arcmin^-1). As indicated by Gautier et al. (1992), the power spectrum of the dust emission is usually well described by a power law proportional to $k^{-3 \pm0.2}$. A more detailed study of the statistical properties of the interstellar cirrus HI 21 cm emission has been carried out by Miville-Deschênes (1999). In this analysis it is shown that there are limited variations of the spectral index of the power law from field to field but, what is most important for the present work, the power spectrum of cirrus emission for scales smaller than $12.5^\circ$ is always characterized by a single power law with no break. Therefore, it is unlikely that the power excess observed here at small scales is of interstellar origin.

We could also wonder if this break is of instrumental origin. It was shown by Wheelock et al. (1993) that the response of the IRAS detectors are affected by memory effects. This produces variations of the detector response as a function of scale. This effect is more important at small scales (under a few tens of arcminutes) but Wheelock et al. (1993) have shown that the amplitude of the fluctuations at these scales were underestimated. This effect will thus produce a drop of the power spectrum at small scale and cannot explain the power excess observed here.

On the other hand, instrumental noise could produce such an excess in the power spectrum. But, as the IRAS ISSA maps result from redundant individual observations, we were able to estimate the contribution of the noise to the power spectrum. We are aware that our estimate of the noise rely on the fact that the individual HCONs are independant. This is not perfectly true as, in the construction of the HCONs, an offset was added to each scan to minimize the difference between different observations of the same position. Therefore the noise level estimated by subtracting HCONs may be underestimated at the scale of a scan, which is a few degrees. But at this angular scale the signal is completely dominated by the cirrus emission, even in the low brightness regions selected for our analysis. At the scale of a few arcminutes where the CIB is detected, the noise contribution to the power spectrum has been removed accurately.

Figure 4: Redshift distribution of the sources making the CIB and the fluctuations at 60, 100 and 170 $\mu$m. The first two-left panels are for the total contributions, the last two-right panels, for sources with flux below 1 Jy (Lagache et al. 2002).

In fact it appears that most of the power excess can be attributed to the numerous extra-galactic point sources that are present in such a low cirrus emission field. When the strong ( $I_{100 ~ \mu\rm m} > 1$ Jy) point sources are removed from the ISSA maps, we recover a power spectrum typical of cirrus emission at low spatial frequencies but with still a power excess at small scales (k > 0.02 arcmin^-1) that can be attributed to the unresolved cosmic infrared background. Moreover, the residue has homogeneous properties over the sky, consistent with CIB.

Knox et al. (2001) computed the expected power spectrum of the CIB at several frequencies ( $\nu \le 1060$ GHz), exploiting the far-IR volume emissivity derived from the count models of Guiderdoni et al. (1998) and assuming a bias b=3, constant with redshift. They concluded that the clustering-induced fluctuations can match those of the CMB at $\ell \lesssim 300$. They also predict a shape of the CIB power spectrum peaking around scales of 1-3 degrees. This broad maximum, if present, is at the limit of our frequency range where the noise is becoming large, making the detection of the clustering very difficult. The power spectra of the CIB at 60 and 100 $\mu$m are compatible with a Poissonian distribution with levels $\sim$ $1.6\times10^{3}$ Jy²/sr and $\sim$ $5.8\times10^{3}$ Jy²/sr respectively.

5.2 CIB intensity and anisotropy amplitudes color ratio

The CIB rms fluctuations in the IRAS maps corresponding to the white noise power spectra are:

$$\sigma^{2}= \int P_{\rm cib}(k) 2 \pi k {\rm d}k \quad {\rm Jy}^2/{\rm sr}^2$$ (7)

giving $\sigma=0.048$ MJy/sr and $\sigma=0.09$ MJy/sr at 60 and 100 $\mu$m respectively. As shown by Gispert et al. (2000), the CIB at different wavelengths is dominated by sources at different redshifts, larger wavelengths being dominated by more distant sources. The same applies to the fluctuations: lower frequencies probe higher redshifts (e.g. Knox et al. 2001) The fluctuations at 60 $\mu$m are dominated by nearby bright objects. When we remove these objects, the residual fluctuations are quite low. At 100 $\mu$m, the contribution of higher redshift objects is increasing, leading to higher level of residual fluctuations. Therefore, the ratio of the 60 to 100 $\mu$m fluctuation is qualitatively consistent with what is expected.

This is illustrated more quantitatively in Fig. 4 on the panels showing the redshift distribution of sources contributing to the CIB intensity and the fluctuations (Lagache et al. 2002). It is clear from these figures that the z-distribution of the fluctuations is bimodal, with one contribution at redshift lower than 0.25 and the other one centered at redshift around 1. For all sources the ratio of nearby to moderate-redshift source contribution to the fluctuations is equal to 3.8, 1.8 and 1.3 at 60, 100, 170 $\mu$m respectively, illustrating that fluctuations at larger wavelengths are dominated by more distant sources. When the brightest sources are removed the ratio of nearby to moderate-redshift contribution becomes equal to 1.7, 1 and 0.74 at 60, 100, 170 $\mu$m. In this case, at 100 $\mu$m the contribution to the fluctuations of nearby and moderate-redshift sources is the same, becoming lower at higher wavelength. At 60 $\mu$m, the fluctuations are still dominated by the nearby objects.

For the three wavelengths, the CIB is mainly due to sources at redshift around 1. A detailed analysis of the CIB fluctuations at 100 and 170 $\mu$m (which is beyond the scope of this paper) will give information on the distribution of sources at $z \sim 1$ making the bulk of the CIB. This is particularly true at 170 $\mu$m where sources with flux lower than 4 $\sigma=135$ mJy can be removed (Dole et al. 2001), leading fluctuations highly dominated by the moderate-z sources.

We can compute the ratio of CIB fluctuations to intensity ( $R_{\lambda} = \frac{\sigma_{\lambda}}{I_{\lambda}}$) at 100 $\mu$m and compare it with the previous determination at 170 $\mu$m. To compute R₁₀₀ and R₁₇₀, we use:

• a CIB intensity at 100 $\mu$m of 0.5 MJy/sr (Renault et al. 2001), leading to R₁₀₀ = 0.18
• a CIB intensity at 170 $\mu$m of 1 MJy/sr Lagache & Dole (2001). For the fluctuations, we assume that a cut of 1 Jy at 60 and 100 $\mu$m corresponds to the same cut at 170 $\mu$m. We obtain rms fluctuations of 0.12 MJy/sr (corresponding to $\sim$11 000 Jy²/sr, Puget & Lagache 2002). This gives R₁₇₀ = 0.12.

The ratio R is decreasing between 100 and 170 $\mu$m, as expected. At 60 $\mu$m, the minimal hypothesis is to consider that R₆₀ is equal to R₁₀₀=0.18 which gives an upper limit of 0.27 MJy/sr on the CIB intensity. An illustrative logarithmic extrapolation of R out to 60 $\mu$m gives R₆₀=0.27 leading to an estimate of the CIB intensity at 60 $\mu$m of 0.18 MJy/sr, which is significantly smaller than the previous determination of Finkbeiner et al. (2000) of 0.56 MJy/sr.

5.3 Implication for the component separation

This work suggests that the high latitude IRAS maps, in the lowest cirrus regions, cannot be used as a tracer of the interstellar extinction structure as proposed by (Schlegel et al. 1998). In fact it should be noted that it is only above an intensity of order of 10 MJy/sr at 100 $\mu$m that the CIB fluctuations are lower than the cirrus contribution at the smallest scales ( $k \sim 0.2$ arcmin^-1).

Present and future CMB observations, above 100 GHz, with high sensitivity bolometers need to remove foreground contributions (cirrus and CIB fluctuations). The CIB spectrum being significantly "colder'' than the cirrus spectrum ( ${I_{\rm cirrus}100~\rm\mu m} / {I_{\rm cirrus}1~\rm mm} \sim 30$; ${I_{\rm CIB}100~\rm\mu m} / {I_{\rm CIB} 1~\rm mm} \sim 5$), the relative contribution of the CIB will increase with wavelength. It is thus expected that at 1 mm the range in l space dominated by the CIB will be much more extended than sees at 100 $\mu$m. This question will be dealt in a forthcoming paper.

Acknowledgements

The Fond FCAR du Québec provided funds to support this research project.