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Astron. Astrophys. 346, 1-6 (1999)

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3. Power spectrum for sources

We shall quote our results in `temperature' units as is usual in CMB anisotropy studies. For fluctuations about a mean, the conversion factor from temperature to intensity (or flux) is


where [FORMULA] is the Planck function, k is Boltzmann's constant, [FORMULA]GHz is the `dimensionless frequency' and [FORMULA].

If we assume that the number of sources of a given flux is independent of the number at a different flux, and if the angular two-point function of the point-sources is [FORMULA], then the angular power spectrum, [FORMULA], contributed by these sources is


assuming that all sources with [FORMULA] are removed. Here, [FORMULA] is the background contributed by sources below [FORMULA] as before. Following the conventional notation (essentially introduced by Peebles 1973), [FORMULA] is the Legendre transform of the correlation function [FORMULA] produced by the sources and [FORMULA] is the Legendre transform of [FORMULA]:


with [FORMULA] the Legendre polynomial of order [FORMULA]. The first term in Eq. (5) is the usual Poisson shot-noise term (see Peebles 1980Sect. 46, or Tegmark & Efstathiou 1996), the second is due to clustering, assuming that the clustering is independent of flux.

Integrating by parts the Poisson term can be rewritten


At low flux it becomes [FORMULA], if the counts have slope [FORMULA] at the faint end.

The shot-noise component of the angular power spectrum of our fiducial model is shown in Fig. 4, along with the primary CMB signal and the projected noise in the Planck 353 GHz channel. The Poisson fluctuations are calculated using our fiducial model and assuming that we cut out all sources brighter than [FORMULA] mJy. Note that much of the fluctuation power comes from sources fainter than 50 mJy. The uncertainty in the normalization of these curves is directly proportional to our uncertainty in the counts, [FORMULA], which we have normalized to the data of Table 1. Clearly [FORMULA] is uncertain to at least a factor of 2. However, increasing the normalization by this amount would overproduce the FIB unless the faint end slope is also modified.

[FIGURE] Fig. 4. The Poisson component of the angular power spectrum of the point sources, in dimensionless units, for a range of flux cuts (i.e. removing all sources brighter than [FORMULA]). Higher flux cuts give larger fluctuation levels. To compare with the level of primary anisotropy expected, the prediction for a standard CDM spectrum is also shown, normalized to COBE. The thick solid line is the expected contribution to the power spectrum from noise in the 353 GHz channel of the Planck HFI.

As an aside we mention that though the noise and Poisson component of the sources appear to have similar power spectra, they are nonetheless quite different entities. The sources are on the sky, and thus contribute to the flux in every observation of that pixel, whereas the noise varies from observation to observation and by assumption is uncorrelated with the signal in the pixel observed. Given many observations of a given direction on the sky (as expected for Planck), the noise properties can be separated from the sky signal, even if they have the same power spectra. The estimated instrumental noise can be subtracted from the measured power spectrum, with any point source contribution being evident as an extra [FORMULA] component. We therefore expect that analysis of the Planck data set will include fitting for an excess white noise component, which would be most likely due to unclustered point sources.

We now turn to the other contribution to Eq. (5). Unfortunately there is essentially no information about the clustering of the SCUBA sources at present. Hence the most conservative assumption would be no clustering at all, but that is obviously unreasonable. Hence, although we tried to be as conservative as possible when discussing the shot-noise power spectrum, for clustering we will just make some simple guess. If we approximate [FORMULA] then [FORMULA]. Assuming either that the sources cluster like galaxies today or as Lyman-break galaxies (Giavalisco et al. 1998) at [FORMULA], we would expect [FORMULA]-0.9. As an illustrative example we can assume that the sources which make up much of the FIB cluster like these Lyman-break galaxies at [FORMULA]. We suspect that this may in fact be close to reality, since it is equivalent to assuming that the population is a highly biased one, collapsing early. On the other hand the SCUBA sources will span a wider redshift range than galaxies selected by the UV-dropout technique, thereby washing out the angular correlations to some extent. But, certainly the SCUBA-type sources are likely to be more highly clustered than IRAS galaxies at low redshift. Our example can be probably considered an optimistic one in terms of clustered power. If the objects are less biased than the Lyman-break galaxies (LBGs) by a factor of [FORMULA] then one reduces [FORMULA], and hence the clustered contribution to [FORMULA], by [FORMULA].

With our assumption [FORMULA] and


On large angles [FORMULA] is expected to drop below the power-law behaviour assumed in Eq. (9). The scale of non-linearity approximately marks this transition. If we assume the power spectrum is a power-law with index n then [FORMULA]. Thus we expect that on larger angular scales [FORMULA] will flatten and gradually turn over to [FORMULA]. To take this into account we cause [FORMULA] to become constant 2 for [FORMULA], where we have chosen this multipole because it is a round number, and not because we think it has any physical significance. Explicitly we approximate using


We show the amplitude of the clustering signal in Fig. 5. Note that the contribution due to source clustering dominates over the Poisson term on the range of angular scales relevant to CMB anisotropies. In fact if the clustering proves to be this strong then Planck may be able to measure the power spectrum of the IR sources over a range of angular scales. We imagine that such a component will be included as one of the foreground templates to be fitted for in a full Planck analysis - the template would include frequency dependence, as well as a power spectrum which is white noise with perhaps one or two additional parameters to describe the clustering part. Exactly how to model this component may change as we learn more from SCUBA and other instruments. Further understanding of the clustering of these sources is clearly an important direction for future research.

[FIGURE] Fig. 5. As in Fig. 4, showing a flux cut of 100 mJy only, and including the component due to clustering. We have modelled the clustering using Eq. (10).

One other issue is the variance in these power spectra estimates. Assuming that the point sources are a Poisson sample, it is straightforward to estimate the cosmic variance associated with this component of the angular power spectrum 3. The variance is a sum of two terms, one due to the finite number of modes sampled by any given [FORMULA] and the other from the Poisson nature of the process. So we have


The first term of Eq. (12) is the usual result for Gaussian fluctuations, the second term is the extra variance associated with the Poisson sampling. If only a fraction [FORMULA] of the sky is observed then the first term is increased by [FORMULA] while the Poisson term is unchanged. For now it seems safe to assume that the uncertainty in our modelling of the sources (i.e. the normalization and shape of [FORMULA]) is larger than the estimate of Eq. (12), so we can neglect the latter.

We expect the intensity of these sources to decrease for wavelengths longward of [FORMULA]m. Assuming the flux decreases as [FORMULA], similar to the slope of the FIB, we have calculated the contribution at the next lowest Planck frequency: 217 GHz. As expected the contribution is considerably lower, as shown in Fig. 6. If we use information on the spectra of individual galaxies which have been detected by SCUBA, (e.g. Ivison et al. 1998, Hughes et al. 1998), then the derived slope may be steeper still, leading to lower 217 GHz contributions. Obviously at even lower frequencies, the signal will be correspondingly reduced. At the next higher frequency channel, 545 GHz, the sources are obviously brighter. We find that the Poisson contribution with a 100 mJy cut dominates the noise by more than 2 orders of magnitude over the range [FORMULA], with the clustered contribution potentially larger still.

[FIGURE] Fig. 6. As in Fig. 5, except at 217 GHz assuming the fluxes scale as [FORMULA]. We have kept the flux cut at 100 mJy to isolate the effect of changing frequency, however in principle the higher frequency channels could be used to isolate sources with 217 GHz flux substantially less than 100 mJy. This would lower the power spectra even further.

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Online publication: May 6, 1999