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

2. Counts and the far infrared background

Guided by models of galaxy formation (e.g. Toffolatti et al. 1998, Guiderdoni et al. 1998, Blain et al. 1998), we model as a double-power-law,

(see also Borys et al. 1998), where for convenience we take mJy. Such a parameterization will certainly not be valid for all values of S, but it will be adequate for our purposes. Matching to the general behaviour of successful models, and normalizing specifically to the Hubble Deep Field (HDF) counts, we obtain `fiducial' values for these parameters of , and . This fit, plus the data of Table 1, are shown in Fig. 1. Some models show the counts to steepen even more at the bright end, but this has little effect on any results, since it is already steep enough that any upper flux cut is not very important. We also show in Fig. 1 an estimate for the number of pixels over the whole sky at 353GHz, to indicate where we expect one source per pixel. We have assumed an oversampling by a factor of 10 pixels per beam as an illustrative number. With our adopted source counts model, Planck will then have about one 20 mJy source per beam (which then sets the basic level of `confusion noise').

Fig. 1. The number of objects brighter than flux as a function of . The curve is Eq. (1) with our fiducial parameters. The data are taken from Table 1, with the second and third points offset for clarity. The error bars are (68% Bayesian confidence region) errors based on Poisson statistics. The horizontal dashed line is an estimate of the number of pixels on the whole sky for the Planck 353 GHz channel, assuming 10 pixels per beam.

In practice Planck will only be able to detect individual sources with at best a flux cut of mJy using data from the 353 GHz channel alone. With extra information from the higher frequency channels (as well as information from other instruments, at least in some regions of the sky), it should be possible to remove all sources down to a few mJy. The SCUBA counts constrain the model at somewhat lower flux levels than these, however in our model the counts at the SCUBA flux levels contribute significantly to the IR background and hence the CMB fluctuations if the sources are clustered (see below). In any case the precise flux cut for Planck is not currently easy to estimate, and so we have erred on the side of conservatism; if the flux cut ends up being higher than we are assuming here, then the fluctuations will only be larger.

By describing the counts in this phenomenological way we avoid any direct modelling of galaxy formation, evolution and spectral synthesis. Currently there are too many free parameters in these `semi-analytic' models to yield a great deal of insight. Instead we prefer to use simple model fits to observables on the sky, which are motivated by the current data. Because of this we are considering only the two-dimensional distribution of objects on the sky, with no requirement on the radial distribution.

The contribution of these sources to the FIB is just the total flux per unit solid angle, or

which can be integrated by parts to yield

(a little care has to be taken with minus signs, since conventionally ). The faint end limit for constant slope is just . We show in Fig. 2 the contribution to the integrated background light as a function of . Notice that the sources at the flux levels probed by SCUBA contribute significantly to the background. As we will see below, it is those clustered sources which contribute most to the background that may be of greatest interest to us here. In Fig. 3 we show the contribution to the FIB, integrating to , i.e. the total background, as a function of the faint-end slope . The FIB was first detected by Puget et al. (1996), and has recently been measured by Fixsen et al. (1998). Their value is shown in Figs. 2, 3 as the hatched region. In our fiducial model the sub-mm sources account for all of the FIB.

Fig. 2. The integrated flux as a function of the upper flux cut, with mJy, , , and normalized to the HDF counts. The hatched region is the FIB detection of Fixsen et al. (1998). Note that in our fiducial model much of the integrated flux comes from sources near the flux levels detected by SCUBA.

Fig. 3. The total integrated flux as a function of the faint end slope, of Eq. (1), with the other parameters fixed at their fiducial values. The hatched region is the FIB detection of Fixsen et al. (1998).

Let us acknowledge that the real situation may be more complicated. The counts may come from a number of separate populations, and so of course there could be features in the actual curve. In addition there is the possibility that some more diffuse emission contributes to the FIB, and is not accounted for in these counts. Indeed there are some early indications (Hughes et al. 1998, Borys et al. 1998) that the counts may be flatter at the faint end than the form we have adopted. Again we have been conservative here; lower faint-end slopes would require higher overall normalization in order to match the background, implying stronger fluctuations.

© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999
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