7 Source counts

  
7.1 Completeness

Simulations allow us to derive the completeness, that is the ratio at a given flux between the number of added sources and the number of detected ones. The completeness is plotted in Fig. 12. Our sample is complete above 500 mJy, and is about 90% (respectively about 85%) complete above 225 mJy (respectively 180 mJy). We thus correct the surface source density for this incompleteness.

  
7.2 Malmquist-Eddington bias

Uncertainties in the flux determination introduce an excess in the number of counts, known as the Malquist-Eddington bias. We characterize it with the results of the simulations, by comparing the effect of a flux dispersion on a known input source count model: a simple power law. Figure 13 shows the ratio of an input source count model, to the simulated observations of this model. We apply the appropriate correction to the data: at 225 mJy (respectively 180 mJy) the raw counts have to be decreased by 20% (respectively 30%). We check that these values are not more sensitive than 5% (respectively 10%) at $5 \sigma_{\rm c}$ (respectively $4 \sigma _{\rm c}$) to the power law of the input model in the range 3.0-3.6.

7.3 FIRBACKsource counts

Figure 14 shows the differential source counts at 170 $\mu$m coming from the FIRBACKsurvey (3.89 sq. deg), with 106 sources between 180 ($4 \sigma$) and 2400 mJy. The horizontal error bar gives the flux uncertainty, and the vertical error bar the Poisson noise in $\sqrt{n}$ where n is the number of sources in the bin.

The statistics of sources used for source counts before any correction is given in Table 4. The integral (respectively differential) source count values are given in Table 5 (respectively Table 6). Note that for the differential counts we took only 5 sources in the last flux box, corresponding to highest fluxes (between 500 and 700 mJy).

The two points at high flux levels are compatible with no evolution since we can adjust a horizontal line inside the error bars. The slope of the differential source counts is not constant, but can reasonably be fitted by a linear of slope $3.3 \pm 0.6$ between 180 and 500 mJy.