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Aims.Because of their limited angular resolution, far-infrared telescopes are usually affected by the confusion phenomenon. Since several galaxies can be located in the same instrumental beam, only the brightest objects emerge from the fluctuations caused by fainter sources. The PRobe far-Infrared Mission for Astrophysics imager (PRIMAger) will observe the mid- and far-infrared (25–235 μm) sky both in intensity and polarization. We aim to provide predictions of the confusion level and its consequences for future surveys. Methods.We produced simulated PRIMAger maps affected only by the confusion noise using the simulated infrared extragalactic sky (SIDES) semi-empirical simulation. We then estimated the confusion limit in these maps and extracted the sources using a basic blind extractor. By comparing the input galaxy catalog and the extracted source catalog, we derived various performance metrics as completeness, purity, and the accuracy of various measurements (e.g., the flux density in intensity and polarization or the polarization angle). Results.In intensity maps, we predict that the confusion limit increases rapidly with increasing wavelength (from 21 μJy at 25 μm to 46 mJy at 235 μm). The confusion limit in polarization maps is more than two orders of magnitude lower (from 0.03 mJy at 96 μm to 0.25 mJy at 235 μm). Both in intensity and polarization maps, the measured (polarized) flux density is dominated by the brightest galaxy in the beam, but other objects also contribute in intensity maps at longer wavelengths (∼30% at 235 μm). We also show that galaxy clustering has a mild impact on confusion in intensity maps (up to 25%), while it is negligible in polarization maps. In intensity maps, a basic blind extraction will be sufficient to detect galaxies at the knee of the luminosity function up toz ∼ 3 and 10¹¹M_⊙main-sequence galaxies up toz ∼ 5. In polarization for the most conservative sensitivity forecast (payload requirements), ∼200 galaxies can be detected up toz = 1.5 in two 1500 h surveys covering 1 deg²and 10 deg². For a conservative sensitivity estimate, we expect ∼8000 detections up toz = 2.5, opening a totally new window on the high-zdust polarization. Finally, we show that intensity surveys at short wavelengths and polarization surveys at long wavelengths tend to reach confusion at similar depth. There is thus a strong synergy between them. 

We present the first [C II] 158 μ m luminosity function (LF) at z  ∼ 5 from a sample of serendipitous lines detected in the ALMA Large Program to INvestigate [C II] at Early times (ALPINE). A study of the 118 ALPINE pointings revealed several serendipitous lines. Based on their fidelity, we selected 14 lines for the final catalog. According to the redshift of their counterparts, we identified eight out of 14 detections as [C II] lines at z  ∼ 5, along with two as CO transitions at lower redshifts. The remaining four lines have an elusive identification in the available catalogs and we considered them as [C II] candidates. We used the eight confirmed [C II] and the four [C II] candidates to build one of the first [C II] LFs at z  ∼ 5. We found that 11 out of these 12 sources have a redshift very similar to that of the ALPINE target in the same pointing, suggesting the presence of overdensities around the targets. Therefore, we split the sample in two (a “clustered” and “field” subsample) according to their redshift separation and built two separate LFs. Our estimates suggest that there could be an evolution of the [C II] LF between z  ∼ 5 and z  ∼ 0. By converting the [C II] luminosity to the star-formation rate, we evaluated the cosmic star-formation rate density (SFRD) at z  ∼ 5. The clustered sample results in a SFRD ∼10 times higher than previous measurements from UV–selected galaxies. On the other hand, from the field sample (likely representing the average galaxy population), we derived a SFRD ∼1.6 higher compared to current estimates from UV surveys but compatible within the errors. Because of the large uncertainties, observations of larger samples will be necessary to better constrain the SFRD at z  ∼ 5. This study represents one of the first efforts aimed at characterizing the demography of [C II] emitters at z  ∼ 5 using a mm selection of galaxies.