Confusion of extragalactic sources in the far infrared: a baseline assessment of the performance of PRIMAger in intensity and polarization

Because of their limited angular resolution, far-infrared telescopes are usually affected by 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\,$\mu$m) sky both in intensity and polarization. We aim to provide predictions of the confusion level and its consequences for future surveys.} {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).} {In intensity \textbf{maps}, we predict that the confusion limit increases rapidly with increasing wavelength (from 21\,$\mu$Jy at 25\,$\mu$m to 46\,mJy at 235\,$\mu$m). The confusion limit in polarization \textbf{maps} is more than two orders of magnitude lower (from 0.03\,mJy at 96\,$\mu$m to 0.25\,mJy at 235\,$\mu$m). \textbf{Both in intensity and polarization maps}, the measured \textbf{(polarized)} flux density is dominated by the brightest galaxy in the beam, but other objects also contribute \textbf{in intensity maps} at longer wavelength ($\sim$30\,\% at 235\,$\mu$m). We also show that galaxy clustering has a mild impact on confusion in intensity \textbf{maps} (up to 25\,\%), while it is negligible in polarization \textbf{maps}. In intensity \textbf{maps}, a basic blind extraction will be sufficient to detect galaxies at the knee of the luminosity function up to z$\sim$3 and 10$^{11}$\,M$_\odot$ main-sequence galaxies up to z$\sim$5. In polarization for the most conservative sensitivity forecast \textbf{(payload requirements)}, $\sim$200 galaxies can be detected up to z=1.5 in two 1\,500\,h surveys covering 1\,deg$^2$ and 10\,deg$^2$. \textbf{For a conservative sensitivity estimate,} we expect $\sim$8\,000 detections up to z=2.5 opening a totally new window on the high-z dust polarization. Finally, we show that intensity surveys at short wavelength and polarization surveys at long wavelength tend to reach confusion at similar depth. There is thus a strong synergy between them.