Constraints on galaxy formation from the cosmic-far-infrared-background\,$-$\,optical-imaging cross-correlation using \textit{Herschel} and UNIONS

Using {\it Herschel}-SPIRE imaging and the Canada-France Imaging Survey (CFIS) Low Surface Brightness data products from the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS), we present a cross-correlation between the cosmic far-infrared background and cosmic optical background fluctuations. The cross-spectrum is measured for two cases: all galaxies are kept in the images; or all individually-detected galaxies are masked to produce `background' maps. We report the detection of the cross-correlation signal at $\gtrsim 18\,\sigma$ ($\gtrsim 14\,\sigma$ for the background map). The part of the optical brightness variations that are correlated with the submm emission translates to an rms brightness of $\simeq 32.5\,{\rm mag}\,{\rm arcsec}^{-2}$ in the $r$ band, a level normally unreachable for individual sources. A critical issue is determining what fraction of the cross-power spectrum might be caused by emission from Galactic cirrus. For one of the fields, the Galactic contamination is 10 times higher than the extragalactic signal; however, for the other fields, the contamination is around 20~per cent. An additional discriminant is that the cross-power spectrum is of the approximate form $P(k)\propto 1/k$, much shallower than that of Galactic cirrus. We interpret the results in a halo-model framework, which shows good agreement with independent measurements for the scalings of star-formation rates in galaxies. The approach presented in this study holds great promise for future surveys such as FYST/CCAT-prime combined with {\it Euclid} or the Vera Rubin Observatory (LSST), which will enable a detailed exploration of the evolution of star formation in galaxies.