Constraints on galaxy formation from the
  cosmic-infrared-background$\,-\,$optical-imaging cross-correlation using
  $\boldsymbol{Herschel}$ 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 infrared background and cosmic optical background fluctuations. With a
combined sky area of $91\,{\rm deg}^2$ and a minimum resolved scale of
$18\,$arcsec, 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
$\gtrsim 22\,\sigma$ ($\gtrsim 18\,\sigma$ 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 approximately a factor of 10 higher than the extragalactic
signal, with the contamination being estimated using a linear regression from
several external survey maps; however, for the other fields, the contamination
is typically 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.
