An optimal ALMA image of the Hubble Ultra Deep Field in the era of JWST: obscured star formation and the cosmic far-infrared background

We combine archival ALMA data targeting the Hubble Ultra Deep Field (HUDF) to produce the deepest currently attainable 1-mm maps of this key region. Our deepest map covers 4.2arcmin^2, with a beamsize of 1.49''x1.07'' at an effective frequency of 243GHz (1.23mm). It reaches an rms of 4.6uJy/beam, with 1.5arcmin^2 below 9.0uJy/beam, an improvement of >5% (and up to 50% in some regions) over the best previous map. We also make a wider, shallower map, covering 25.4arcmin^2. We detect 45 galaxies in the deep map down to 3.6sigma, 10 more than previously detected, and 39 of these galaxies have JWST counterparts. A stacking analysis on the positions of ALMA-undetected JWST galaxies with z<4 and stellar masses from 10^8.4 to 10^10.4 M_sun yields 10% more signal compared to previous stacking analyses, and we find that detected sources plus stacking contribute (10.0+/-0.5)Jy/deg^2 to the cosmic infrared background (CIB) at 1.23mm. Although this is short of the (uncertain) background level of about 20Jy/deg^2, we show that our measurement is consistent with the background if the HUDF is a mild (~2sigma) negative CIB fluctuation, and that the contribution from faint undetected objects is small and converging. In particular, we predict that the field contains about 60 additional 15uJy galaxies, and over 300 galaxies at the few uJy level. This suggests that JWST has detected essentially all of the galaxies that contribute to the CIB, as anticipated from the strong correlation between galaxy stellar mass and obscured star formation.