  We present new determinations of the local sub-mm luminosity functions,
solving the ``sub-mm Olbers' Paradox.'' We also present predictions of source
counts and luminosity functions in current and future far-infrared to sub-mm
surveys. Using the sub-mm colour temperature relations from the SCUBA Local
Universe Galaxy Survey, and the discovery of excess 450 micron excess emission
in these galaxies, we interpolate and extrapolate the IRAS detections to make
predictions of the SEDs of all 15411 PSC-z galaxies from 50-1300 microns.
Despite the long extrapolations we find excellent agreement with (a) the 90
micron luminosity function of Serjeant et al. (2001), (b) the 850 micron
luminosity function of Dunne et al. (2000), (c) the mm-wave photometry of
Andreani & Franceschini (1996); (d) the asymptotic differential and integral
source count predictions at 50-1300 microns by Rowan-Robinson (2001). We find
the local 850 micron sub-mm luminosity density converges to (7.3+/-0.2)x10^{19}
h_{65} W/Hz/Mpc^3. Remarkably, the local spectral luminosity density and the
extragalactic background light together strongly constrain the cosmic star
formation history for a wide class of evolutionary assumptions. We find that
the extragalactic background light, the 850 micron 8mJy source counts, and the
Omega_* constraints all independently point to a decline in the comoving star
formation rate at z>1. In order to reconcile this with direct determinations,
we suggest either there is a top-heavy initial mass function at high redshifts,
and/or there is stronger evolution in the more luminous far-infrared galaxies
than seen in the population as a whole.
