The local sub-mm luminosity functions and predictions from Spitzer to   Herschel
  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. 
Stephen Serjeant (Kent) and Diana Harrison (Imperial College;   Cavendish, Cambridge)