The local submillimetre luminosity functions and predictions from Spitzer to Herschel

We present new determinations of the local submillimetre (submm) luminosity functions, solving the 'submm Olbers' paradox'. We also present predictions of source counts and luminosity functions in current and future far-infrared to submm surveys. Using the submm colour temperature relations from the Submillimetre Common User Bolometer Array (SCUBA) Local Universe Galaxy Survey, and the discovery of 450-mum excess emission in these galaxies, we interpolate and extrapolate the IRAS detections to make predictions of the spectral energy distributions of all 15 411 PSCz galaxies from 50 to 1300 mum. Despite the long extrapolations, we find excellent agreement with (i) the 90-mum luminosity function of Serjeant et al., (ii) the 850-mum luminosity function of Dunne et al., (iii) the millimetre-wave photometry of Andreani and Franceschini, and (iv) the asymptotic differential and integral source count predictions at 50-1300 mum by Rowan-Robinson. We find that the local 850-mum submm luminosity density converges to 7.3 +/- 0.2 x 10(19) h(65) W Hz(-1) 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-mum 8-mJy 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 that 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.