The Radio-to-Submm Spectral Index as a Redshift Indicator

We present models of the 1.4 GHz to 350 GHz spectral index, alpha(350/1.4), for starburst galaxies as a function of redshift. The models include a semi-analytic formulation, based on the well quantified radio-to-far infrared correlation for low redshift star forming galaxies, and an empirical formulation, based on the observed spectrum of the starburst galaxies M82 and Arp 220. We compare the models to the observed values of alpha(350/1.4) for starburst galaxies at low and high redshift. We find reasonable agreement between the models and the observations, and in particular, that an observed spectral index of alpha(350/1.4) > +0.5 indicates that the target source is likely to be at high redshift, z > 1. The evolution of alpha(350/1.4) with redshift is mainly due to the very steep rise in the Raleigh-Jeans portion of the thermal dust spectrum shifting into the 350 GHz band with increasing redshift. We also discuss situations where this relationship could be violated. We then apply our models to examine the putative identifications of submm sources in the Hubble Deep Field, and conclude that the submm sources reported by Hughes et al. are likely to be at high redshifts, z > 1.5.