The radio-to-submillimeter spectral index as a redshift indicator

We present models of the 1.4 to 350 GHz spectral index alpha(1.4)(350) for starburst galaxies as a function of redshift. The models include a semianalytic 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(1.4)(350) 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(1.4)(350) greater than or equal to +0.5 indicates that the target source is likely to be at high redshift, z greater than or equal to 1. The evolution of alpha(1.4)(350) with redshift is mainly due to the very steep rise in the Rayleigh-Jeans portion of the thermal dust spectrum shifting into the 350 GHz band with increasing redshift. We also discuss situations in which this relationship could be violated. We then apply our models to examine the putative identifications of submillimeter sources in the Hubble Deep Field and conclude that the submillimeter sources reported by Hughes et al. are likely to be at high redshifts, z greater than or equal to 1.5.