The Scatter in the Relationship between Redshift and the Radio‐to‐Submillimeter Spectral Index

We derive the scatter in the relationship between redshift and the radio-to-submillimeter spectral index, α , using the observed spectral energy distributions of 17 low-redshift star-forming galaxies. A mean galaxy model is derived, along with the rms scatter in α . The scatter is roughly constant with redshift. Constant rms scatter, combined with the flattening of the mean α - z relationship with increasing redshift, leads to increasing uncertainty for redshift estimates at high redshifts. Normalizing by the dust temperature in the manner proposed by Blain decreases the scatter in α for most of the sample, but does not remove outliers, and free-free absorption at rest frequencies above 1.4 GHz is not likely to be a dominant cause of scatter in the α - z relationship. We rederive the cumulative redshift distribution of the 14 field galaxies in a recent submillimeter and radio source sample of Smail et al. The most likely median redshift for the distribution is 2.7, with a conservative lower limit of z = 2, as was also found by Smail et al. based on the original models. The normalization and shape of the redshift distribution for the faint submillimeter sources are consistent with those expected for forming elliptical galaxies.