The Scatter in the Relationship between Redshift and the Radio-to-Submm Spectral Index

We derive the scatter in the relationship between redshift and radio-to-submm spectral index, alpha^{350}_{1.4}, 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 alpha^{350}_{1.4}. The scatter is roughly constant with redshift. Constant rms scatter, combined with the flattening of the mean alpha^{350}_{1.4} -- 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 alpha^{350}_{1.4} 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 for scatter in the alpha^{350}_{1.4} -- z relationship. We re-derive the cumulative redshift distribution of the 14 field galaxies in a recent submm 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 alpha^{350}_{1.4} -- z models. The normalization and shape of the redshift distribution for the faint submm sources are consistent with those expected for forming elliptical galaxies.