MIGHTEE: exploring the relationship between spectral index, redshift, and radio luminosity

It has been known for many years that there is an apparent trend for the spectral index ($\alpha$) of radio sources to steepen with redshift z, which has led to attempts to select high-redshift objects by searching for radio sources with steep spectra. In this study, we use data from the MeerKAT, Low Frequency Array survey, Giant Metre-wave Radio Telescope survey (GMRT), and uGMRT telescopes, particularly using the MeerKAT International GHz Tiered Extragalactic Exploration (MIGHTEE) and superMIGHTEE surveys, to select compact sources over a wide range of redshifts and luminosities. We investigate the relationship between spectral index, luminosity and redshift and compare our results to those of previous studies. Although there is a correlation between $\alpha$ and z in our sample for some combinations of frequency where good data are available, there is a clear offset between the $\alpha$–z relations in our sample and those derived previously from samples of more luminous objects; in other words, the $\alpha$–z relation is different for low and high-luminosity sources. The relationships between $\alpha$ and luminosity are also weak in our sample but in general the most luminous sources are steeper-spectrum and this trend is extended by samples from previous studies. In detail, we argue that both a $\alpha$–luminosity relation and an $\alpha$–z relation can be found in the data, but it is the former that drives the apparent $\alpha$–z relation observed in earlier work, which only appears because of the strong redshift–luminosity relation in bright, flux density-limited samples. Steep-spectrum selection should be applied with caution in searching for high-z sources in future deep surveys.