Modelling the luminosities and sizes of radio sources: radio luminosity function at z = 6

We present a model to predict the luminosity function for radio galaxies and their linear size distribution at any redshift. The model takes a black hole mass function and Eddington ratio distribution as input and tracks the evolution of radio sources, taking into account synchrotron, adiabatic and inverse Compton energy losses. We first test the model at z = 2 where plenty of radio data are available and show that the radio luminosity function (RLF) is consistent with observations. We are able to reproduce the break in luminosity function that separates locally the Fanaroff-Riley class I and Fanaroff-Riley class I radio sources. Our prediction for linear size distribution at z = 2 matches the observed distribution too. We then use our model to predict an RLF and linear size distribution at z = 6, as this is the epoch when radio galaxies can be used as probes of reionization. We demonstrate that higher inverse Compton losses lead to shorter source lifetimes and smaller sizes at high redshifts. The predicted sizes are consistent with the generally observed trend with redshift. We evolve the z = 2 RLF based on observed quasar space densities at high redshifts, and show that our RLF prediction at z = 6 is consistent. Finally, we predict the detection of 0.63, 0.092 and 0.0025 z ≥ 6 sources deg² at flux density limits of 0.1, 0.5 and 3.5 mJy. We assess the trade-off between coverage area and depth and show that LOFAR surveys with flux density limits of 0.1 and 0.5 mJy are the most efficient at detecting a large number of z ≥ 6 radio sources.