Flux dependence of redshift distribution and clustering of LOFAR radio sources

In this work we study the flux density dependence of the redshift distribution of low-frequency radio sources observed in the LOFAR Two-metre Sky Survey (LoTSS) deep fields and apply it to estimate the clustering length of the large-scale structure of the Universe, examining flux density limited samples (1 mJy, 2 mJy, 4 mJy and 8 mJy) of LoTSS wide field radio sources. We utilise and combine the posterior probability distributions of photometric redshift determinations for LoTSS deep field observations from three different fields (Bo\"otes, Lockman hole and ELAIS-N1, together about $26$ square degrees of sky), which are available for between $91\%$ to $96\%$ of all sources above the studied flux density thresholds and observed in the area covered by multi-frequency data. We estimate uncertainties by a bootstrap method. We apply the inferred redshift distribution on the LoTSS wide area radio sources from the HETDEX field (LoTSS-DR1; about $424$ square degrees) and make use of the Limber approximation and a power-law model of three dimensional clustering to measure the clustering length, $r_0$, for various models of the evolution of clustering. We find that the redshift distributions from all three LoTSS deep fields agree within expected uncertainties. We show that the radio source population probed by LoTSS at flux densities above $1$ mJy has a median redshift of at least $0.9$. At $2$ mJy, we measure the clustering length of LoTSS radio sources to be $r_0 = (10.1\pm 2.6) \ h^{-1}$Mpc in the context of the comoving clustering model. Our findings are in agreement with measurements at higher flux density thresholds at the same frequency and with measurements at higher frequencies in the context of the comoving clustering model.