Cosmology from LOFAR Two-metre Sky Survey Data Release 2: Cross-correlations with luminous red galaxies from eBOSS

We cross-correlated galaxies from the LOw-Frequency ARray (LOFAR) Two-metre Sky Survey (LoTSS) second data release (DR2) radio source with the extended Baryon Oscillation Spectroscopic Survey (eBOSS) luminous red galaxy (LRG) sample to extract the baryon acoustic oscillation (BAO) signal and constrain the linear clustering bias of radio sources in LoTSS DR2. In the LoTSS DR2 catalogue, employing a flux density limit of $1.5$ mJy at the central LoTSS frequency of 144 MHz and a signal-to-noise ratio (S/N) of $7.5$, additionally considering eBOSS LRGs with redshifts between 0.6 and 1, we measured both the angular LoTSS-eBOSS cross-power spectrum and the angular eBOSS auto-power spectrum. These measurements were performed across various eBOSS redshift tomographic bins with a width of $\Delta z=0.06$. By marginalising over the broadband shape of the angular power spectra, we searched for a BAO signal in cross-correlation with radio galaxies, and determine the linear clustering bias of LoTSS radio sources for a constant-bias and an evolving-bias model. Using the cross-correlation, we measured the isotropic BAO dilation parameter as $\alpha=1.01\pm 0.11$ at $z_{\rm eff}=0.63$. By combining four redshift slices at $z_{\rm eff}=0.63, 0.69, 0.75$, and $0.81$, we determined a more constrained value of $\alpha = 0.968^{+0.060}_{-0.095}$. For the entire redshift range of $z_{\rm eff}=0.715$, we measured $b_C = 2.64 \pm 0.20$ for the constant-bias model, $b(z)=b_C$, and then $b_D = 1.80 \pm 0.13$ for the evolving-bias model, $b(z) = b_D / D(z)$, with $D(z)$ denoting the growth rate of linear structures. Additionally, we measured the clustering bias for individual redshift bins.