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These measurements were performed across various eBOSS redshift tomographic bins with a width of \u0394<jats:italic>z<\/jats:italic>\u2004=\u20040.06. 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.<\/jats:p>\n<jats:p><jats:italic>Results.<\/jats:italic> Using the cross-correlation, we measured the isotropic BAO dilation parameter as <jats:italic>\u03b1<\/jats:italic>\u2004=\u20041.01\u2005\u00b1\u20050.11 at <jats:italic>z<\/jats:italic><jats:sub>eff<\/jats:sub>\u2004=\u20040.63. By combining four redshift slices at <jats:italic>z<\/jats:italic><jats:sub>eff<\/jats:sub>\u2004=\u20040.63,\u20060.69,\u20060.75, and 0.81, we determined a more constrained value of <jats:italic>\u03b1<\/jats:italic>\u2009=\u20090.968<jats:sub>\u22120.095<\/jats:sub><jats:sup>+0.060<\/jats:sup>. For the entire redshift range of <jats:italic>z<\/jats:italic><jats:sub>eff<\/jats:sub>\u2004=\u20040.715, we measured <jats:italic>b<\/jats:italic><jats:sub><jats:italic>C<\/jats:italic><\/jats:sub>\u2004=\u20042.64\u2005\u00b1\u20050.20 for the constant-bias model, <jats:italic>b<\/jats:italic>(<jats:italic>z<\/jats:italic>0) = <jats:italic>b<\/jats:italic><jats:sub><jats:italic>C<\/jats:italic><\/jats:sub>, and then <jats:italic>b<\/jats:italic><jats:sub><jats:italic>D<\/jats:italic><\/jats:sub>\u2004=\u20041.80\u2005\u00b1\u20050.13 for the evolving-bias model, <jats:italic>b<\/jats:italic>(<jats:italic>z<\/jats:italic>) = <jats:italic>b<\/jats:italic><jats:sub><jats:italic>D<\/jats:italic><\/jats:sub>\/<jats:italic>D<\/jats:italic>(<jats:italic>z<\/jats:italic>), with <jats:italic>D<\/jats:italic>(<jats:italic>z<\/jats:italic>) denoting the growth rate of linear structures. Additionally, we measured the clustering bias for individual redshift bins.<\/jats:p>\n<jats:p><jats:italic>Conclusions.<\/jats:italic> We detected the cross-correlation of LoTSS radio sources and eBOSS LRGs at a 9.2<jats:italic>\u03c3<\/jats:italic> statistical significance for one single redshift bin and at a 14.7<jats:italic>\u03c3<\/jats:italic> significance when the four redshift bins were combined. For the BAO signal, we achieved a significance of 2.2<jats:italic>\u03c3<\/jats:italic> for a single redshift bin, 2.7<jats:italic>\u03c3<\/jats:italic> for the combined cross-correlation and eBOSS auto-correlation, and 4<jats:italic>\u03c3<\/jats:italic> for the combined analysis of four redshift bins in the cross-correlation, when assuming a Gaussian distribution for the BAO dilation parameter.<\/jats:p>","DOI":"10.1051\/0004-6361\/202451958","type":"journal-article","created":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T18:43:52Z","timestamp":1745347432000},"page":"A58","source":"Crossref","is-referenced-by-count":2,"title":["Cosmology from LOFAR Two-metre Sky Survey Data Release 2: Cross-correlations with luminous red galaxies from 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