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Aims. We combined the LOw-Frequency ARray (LOFAR) Two-metre Sky Survey (LoTSS) second data release (DR2) catalogue with gravitational lensing maps from the cosmic microwave background (CMB) to place constraints on the bias evolution of LoTSS-detected radio galaxies, and on the amplitude of matter perturbations. Methods. We constructed a flux-limited catalogue from LoTSS DR2, and analysed its harmonic-space cross-correlation with CMB lensing maps fromPlanck,C_ℓ^gk, as well as its auto-correlation,C_ℓ^gg. We explored the models describing the redshift evolution of the large-scale radio galaxy bias, discriminating between them through the combination of bothC_ℓ^gkandC_ℓ^gg. Fixing the bias evolution, we then used these data to place constraints on the amplitude of large-scale density fluctuations, parametrised byσ₈. Results. We report the significance of theC_ℓ^gksignal at a level of 26.6σ. We determined that a linear bias evolution of the formb_g(z) =b_g,D/D(z), whereD(z) is the growth rate, is able to provide a good description of the data, and we measuredb_g,D= 1.41 ± 0.06 for a sample that is flux limited at 1.5 mJy, for scalesℓ< 250 forC_ℓ^gg, andℓ< 500 forC_ℓ^gk. At the sample’s median redshift, we obtainedb(z= 0.82) = 2.34 ± 0.10. Usingσ₈as a free parameter, while keeping other cosmological parameters fixed to thePlanckvalues, we found fluctuations of σ₈= 0.75_−0.04^+0.05. The result is in agreement with weak lensing surveys, and at 1σdifference withPlanckCMB constraints. We also attempted to detect the late-time-integrated Sachs-Wolfe effect with LOFAR data; however, with the current sky coverage, the cross-correlation with CMB temperature maps is consistent with zero. Our results are an important step towards constraining cosmology with radio continuum surveys from LOFAR and other future large radio surveys. 

ABSTRACT Covering $$\sim 5600\, \deg ^2$$ to rms sensitivities of ∼70−100 $$\mu$$Jy beam−1, the LOFAR Two-metre Sky Survey Data Release 2 (LoTSS-DR2) provides the largest low-frequency (∼150 MHz) radio catalogue to date, making it an excellent tool for large-area radio cosmology studies. In this work, we use LoTSS-DR2 sources to investigate the angular two-point correlation function of galaxies within the survey. We discuss systematics in the data and an improved methodology for generating random catalogues, compared to that used for LoTSS-DR1, before presenting the angular clustering for ∼900 000 sources ≥1.5 mJy and a peak signal-to-noise ≥ 7.5 across ∼80 per cent of the observed area. Using the clustering, we infer the bias assuming two evolutionary models. When fitting angular scales of $$0.5 \le \theta \lt 5{^\circ }$$, using a linear bias model, we find LoTSS-DR2 sources are biased tracers of the underlying matter, with a bias of $$b_{\rm C}= 2.14^{+0.22}_{-0.20}$$ (assuming constant bias) and $$b_{\rm E}(z=0)= 1.79^{+0.15}_{-0.14}$$ (for an evolving model, inversely proportional to the growth factor), corresponding to $$b_{\rm E}= 2.81^{+0.24}_{-0.22}$$ at the median redshift of our sample, assuming the LoTSS Deep Fields redshift distribution is representative of our data. This reduces to $$b_{\rm C}= 2.02^{+0.17}_{-0.16}$$ and $$b_{\rm E}(z=0)= 1.67^{+0.12}_{-0.12}$$ when allowing preferential redshift distributions from the Deep Fields to model our data. Whilst the clustering amplitude is slightly lower than LoTSS-DR1 (≥2 mJy), our study benefits from larger samples and improved redshift estimates.