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We combine the LOw-Frequency ARray (LOFAR) Two-metre Sky Survey (LoTSS)second data release (DR2) catalogue with gravitational lensing maps from theCosmic Microwave Background (CMB) to place constraints on the bias evolution ofLoTSS radio galaxies, and on the amplitude of matter perturbations. Weconstruct a flux-limited catalogue, and analyse its harmonic-spacecross-correlation with CMB lensing maps from Planck, $C_\ell^{g\kappa}$, aswell as its auto-correlation, $C_\ell^{gg}$. We explore the models describingthe redshift evolution of the large-scale radio galaxy bias, discriminatingbetween them through the combination of both $C_\ell^{g\kappa}$ and$C_\ell^{gg}$. Fixing the bias evolution, we then use these data to placeconstraints on the amplitude of large scale density fluctuations. We report thesignificance of the $C_\ell^{g\kappa}$ signal at a level of $26.6\sigma$. Wedetermine that a linear bias evolution of the form $b_g(z) = b_{g,D} / D(z)$,where $D(z)$ is the growth rate, is able to provide a good description of thedata, and measure $b_{g,D} = 1.41 \pm 0.06$ for a sample flux-limited at$1.5\,{\rm mJy}$, for scales $\ell < 250$ for $C_\ell^{gg}$, and $\ell < 500$for $C_\ell^{g\kappa}$. At the sample's median redshift, we obtain $b(z = 0.82)= 2.34 \pm 0.10$. Using $\sigma_8$ as a free parameter, while keeping othercosmological parameters fixed to the Planck values, we find fluctuations of$\sigma_8 = 0.75^{+0.05}_{-0.04}$. The result is in agreement with weak lensingsurveys, and at $1\sigma$ difference with Planck CMB constraints. We alsoattempt to detect the late-time integrated Sachs-Wolfe effect with LOFAR, butwith the current sky coverage, the cross-correlation with CMB temperature mapsis consistent with zero. Our results are an important step towards constrainingcosmology with radio continuum surveys from LOFAR and other future large radiosurveys.

Cosmology from LOFAR Two-metre Sky Survey Data Release 2: Cross-correlation with Cosmic Microwave Background