Cosmology from LOFAR Two-metre Sky Survey Data Release 2: Cross-correlation with the cosmic microwave background

We combine 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 radio galaxies, and on the amplitude of matter perturbations. We construct a flux-limited catalogue, and analyse its harmonic-space cross-correlation with CMB lensing maps from Planck, $C_\ell^{g\kappa}$, as well as its auto-correlation, $C_\ell^{gg}$. We explore the models describing the redshift evolution of the large-scale radio galaxy bias, discriminating between them through the combination of both $C_\ell^{g\kappa}$ and $C_\ell^{gg}$. Fixing the bias evolution, we then use these data to place constraints on the amplitude of large scale density fluctuations. We report the significance of the $C_\ell^{g\kappa}$ signal at a level of $26.6\sigma$. We determine 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 the data, 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 other cosmological 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 lensing surveys, and at $1\sigma$ difference with Planck CMB constraints. We also attempt to detect the late-time integrated Sachs-Wolfe effect with LOFAR, but 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.