AMICO galaxy clusters in KiDS-1000: Cosmological constraints and mass calibration from counts and weak lensing

Aims. We present the joint modelling of weak-lensing and count measurements of the galaxy clusters detected with the Adaptive Matched Identifier of Clustered Objects (AMICO) code, in the fourth data release of the Kilo Degree Survey (KiDS-1000). The analysed sample comprises approximately 8000 clusters that cover an effective area of 839 deg2 and extend up to a redshift of z = 0.8. This modelling provides the first mass calibration of this cluster sample, as well as the first cosmological constraints derived from it. Methods. We derived stacked cluster weak-lensing and count measurements in bins of redshift and intrinsic richness, λ*. To define the background galaxy samples for the stacked profiles, we used a combination of selections based on photometric redshifts (photo-zs) and colours. Then, based on self-organising maps, we reconstructed the true redshift distributions of the background galaxy samples. In the joint modelling of weak lensing and counts, we accounted for the systematic uncertainties arising from impurities in the background and cluster samples, biases in the cluster z and λ*, projection effects, halo orientation and miscentring, truncation of cluster halo mass distributions, matter correlated with cluster haloes, multiplicative shear bias, baryonic matter, geometric distortions in the lensing profiles, uncertainties in the theoretical halo mass function, and super-sample covariance. In addition, we employed a blinding strategy based on perturbing the cluster sample completeness. Results. The improved statistics and photometry, along with the refined analysis compared to the previous KiDS data release, KiDS-DR3, led to a halving of the uncertainties on Ωm and σ8, as we obtained $ \Omega_{\mathrm{m}}={0.218}^{+{0.024}}_{-{0.021}} $ and $ \sigma_8={0.86}^{+{0.03}}_{-{0.03}} $, despite a more extensive modelling of systematic uncertainties. The constraint on S8 ≡ σ8(Ωm/0.3)0.5, $ S_8={0.74}^{+{0.03}}_{-{0.03}} $, is in excellent agreement with recent cluster count and KiDS-1000 cosmic shear analyses, while it shows a 2.8σ tension with Planck cosmic microwave background results. The constraints on the log λ* − log M200 relation imply a mass precision of 8%, on average, which is an improvement of three percentage points compared to KiDS-DR3. In addition, the result on the intrinsic scatter of the log λ* − log M200 relation, $ \sigma_{\mathrm{intr}}={0.052}^{+{0.023}}_{-{0.015}} $, confirms that λ* is an excellent mass proxy.