Planck 2013 results. XX. Cosmology from Sunyaev-Zeldovich cluster counts

We present constraints on cosmological parameters using number counts as a function of redshift for a sub-sample of 189 galaxy clusters from the Planck SZ (PSZ) catalogue. The PSZ is selected through the signature of the Sunyaev--Zeldovich (SZ) effect, and the sub-sample used here has a signal-to-noise threshold of seven, with each object confirmed as a cluster and all but one with a redshift estimate. We discuss the calculation of the expected cluster counts as a function of cosmological parameters, the completeness of the sample, and the likelihood construction method. Using a relation between mass M and SZ signal Y based on comparison to X-ray measurements, we derive constraints on the power spectrum amplitude sigma\_8 and matter density parameter $\backslash$Omega\_m in a flat LCDM model. We test the robustness of our estimates and find that possible biases in the Y-M relation and the halo mass function appear larger than the statistical uncertainties from the cluster sample. Assuming a bias between the X-ray determined mass and the true mass of 20\%, motivated by comparison of the observed mass scaling relations to those from a set of numerical simulations, we find that sigma\_8(Omega\_m/0.27)\^{}0.3=0.78+-0.01, with one-dimensional ranges sigma\_8=0.77+-0.02 and Omega\_m=0.29+-0.02. The values of the cosmological parameters are degenerate with the mass bias, and it is found that the larger values of sigma\_8 and Omega\_m preferred by the Planck's measurements of the primary CMB anisotropies can be accommodated by a mass bias of about 45\%. Alternatively, consistency with the primary CMB constraints can be achieved by inclusion of processes that suppress power on small scales, such as a component of massive neutrinos. We place our results in the context of other determinations of cosmological parameters, and discuss issues that need to be resolved in order to make progress in this field.