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 completeness of the sample
and our construction of a likelihood analysis. Using a relation between mass
M and SZ signal Y calibrated to X-ray measurements, we derive constraints
on the power spectrum amplitude σ8 and matter density parameter
Ωm in a flat ΛCDM model. We test the robustness of
our estimates and find that possible biases in the Y--M relation and the
halo mass function are larger than the statistical uncertainties from the
cluster sample. Assuming the X-ray determined mass to be biased low relative to
the true mass by between zero and 30%, motivated by comparison of the observed
mass scaling relations to those from a set of numerical simulations, we find
that σ8=0.75±0.03, Ωm=0.29±0.02, and
σ8(Ωm/0.27)0.3=0.764±0.025. The value of
σ8 is degenerate with the mass bias; if the latter is fixed to a value
of 20% we find σ8(Ωm/0.27)0.3=0.78±0.01 and a
tighter one-dimensional range σ8=0.77±0.02. We find that the larger
values of σ8 and Ωm preferred by Planck's
measurements of the primary CMB anisotropies can be accommodated by a mass bias
of about 40%. Alternatively, consistency with the primary CMB constraints can
be achieved by inclusion of processes that suppress power on small scales
relative to the ΛCDM model, such as a component of massive neutrinos
(abridged).Comment: 20 pages, accepted for publication by A&