  The angular power spectrum of the SZ effect, C_l, is a powerful probe of
cosmology. It is easier to detect than individual clusters in the field, is
insensitive to observational selection effects and does not require a
calibration between cluster mass and flux, reducing the systematic errors which
dominate the cluster-counting constraints. It receives a dominant contribution
from cluster region between 20-40% of the virial radius and is thus insensitive
to the poorly known gas physics in the cluster centre, such as cooling or
(pre)heating. In this paper we derive a refined analytic prediction for C_l
using the universal gas-density and temperature profile and the dark-matter
halo mass function. The predicted C_l has no free parameters and fits all of
the published hydrodynamic simulation results to better than a factor of two
around l=3000. We find that C_l scales as (sigma_8)^7 times (Omega_b h)^2 and
is almost independent of all of the other cosmological parameters. This differs
from the local cluster abundance studies, which give a relation between sigma_8
and Omega_m. We also compute the covariance matrix of C_l using the halo model
and find a good agreement relative to the simulations. We estimate how well we
can determine sigma_8 with sampling-variance-limited observations and find that
for a several-square-degree survey with 1-2 arcminute resolution one should be
able to determine sigma_8 to within a few percent, with the remaining
uncertainty dominated by theoretical modelling. If the recent excess of the CMB
power on small scales reported by the CBI experiment is due to the SZ effect,
then we find sigma_8(Omega_b h/0.029)^0.3 = 1.04 +- 0.12 at the 95% confidence
level (statistical) and with a residual 10% systematic (theoretical)
uncertainty.
