  (Abridged) The amplitude of the thermal Sunyaev Zel'dovich effect (tSZ) power
spectrum is extremely sensitive to the abundance of galaxy clusters and
therefore to fundamental cosmological parameters that control their growth,
such as sigma_8 and Omega_m. Here we explore the sensitivity of the tSZ power
spectrum to important non-gravitational ('sub-grid') physics by employing the
cosmo-OWLS suite of large-volume cosmological hydrodynamical simulations, run
in both the Planck and WMAP7 best-fit cosmologies. On intermediate and small
angular scales (ell > ~1000, or theta < ~10 arcmin), accessible with the South
Pole Telescope and the Atacama Cosmology Telescope, the predicted tSZ power
spectrum is highly model dependent, with AGN feedback having a particularly
large effect. However, at large scales, observable with the Planck telescope,
the effects of sub-grid physics are minor. Comparing the simulations with
observations, we find a significant amplitude offset on all measured angular
scales (including large scales), if the Planck best-fit cosmology is assumed by
the simulations. This is shown to be a generic result for all current tSZ
models. By contrast, if the WMAP7 cosmology is adopted, there is full
consistency with the Planck power spectrum measurements on large scales and
agreement at the 2 sigma level with the SPT/ACT measurements at intermediate
scales for our fiducial AGN model, which Le Brun et al. (2014) have shown
reproduces the 'resolved' properties of the local cluster population remarkably
well. These findings strongly suggest that there are significantly fewer
massive galaxy clusters than expected for the Planck best-fit cosmology, which
is consistent with recent measurements of the tSZ number counts. Our findings
therefore pose a significant challenge to the cosmological parameter values
preferred (and/or the model adopted) by the Planck primary CMB analyses.
