  Thermal Sunyaev-Zeldovich (tSZ) power spectrum is one of the most sensitive
methods to constrain cosmological parameters, scaling as the amplitude
$\sigma_8^8$. It is determined by the integral over the halo mass function
multiplied by the total pressure content of clusters, and further convolved by
the cluster gas pressure profile. It has been shown that various feedback
effects can change significantly the pressure profile, possibly even pushing
the gas out to the virial radius and beyond, strongly affecting the tSZ power
spectrum at high $l$. Energetics arguments and SZ-halo mass scaling relations
suggest feedback is unlikely to significantly change the total pressure
content, making low $l$ tSZ power spectrum more robust against feedback
effects. Furthermore, the separation between the cosmic infrared background
(CIB) and tSZ is more reliable at low $l$. Low $l$ modes are however probing
very small volumes, giving rise to very large non-gaussian sampling variance
errors. By computing the trispectrum contribution we identify $90<l<350$ as the
minimum variance scale where the combined error is minimized. Using the
measurement at this $l$ provides constraints that are almost as strong as using
the entire power spectrum. We extend the work by the Planck collaboration by
including a full trispectrum and feedback effects in the analysis. We perform a
Markov-Chain Monte Carlo analysis over the two dimensional parameter space and
find constraints on $\sigma_8$ by marginalizing over the feedback nuisance
parameter.We obtain $\sigma_8
=0.820^{+0.021}_{-0.009}\left(\Omega_m/0.31\right)^{0.4}$, when fixing other
parameters to Planck cosmology values. Our results suggest it is possible to
obtain competitive cosmological constraints from tSZ without cluster redshift
information, and that the current tSZ power spectrum shows no evidence for a
low amplitude of $\sigma_8$.
