  Non-Gaussian (NG) statistics of the thermal Sunyaev-Zeldovich (tSZ) effect
carry significant information which is not contained in the power spectrum.
Here, we perform a joint Fisher analysis of the tSZ power spectrum and
bispectrum to verify how much the full bispectrum can contribute to improve
parameter constraints. We go beyond similar studies of this kind in several
respects: first of all, we include the complete power spectrum and bispectrum
(auto- and cross-) covariance in the analysis, computing all NG contributions;
furthermore we consider a multi-component foreground scenario and model the
effects of component separation in the forecasts; finally, we consider an
extended set of both cosmological and intra-cluster medium parameters. We show
that the tSZ bispectrum is very efficient at breaking parameter degeneracies,
making it able to produce even stronger cosmological constraints than the tSZ
power spectrum: e.g. the standard deviation on $\sigma_8$ shrinks from
$\sigma^\text{PS}(\sigma_8)=0.35$ to $\sigma^\text{BS}(\sigma_8)=0.065$ when we
consider a multi-parameter analysis. We find that this is mostly due to the
different response of separate triangle types (e.g. equilateral and squeezed)
to changes in model parameters. While weak, this shape dependence is clearly
non-negligible for cosmological parameters, and it is even stronger, as
expected, for intra-cluster medium parameters.
