  We compute the production rate of the energy density carried by gravitational
waves emitted by a Standard Model plasma in thermal equilibrium, consistently
to leading order in coupling constants for momenta $k\sim \pi T$. Summing up
the contributions from the full history of the universe, the highest
temperature of the radiation epoch can be constrained by the so-called $N_{\rm
eff}$ parameter. The current theoretical uncertainty $\Delta N_{\rm eff} \le
10^{-3}$ corresponds to $T_{\rm max} \le 2\times 10^{17}$ GeV. In the course of
the computation, we show how a subpart of the production rate can be determined
with the help of standard packages, even if subsequently an IR subtraction and
thermal resummation need to be implemented.
