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 $ksim pi T$. 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 $T_{rm max} le 2times 10^{17}$ 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.
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