Any plasma in thermal equilibrium emits gravitational waves, caused by physical processes such as macroscopic hydrodynamic fluctuations and microscopic particle collisions. We will show that, for the largest wavelengths, the emission rate is due to the former process and is proportional to the shear viscosity of the plasma. In the Standard Model at T > 160 GeV, the shear viscosity is dominated by the most weakly interacting particles, right-handed leptons, and is relatively large. We estimate the order of magnitude of the corresponding spectrum of gravitational waves. At smaller wavelengths the leading contribution is given by particle collisions, which we also estimate at leading logarithmic order. Even though at small frequencies (corresponding to the sub-Hz range relevant for planned observatories such as eLISA) this SM background is tiny compared with that from non-equilibrium sources, we conclude that the total energy carried by the high-frequency part of the spectrum is non-negligible if the production continues for a long time. Finally, we suggest that this may constrain (weakly) the highest temperature of the radiation epoch. Observing the high-frequency part directly sets a very ambitious goal for future generations of GHz-range detectors.
