  The thermal plasma filling the early universe generated a stochastic
gravitational wave background that peaks in the microwave frequency range
today. If the graviton production rate is expressed as a series in a
fine-structure constant, $\alpha$, and the temperature over the Planck mass,
$T^2_{ } / m_{\rm pl}^2$, then the lowest-order contributions come from single
($\sim \alpha T^2_{ }/m_{\rm pl}^2$) and double ($\sim T^4_{ }/m_{\rm pl}^4$)
graviton production via $2\to 2$ scatterings. We show that in the Standard
Model, single-graviton production dominates if the maximal temperature is
smaller than $4\times 10^{18}_{ }$ GeV. This justifies previous calculations
which relied solely on single-graviton production. We mention Beyond the
Standard Model scenarios in which the single and double-graviton contributions
could be of comparable magnitudes. Finally, we elaborate on what these results
imply for the range of applicability of General Relativity as an effective
theory.
