An Upper Limit on the Initial Temperature of the Radiation-Dominated Universe

Gravitational waves (GWs) are produced by colliding particles through the gravitational analogue of electromagnetic bremsstrahlung. We calculate the contribution of free-free emission in the radiation-dominated Universe to the stochastic GW background. We find that the energy density of the resulting GW radiation is heavily dependent on the number of elementary particles, $N_{\mathrm{t}\mathrm{o}\mathrm{t}}$, and the maximum initial temperature, $T_{\mathrm{m}\mathrm{a}\mathrm{x}}$. We rule out $N_{\mathrm{t}\mathrm{o}\mathrm{t}}\gtrsim N_{\mathrm{S}\mathrm{M}}$ for $T_{$\mathrm{m}\mathrm{a}\mathrm{x}$}\sim T_{$\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{k}$}\approx10^{19}$GeVand$N_{$\mathrm{t}\mathrm{o}\mathrm{t}$}\gtrsim10^{13}\times N_{$\mathrm{S}\mathrm{M}$}$for$T_{$\mathrm{m}\mathrm{a}\mathrm{x}$}\sim10^{16}$GeV,where$N_{$\mathrm{S}\mathrm{M}$}$ is the number of particles in the Standard Model. In the case of inflation, existing cosmological data constrain $T_{\mathrm{m}\mathrm{a}\mathrm{x}}\lesssim {10}^{16}$ GeV. However, alternative models to inflation such as bouncing cosmologies allow for $T_{\mathrm{m}\mathrm{a}\mathrm{x}}$ near $T_{\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{k}}$. At the energy scales we are considering, the extra number of particles arise naturally in models of extra dimensions.