  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,
$N_{\mathrm{tot}}$, and the maximum initial temperature, $T_{\mathrm{max}}$. We
rule out $N_{\mathrm{tot}}\gtrsim N_{\mathrm{SM}}$ for $T_{\mathrm{max}}\sim
T_{\mathrm{Planck}}\approx10^{19}$ GeV and
$N_{\mathrm{tot}}\gtrsim10^{13}\times N_{\mathrm{SM}}$ for
$T_{\mathrm{max}}\sim10^{16}$ GeV, where $N_{\mathrm{SM}}$ is the number of
particles in the Standard Model. In the case of inflation, existing
cosmological data constrain $T_{\mathrm{max}}\lesssim10^{16}$ GeV. However,
alternative models to inflation such as bouncing cosmologies allow for
$T_{\mathrm{max}}$ near $T_{\mathrm{Planck}}$. At the energy scales we are
considering, the extra number of particles arise naturally in models of extra
dimensions.
