Bound on Induced Gravitational Waves during Inflation Era

Abstract: We put the upper bound on the gravitational waves (GWs) induced by the scalar-field fluctuations during the inflation. In particular, we focus on the case where the scalar fluctuations get amplified within some subhorizon scales by some mechanism during the inflation. Since the energy conservation law leads to the upper bound on the energy density of the scalar fluctuations, the amplitudes of the scalar fluctuations are constrained and therefore the induced GWs are also. Taking into account this, we derive the upper bound on the induced GWs. As a result, we find that the GW power spectrum must be $\mathcal P_h \lesssim \mathcal O(\epsilon^2 (k/k_)^2)$ up to the logarithmic factor, where $\epsilon$ is the slow-roll parameter and $k_$ is the peak scale of the scalar-field fluctuations.