Missing one-loop contributions in secondary gravitational waves

Abstract

We find several missing one-loop-order contributions in previous considerations about secondary gravitational waves induced at nonlinear order in cosmological perturbations. We consider a consistent perturbative expansion to third-order in cosmological perturbations, including higher-order interactions and iterative solutions ignored in the previous literature. Tensor fluctuations induced by the source with two scalar and one tensor perturbations are correlated with the first-order tensor fluctuation and thus give a one-loop-order correction to the tensor power spectrum. The missing loop correction is \textit{scale-invariant} and \textit{negative} in the superhorion region, which secondarily reduces the initial primordial tensor power spectrum prior to the horizon re-entry. Such an IR behavior is very different from the auto-spectrum of second-order induced tensor modes discussed in the previous literature and can be important for the actual gravitational wave measurements. For a sharp peak of scalar fluctuations with $A_\zeta=10^{-2}$ at $k_*=10^{5}h/{\rm Mpc}$ motivated by the LIGO/Virgo events, we show that the tensor power spectrum at the cosmic microwave background scale reduces by at most 35%. Hence, the polarization B-mode might not be seen because of the reduction of the original tensor spectrum due to the secondary effect of primordial black hole formation.