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 scale invariant and negative in the superhorizon region, which secondarily reduces the initial primordial tensor-power spectrum prior to the horizon reentry. Such an IR behavior is very different from the autospectrum 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 ${A}_{\ensuremath{\zeta}}={10}^{\ensuremath{-}2}$ at ${k}_{*}={10}^{5}\text{ }h/\mathrm{Mpc}$ 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.