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 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 ζ = 10 −2 at k * = 10 5 h/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.

Ruling Out Primordial Black Hole Formation From Single-Field Inflation

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Primordial Gravitational Wave- and Curvature Perturbation-Induced Energy Density Perturbations

“…Using Equations ( 28)- (32), we can calculate the power spectra of second-order induced scalar and energy density perturbations. The induced density perturbations will affect many physical processes at small scales, such as the formation of PBH [68] and high-order GW background [59]. By observing PBHs or higher-order GWs, we can constrain the power spectra of second-order induced density perturbations, thereby constraining inflationary models and the physical properties of primordial scalar and gravitational waves at small scales.…”

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