Infrared Universality of Scalar Induced Gravitational Waves Beyond Second Order

Abstract

Primordial black hole dark matter is formed from the collapse of enhanced primordial density fluctuations. The formation of primordial black holes is accompanied by scalar induced gravitational waves whose infrared scaling only depends on the equation of state, $w$, of the background at leading order. Establishing whether this characteristic scaling remains robust under corrections of arbitrary perturbative order, including in the presence of non-Gaussianities, requires a physical understanding of the infrared scaling. In this work, we provide such a physical interpretation. We find that the stress of sound waves with nearly equal frequencies produces a slow beat. Although the incoming waves oscillate rapidly, this slow beat continues to source the long wavelength tensor perturbation before its horizon entry, thus generating an infrared scaling that only depends on the expanding background. We show that higher order interactions generate freely propagating waves that finally lead to the same slow beat deep inside the horizon. Since non-Gaussianities would not change the speed of the freely propagating waves, they do not modify the infrared scaling. As the source with a slow beat has enough time to accumulate before the horizon entry of the infrared tensor mode, it records the background expansion and thus the infrared scaling could provide a robust probe of the cosmological equation of state.