The Fate of Large Scale White Noise in Second-Order Cosmological Perturbation Theory

Abstract

Working to second order in cosmological perturbation theory, we reconsider the Large Scale White Noise (LSWN) effect proposed in [2511.13866] and [2511.15803]. We demonstrate that the kurvature density variable $\Delta \rho$ does indeed develop LSWN at second order. However, contrary to the claims of [2511.15803], we show that this is not inherited as an infrared (IR) pole in the second-order comoving curvature perturbation $R_2$. This apparent enhancement was an artifact of using a linear Poisson equation to relate two genuinely second order quantities. Further, we show that $R_2$ is protected from developing any such IR enhancement: $\lim_{k \rightarrow 0} \left( k^2 R_2 \right) = 0$, which follows from a conservation law in the soft limit. The constraint proposed in [2511.15803] and its implications for the small-scale primordial power spectrum are therefore invalid.