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: ${\lim }_{k\to 0}\left(k^2{R}_2\right)=0$, 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.

Comments: 25 pages, 1 figure