Not R Kurvature: Beating Large-Scale White Noise

Abstract

Kurvature, a recently identified curvature invariant, has been argued to acquire superhorizon, or large-scale, white noise from hard-hard momentum coupling even when matter nonlinearities are small. If kurvature were related directly to cosmological curvature perturbations R through a Poisson equation, this white noise would cause an infrared-divergent variance sensitive to ultraviolet hard-mode physics. However, this relation does not generically hold. Kurvature is not intrinsic 3-curvature: on comoving slices it contains extrinsic-curvature terms, and intrinsic curvature is not related to R by a Poisson equation beyond linear order. We test this inference with second-order perturbation theory in radiation domination, relevant to CMB observables. Quadratic hard-hard composites do generate large-scale white noise in the kurvature density, but the Hamiltonian constraint separates it into intrinsic curvature and extrinsic shear, or equivalently density and expansion. Only the extrinsic terms carry the growing dimensionless kurvature density that mimics an ordinary density fluctuation above the horizon. The direct hard-hard curvature power is ultraviolet convergent, dominated by horizon-scale modes at evaluation, and leaves no IR relic in R from purely ultraviolet modes. By contrast, the Poisson construction of a curvature potential from kurvature is infrared divergent and cutoff sensitive; its white noise arises from extrinsic curvature associated with nonlinear acoustic beat modes in a radiation fluid.