Statistical anisotropy of the curvature perturbation from vector field perturbations

TL;DR

This work extends the δN formalism to encompass vector field perturbations, showing how vector contributions can imprint statistical anisotropy in the primordial curvature perturbation ζ and affect both the power spectrum and bispectrum. It derives tree-level and one-loop expressions for the ζ spectrum, clarifies the role of the longitudinal vector mode, and examines generation mechanisms from vacuum fluctuations, including gauge-coupling and R-coupled scenarios. The paper applies the framework to two concrete models—vector curvaton and vector inflation—finding that a single vector field generally induces anisotropy, while a large number of randomly oriented vectors can restore isotropy; it also provides predictions for the anisotropy parameter g and tensor-to-scalar ratio in these contexts. Overall, the results offer a systematic way to confront vector-field–driven ζ with observational data, potentially revealing or constraining departures from statistical isotropy in the early Universe.

Abstract

The δN formula for the primordial curvature perturbation ζis extended to include vector as well as scalar fields. Formulas for the tree-level contributions to the spectrum and bispectrum of ζare given, exhibiting statistical anisotropy. The one-loop contribution to the spectrum of ζis also worked out. We then consider the generation of vector field perturbations from the vacuum, including the longitudinal component that will be present if there is no gauge invariance. Finally, the δN formula is applied to the vector curvaton and vector inflation models with the tensor perturbation also evaluated in the latter case.