Quadratic Isocurvature Cross-Correlation, Ward Identity, and Dark Matter

TL;DR

This paper calculates the gravitationally induced cross-correlation between isocurvature and curvature perturbations during inflation, revealing a small but persistent correlation that can serve as a test for certain dark matter models.

Contribution

It provides the first detailed computation of the gravitational contributions to isocurvature-curvature cross-correlation, demonstrating gauge invariance and renormalization in inflationary backgrounds.

Findings

Small non-decaying isocurvature-curvature cross-correlation found.

Application to QCD axions and WIMPZILLAs dark matter scenarios.

Use of gravitational Ward identity and soft-{ta} theorem in the analysis.

Abstract

Sources of isocurvature perturbations and large non-Gaussianities include field degrees of freedom whose vacuum expectation values are smaller than the expansion rate of inflation. The inhomogeneities in the energy density of such fields are quadratic in the fields to leading order in the inhomogeneity expansion. Although it is often assumed that such isocurvature perturbations and inflaton-driven curvature perturbations are uncorre- lated, this is not obvious from a direct computational point of view due to the form of the minimal gravitational interactions. We thus compute the irreducible gravitational contributions to the quadratic isocurvature-curvature cross-correlation. We find a small but non-decaying cross-correlation, which in principle serves as a consistency prediction of this large class of isocurvature perturbations. We apply our cross-correlation result to two dark matter isocurvature perturbation scenarios: QCD axions and WIMPZILLAs. On the technical side, we utilize a gravita- tional Ward identity in a novel manner to demonstrate the gauge invariance of the computation. Furthermore, the detailed computation is interpreted in terms of a soft-{\zeta} theorem and a gravitational Ward identity. Finally, we also identify explicitly all the counterterms that are necessary for renormalizing the isocurvature perturbation composite operator in inflationary cosmological backgrounds.