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