  Axions and WIMPZILLAs are well-motivated dark matter candidates with
interesting cos- mological implications, such as isocurvature perturbations and
non-Gaussianities. How- ever, these predicted implications in the literature
are based on the assumption that the cross-correlation between curvature and
CDM isocurvature is negligible. This contribution discusses the
cross-correlation in the axion and the WIMPZILLA scenarios. Particularly, it is
shown that the gravitational Ward identity associated with diffeomorphism
invariance plays an important role in this cross-correlation calculation
confirming the assumption.
