Isocurvature Constraints and Gravitational Ward Identity

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.