  Blue CDM-photon isocurvature perturbations are attractive in terms of
observability and may be typical from the perspective of generic mass relations
in supergravity. We present and apply three theorems useful for blue
isocurvature perturbations arising from linear spectator scalar fields. In the
process, we give a more precise formula for the blue spectrum associated with
the axion model of 0904.3800, which can in a parametric corner give a factor of
O(10) correction. We explain how a conserved current associated with
Peccei-Quinn symmetry plays a crucial role and explicitly plot several example
spectra including the breaks in the spectra. We also resolve a little puzzle
arising from a naive multiplication of isocurvature expression that sheds light
on the gravitational imprint of the adiabatic perturbations on the fields
responsible for blue isocurvature fluctuations.
