Elementary Theorems Regarding Blue Isocurvature Perturbations

TL;DR

This paper presents three theorems that clarify the properties of blue isocurvature perturbations from scalar fields, providing precise formulas and insights into their spectra and gravitational effects.

Contribution

It introduces three new theorems for analyzing blue isocurvature perturbations and refines the spectrum formula for axion models, highlighting the role of Peccei-Quinn symmetry.

Findings

Derived a more accurate blue spectrum formula for axion models.

Illustrated spectra with examples including spectral breaks.

Clarified the gravitational imprint of adiabatic perturbations on isocurvature fields.

Abstract

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.