# Evolution of linear wave dark matter perturbations in the radiation-dominant era

## TL;DR

This paper analyzes the evolution of wave dark matter perturbations during the radiation era, revealing distinct phases and quantum effects that cause deviations from cold dark matter predictions, with implications for axion models.

## Contribution

It provides a detailed analysis of $\psi$DM perturbations in the radiation era, identifying four evolutionary phases and quantum effects causing mode suppression, and compares axion models with $\psi$DM.

## Key findings

- Late-stage $\psi$DM perturbations resemble CDM but with subtle differences.
- Quantum effects lead to significant mode suppression in $\psi$DM.
- Axion models with near-top initial angles predict higher power spectra and cutoff frequencies.

## Abstract

Linear perturbations of the wave dark matter, or $\psi$ dark matter ($\psi$DM), of particle mass $\sim 10^{-22}$eV in the radiation-dominant era are analyzed, and the matter power spectrum at the photon-matter equality is obtained. We identify four phases of evolution for $\psi$DM perturbations, where the dynamics can be vastly different from the counterparts of cold dark matter (CDM). While in late stages after mass oscillation long-wave $\psi$DM perturbations are almost identical to CDM perturbations, some subtle differences remain, let alone intermediate-to-short waves that bear no resemblance with those of CDM throughout the whole evolutionary history. The dissimilarity is due to quantum mechanical effects which lead to severe mode suppression. We also discuss the axion model with a cosine field potential. The power spectrum of axion models are generally almost identical to those of $\psi$DM, but in the extreme case when the initial axion angle is near the field potential top, this axion model predict a power excess over a range of wave number and a higher spectral cutoff than $\psi$DM as if $\psi$DM had a higher particle mass.