Cosmological perturbations for ultra-light axion-like particles in a state of Bose-Einstein condensate

TL;DR

This paper develops a gauge-invariant relativistic framework to analyze cosmological perturbations in ultra-light axion-like particles forming a Bose-Einstein condensate, highlighting differences from perfect fluids and assessing the validity of Newtonian approximations.

Contribution

It introduces a novel gauge-invariant relativistic approach for BEC dark matter perturbations, contrasting with perfect fluid models and evaluating common approximation methods.

Findings

Negative self-coupling does not cause Laplacian instability after BEC formation.

The Newtonian quasi-static approximation is generally valid inside the Hubble radius.

The analysis excludes parametric resonance effects during the transient epoch.

Abstract

For ultra-light scalar particles like axions, dark matter can form a state of the Bose-Einstein condensate (BEC) with a coherent classical wave whose wavelength is of order galactic scales. In the context of an oscillating scalar field with mass $m$, this BEC description amounts to integrating out the field oscillations over the Hubble time scale $H^{-1}$ in the regime $m \gg H$. We provide a gauge-invariant general relativistic framework for studying cosmological perturbations in the presence of a self-interacting BEC associated with a complex scalar field. In particular, we explicitly show the difference of BECs from perfect fluids by taking into account cold dark matter, baryons, and radiation as a Schutz-Sorkin description of perfect fluids. We also scrutinize the accuracy of commonly used Newtonian treatment based on a quasi-static approximation for perturbations deep inside the Hubble radius. For a scalar field which starts to oscillate after matter-radiation equality, we show that, after the BEC formation, a negative self-coupling hardly leads to a Laplacian instability of the BEC density contrast. This is attributed to the fact that the Laplacian instability does not overwhelm the gravitational instability for self-interactions within the validity of the nonrelativistic BEC description. Our analysis does not accommodate the regime of parametric resonance which can potentially occur for a large field alignment during the transient epoch prior to the BEC formation.