Effective theories for a nonrelativistic field in an expanding universe: Induced self-interaction, pressure, sound speed, and viscosity

Abstract

A massive, nonrelativistic scalar field in an expanding spacetime is usually approximated by a pressureless perfect fluid, which leads to the standard conclusion that such a field can play the role of cold dark matter. In this paper, we systematically study these approximations, incorporating subleading corrections. We provide two equivalent effective descriptions of the system, each of which offers its own advantages and insights: (i) A nonrelativistic effective field theory (EFT) with which we show that the relativistic corrections induce an effective self-interaction for the nonrelativistic field. As a byproduct, our EFT also allows one to construct the exact solution, including oscillatory behavior, which is often difficult to achieve from the exact equations. (ii) An effective (imperfect) fluid description, with which we demonstrate that, for a perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) universe: (a) The pressure is small but nonzero (and positive), even for a free theory with no tree-level self-interactions. (b) The sound speed of small fluctuations is also nonzero (and positive), reproducing already known leading-order results, correcting a subdominant term, and identifying a new contribution that had been omitted in previous analyses. (c) The fluctuations experience a negative effective bulk viscosity. The positive sound speed and the negative bulk viscosity act in favor of and against the growth of overdensities, respectively. The net effect may be considered a smoking gun for ultra-light dark matter.

Real scalar field, non-relativistic limit, and cosmological expansion

“…While the authors of ref. [33] agree with us that the non-local field transformation proposed in ref. [29] is not directly applicable for an expanding space-time, they do not generalize it in the way we will describe below, but instead work with an approximate local transformation with the problem that rapidly oscillating terms still appear in the equations of motion.…”

“…Note added: While preparing the present manuscript, we became aware of a recent preprint by Salehian, Namjoo and Kaiser [33] with a similar aim. More specific, the relation between a real relativistic scalar field and a complex, non-relativistic scalar field in the context of a cosmologically expanding space-time is discussed there, as well.…”

Shadow of black hole surrounded by magnetized plasma: Axion-plasmon cloud

“…Besides, the discovery of the Higgs particle is potentially in favor of other scalars since it is in essence the proof for the existence of a spin-zero boson that emerges from a spontaneous symmetry breaking [41]. Meantime, it is essential to make a theoretical framework suitable for low-energy phenomena related to CDM that can safely address the non-relativistic limit of relativistic quantum field theories [42][43][44][45][46][47][48].…”