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.

Beyond Schrödinger-Poisson: nonrelativistic effective field theory for scalar dark matter

“…It was then generalized for curved spacetimes in ref. [24], and more specifically applied to the case of a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe, with the analysis restricted to linearized perturbations. 1 However, one important feature of dark matter is its ability to form dense, nonlinear structures due to gravitational instability in an expanding universe.…”

“…However, as discussed in ref. [24], this strategy is not very helpful for the case of curved spacetimes. An alternative approach is to remove the redundancy in eq.…”

“…(2.5) by adding a constraint on the nonrelativistic field ψ(t, x). One convenient choice for the constraint is [24] e −imtψ + e imtψ * = 0 , (2.6) where the overdot denotes a time derivative. This constraint implies an equation of motion for ψ that is first order in time derivatives.…”

Real scalar field, the nonrelativistic limit, and the cosmological expansion

“…While the authors of Ref. [37] agree with us that the nonlocal 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.…”

“…Finally, it would also be interesting to extend the discussion presented here to fermionic fields, and specifically to study the nonrelativistic limit of Majorana fermions by similar means. [37] with a similar aim. More specifically, the relation between a real relativistic scalar field and a complex, nonrelativistic scalar field in the context of a cosmologically expanding space-time is discussed there, as well.…”