Relativistic Corrections and Structure Formation in Dark Matter Superfluidity

TL;DR

This work extends a non-relativistic dark matter superfluid model to a relativistic cosmological setting and analyzes linear perturbations in FLRW spacetime to assess structure formation. It develops both the relativistic background and a first-order perturbation theory, examining how density fluctuations evolve under phonon dynamics and gravitational coupling. A key finding is that, without baryon coupling and in the weak-field limit, perturbations are suppressed, but in the large-mass limit the model naturally reproduces ΛCDM growth with δ ~ a, indicating compatibility with observed large-scale structure. Overall, the study supports the viability of a unified framework where MOND-like galactic phenomenology emerges from a DM superfluid while cosmology remains consistent with ΛCDM on large scales, pointing to future work on explicit baryon couplings and observational signatures.

Abstract

The theory of dark matter superfluidity has emerged as a compelling framework, in which the dynamics are governed by a non-relativistic $P(X)$ superfluid Lagrangian that naturally leads to Modified Newtonian Dynamics (MOND)-like behavior when coupled to baryons at galactic scales. Notably, at cosmological scales, this effective description reproduces the standard $Λ$ Cold Dark Matter ($Λ$CDM) model at the background level, suggesting that cold dark matter may undergo Bose--Einstein condensation at galactic scales. In this work, we extend the non-relativistic formulation by systematically incorporating relativistic corrections within the Friedmann--Lemaître--Robertson--Walker (FLRW) spacetime. We further perform a linear perturbation analysis in this relativistic setting to investigate the evolution of matter density fluctuations. Our results clarify the viability of the superfluid dark matter scenario in explaining large-scale structure formation and identify the parameter regimes in which it remains consistent with current cosmological observations.

Figures (4)

• Figure 1: Analytically approximated perturbations $\delta \lambda(a)$ and $\delta \rho(a)$ derived from \ref{['afo']} and \ref{['aft']} respectively. The figure illustrates how both perturbations grow rapidly with the expansion of the Universe, highlighting their relative behavior across the scale factor range.
• Figure 2: Small scale factor ($a \ll 1$) behavior of the perturbations \ref{['laj']} and \ref{['lak']}. Both $\delta\rho(a)$ (orange curve) and $\delta\lambda(a)$ (blue curve) oscillate slowly with nearly constant amplitudes, showing that the information in the perturbations is fully visible in this regime before the onset of rapid oscillations and decay at large $a$.
• Figure 3: Large scale factor ($a \gg 1$) behavior of the perturbations \ref{['laj']} and \ref{['lak']}. The orange curve shows $\delta\rho(a)$, whose amplitude decays as $A a^{-3/2}$ (gray dashed envelope) while oscillating rapidly. The blue curve shows $\delta\lambda(a)$, which oscillates at constant amplitude $D$ with a phase shift $x = \arctan(C/B)$. At large $a$, the rapid oscillations of $\delta\rho(a)$ effectively average out, illustrating how information can become hidden in high-frequency, decaying perturbations.
• Figure 4: Oscillatory behavior of the density contrast $\delta(a)$ in the regime $k \gg m$, as described by Equation \ref{['eq:density_contrast_approx']}. The amplitude remains constant while the frequency increases with the scale factor, indicating suppressed structure growth in the absence of baryonic coupling.