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Dark Matter Superfluidity

by Justin Khoury

Submission summary

As Contributors: Justin Khoury
Arxiv Link: (pdf)
Date submitted: 2021-09-29 14:47
Submitted by: Khoury, Justin
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
Approach: Theoretical


In these lectures I describe a theory of dark matter superfluidity developed in the last few years. The dark matter particles are axion-like, with masses of order eV. They Bose-Einstein condense into a superfluid phase in the central regions of galaxy halos. The superfluid phonon excitations in turn couple to baryons and mediate a long-range force (beyond Newtonian gravity). For a suitable choice of the superfluid equation of state, this force reproduces the various galactic scaling relations embodied in Milgrom's law. Thus the dark matter and modified gravity phenomena represent different phases of a single underlying substance, unified through the rich and well-studied physics of superfluidity.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 1 on 2021-10-19 (Invited Report)


In these lectures, the author introduces and motivates the hybrid DM-MOND model superfluid dark matter. I find them to be clear and well-written, providing a good basis to understand superfluid dark matter.

I have a few comments and suggestions:

1. On p. 2, references [2, 3] are not really about the MDAR, as suggested in the text, but about the RAR. Of course the RAR and the MDAR are equivalent, but giving only these references might confuse readers who don't already know this.

2. On p.3, below (4), the author writes that there is a solution $a_\phi = \sqrt{a_b a_0}$. This is usually a good approximation but is strictly true only in spherical symmetry and other special cases. It might be good to note this to avoid confusion for readers who try to rederive this result.

3. On p. 4, I think the exponent in (6) is missing an overall minus sign. Similarly, I think the right-hand side of (8) on p. 5 is missing an overall minus sign.

4. On p. 8, $\psi_0$ is introduced as the condensate wavefunction. I know it is not uncommon to refer to $\psi$ as a wavefunction, but I find this causes a lot of confusion for students since $\psi$ is not actually a wave function, $|\psi|^2$ is not a probability density. Conceptually, $\psi$ is much closer to the quantum/classical electromagnetic field (or its four-potential) than to a QM wavefunction. It would be good to change the naming or briefly explain this to avoid confusion.

5. On p. 14, the author derives the phonon force and the density profile from the Lagrangian (58). But opposite limits are used for the phonon force and the density profile. For the phonon force to look like $\sqrt{a_0 a_b}$ (eq. (61)), the gradient terms (with $\vec{\nabla} \theta$ or $\vec{\nabla} \pi$) in the Lagrangian must dominate the chemical potential terms (with $\mu$). For the density profile to have, e.g., the equation of state claimed in (63) / (64) the opposite limit is needed. Maybe the density profile (but not the phonon force) is supposed to be for the case without any baryons? In this case it should probably be pointed out that the derived density profile does not directly apply to galaxies. In any case, the two opposite assumptions about which terms dominate should be made explicit to avoid confusion.

6. On p. 15, the author writes that it is not surprising that the density profile is cored. I'm not quite sure what this refers to. Is it that Lane-Emden equations generally give cored profiles? A clarification would be helpful.

  • validity: -
  • significance: -
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  • formatting: -
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Anonymous Report 2 on 2021-10-18 (Invited Report)


1. This article gives an introduction to one of the most interesting dark matter models to have been introduced in recent years.

2. It is at a level more accessible to students than anything that can be found in the existing literature.


This is not really a weakness, only a modest wish from someone who will give this reference to some of his students, based on their typical background in his corner of the world:

1. Bose-Einstein condensation, which most students will have been introduced to in a course on statistical physics, is presented in quite some detail. In contrast, some knowledge of effective field theory, which students with only a basic course in quantum field theory will not have been exposed to, is more or less taken for granted.


The acceptance criteria are clearly met: The article provides a correct, very systematic and intelligible presentation of a topic of ongoing interest. It will be a very valuable reference for students (and their thesis advisors).

It may be an oversight of my part, but I cannot see that [68]-[70] in the reference list are referred to in the text. It would perhaps be natural to say something about vortices after concluding in section 6 that the flow is irrotational, and in that case these references would be relevant.

Requested changes

1. Add a reference to an introductory text on effective field theory at the start of section 4.

  • validity: top
  • significance: top
  • originality: top
  • clarity: top
  • formatting: excellent
  • grammar: perfect

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