The role of the chemical potential in coupling superfluid dark matter to baryons

Abstract

Superfluid dark matter postulates that the centers of galaxies contain superfluid condensates. An important quantity regarding these superfluids is their chemical potential µ. Here, we discuss two issues related to this chemical potential. First, there is no exactly conserved quantity associated with this chemical potential due to the symmetry-breaking baryon-phonon coupling. Second, µ is sometimes introduced by shifting the phonon field by µ · t which -again due to the symmetrybreaking baryon-phonon coupling -introduces an explicit time dependence in the Lagrangian. We investigate under which conditions introducing a chemical potential is nevertheless justified and show how to correctly introduce it when these conditions are met. We further propose a model that recovers superfluid dark matter's zero-temperature equations of motion including a chemical potential even if the aforementioned conditions for justifying a chemical potential are not met.The chemical potential of superfluid dark matter U (1) symmetry usually associated with superfluidity [19]. This causes two problems regarding SFDM's chemical potential.The first problem is that SFDM might not have a chemical potential at all. This is because the U (1) symmetry is broken in SFDM and chemical potentials as a statistical physics concept are a consequence of conserved quantities. This problem requires a solution, since SFDM needs a chemical potential for its phenomenology on galactic scales.The other problem is that SFDM's chemical potential µ is sometimes introduced by shifting the phonon field by µ · t (see e.g. Ref.[1] below Eq. (6) for an explicit example). This leads to an explicit time dependence in the U (1)-breaking baryon-phonon coupling. Phenomenologically, this may or may not be problematic depending on the size of µ and the details of the model. But conceptually, a chemical potential is an equilibrium quantity from statistical physics and should not be associated with an explicit time dependence. This problem requires an explanation.The aim of this paper is to clarify these issues regarding SFDM's chemical potential. In the following, we employ units with c = = 1 and the metric signature (+, −, −, −). Small Greek indices run from 0 to 3 and denote spacetime dimensions.We start with an introduction to SFDM in Sec. 2. In Sec. 3, we discuss the nonconservation of the U (1) charge of SFDM and the conditions under which a chemical potential may nevertheless be introduced. We then show how to correctly introduce a chemical potential in case these conditions are met in Sec. 4. Using these results, we distinguish two different non-relativistic limits which may be taken in SFDM in Sec. 5. In Sec. 6, we address possible confusions regarding SFDM's equilibrium energy-momentum tensor. Finally, we propose an alternative model which avoids the problems of SFDM regarding its chemical potential in Sec. 7. We conclude in Sec. 8.

Three problems of superfluid dark matter and their solution

“…This model from Ref. [18] contains fields θ + and θ − similar to our fields θ + and θ − in that one of the fields has an exact shift symmetry θ − → θ − + const., while the shift symmetry of the other field is broken by the baryon coupling. However, this model from Ref.…”

“…However, this model from Ref. [18] solves only the equilibrium problem (see Sec. 2.3), but not the MOND limit problem (see Sec.…”

Aether Scalar Tensor theory confronted with weak lensing data at small accelerations

“…To see this, note that the model is shift-symmetric under ϕ → ϕ + c, Φ → Φ − c for any constant c. In general, to describe equilibrium states, one introduces a chemical potential µ for each symmetry by shifting the Hamiltonian H by H → H − µ Q where Q is the conserved quantity associated with the symmetry. In the AeST model and on the level of the Lagrangian, this corresponds to shifting φ → φ + µ and Φ → Φ − µ (Mistele 2019;Kapusta 1981;Haber & Weldon 1982;Bilic 2008) or equivalently to considering solutions with φ = µ and Φ = −µ. (Note that the parameter m was called µ in Skordis & Złosnik (2021).…”

Observational tests and theoretical developments of hybrid MOND dark matter models

“…Thus, in statistical physics, we would replace the standard Hamiltonian H with the shifted Hamiltonian H − µQ with chemical potential µ. On the level of the Lagrangian, this change of the Hamiltonian corresponds to a shift in time derivatives of θ, namely θ → θ+µ [87][88][89][90]. Equivalently, at least in equilibrium at zero temperature, one can directly consider solutions θ = µ • t. However, introducing µ as described above makes it clear that µ is a chemical potential in the statistical physics sense.…”

“…In Ref. [87], we also estimated a local timescale t loc separately for each point in space. The local equilibrium is valid only on timescales shorter than t −1 loc = m(ᾱΛ/M Pl )(ρ b /ρ SF ).…”