Vortices and rotating solitons in ultralight dark matter

Patrick Valageas; Philippe Brax

REVIEW 3 major objections 4 minor 2 cited by

Vortices and rotating solitons in ultralight dark matter

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read In self-interacting ultralight dark matter, a halo with nonzero spin collapses into a rotating soliton whose rotation is carried by a uniform lattice of quantum vortices, yielding solid-body motion and a maximum spin set by the central…

desk verdict A credible 2D demonstration that rotating solitons in self-interacting ULDM are supported by vortex lattices, with useful new bounds, though the stability proof and numerical tests are weaker than the narrative suggests. read the letter →

arxiv 2501.02297 v2 pith:AL2ZTB5Q submitted 2025-01-04 astro-ph.CO nlin.PS

Philippe Brax , Patrick Valageas This is my paper · ORCID

classification astro-ph.COnlin.PS

keywords ultralightdarkmatterGross-PitaevskiiequationThomas-Fermiregimequantumvorticesvortexlatticerotatingsolitonsolid-bodyrotationangularmomentum

topics Dark Matter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies halo-sized clouds of ultralight dark matter with repulsive self-interactions, governed by the Gross-Pitaevskii equation, and asks what happens when the cloud has net angular momentum. In 2D simulations starting from stochastic initial conditions, a rotating soliton forms in a few dynamical times. The key claim is that the rotation is not due to a large orbital quantum number of the wave function; it is produced by a regular lattice of quantized vortices that, in the continuum limit, becomes solid-body rotation. These rotating solitons are stable minima of the energy at fixed mass and angular momentum, and for a given central density they cannot rotate faster than about 1.343√ρ0 nor be larger than about 1.593 times the static radius. The analytical profiles, vortex densities, and stability bounds match the simulations.

What carries the argument

The key machinery is the multi-vortex wave-function ansatz ψ = $e^{{-iμt/ϵ}}$ √ρ0 ∏_j f(r − r_j) $e^{{iσ_j θ_j}}$, which treats each vortex as a phase singularity carrying circulation 2πϵσ_j. In the limit ϵ→0 the discrete vortices become a smooth vorticity distribution, and extremizing E − μM − ΩL_z yields the solid-body rotation and the balance Φ_N + Φ_I − r²Ω²/2 = μ. The Bessel-function profile and the requirement of finite mass produce the bounds (64)–(65), and the second variation of the energy at fixed angular momentum gives stability.

What would settle it

Solve the second-variation eigenvalue problem for the full solution including the exponential tail, without imposing δρ(R)=0 or δM(R)=0 at a finite radius, and look for a negative eigenvalue; a single negative mode would overturn the claim that all Ω < Ωmax solitons are stable.

Watch

Extended reading notes

Core claim

The central discovery is that the steady rotating state of a self-gravitating Gross-Pitaevskii condensate in the Thomas-Fermi regime is a uniform lattice of unit-circulation vortices embedded in an axisymmetric soliton. In the continuum limit, where the de Broglie wavelength is much smaller than the system size, the lattice is equivalent to a smooth vorticity field and the velocity becomes solid-body rotation, v = Ω r eθ. The density profile is ρ(r) = (ρ0 − Ω²/2π) J0(z0 r/R0) + Ω²/2π, which is broader than the static Bessel profile; requiring finite mass gives Ωmax ≈ 1.343√ρ0 and Rmax ≈ 1.593 R0. A second-variation calculation shows all such configurations with Ω below Ωmax are dynamically stable. Numerical simulations with three values of the de Broglie parameter and several initial anisotropies confirm the deformed profile, the solid-body rotation, the uniform vortex density, and the vortices moving on circular orbits.

Load-bearing premise

The stability proof assumes the soliton density drops exactly to zero at a finite boundary radius, so it does not examine modes living in the exponential tail of the density profile.

Editorial extensions

If this is right

The density profile of a rotating soliton is wider than the static one, so rotation-curve fits that ignore rotation will overestimate the central density or underestimate the radius.
The vortex lattice provides a concrete mechanism for a central soliton to retain a large fraction of the halo's angular momentum instead of expelling it.
There is an upper bound on angular momentum for a given central density; halos with more spin must leave the excess in the outer envelope or form a different configuration.
In 3D, point vortices should become vortex rings, making the prediction testable in full 3D simulations of scalar-field dark matter.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not explored in the paper: the stability proof truncates the density at a finite radius, so a numerical eigenvalue solve on the true exponential tail would test whether any mode living outside that radius is unstable; if one were found, the claim that all Ω < Ωmax solitons are stable would need revision.
An extension the authors leave implicit is that in 3D the point vortices become vortex rings; tracking whether such rings form and stay circular in a 3D simulation would show whether the 2D solid-body picture carries over.
A testable corollary not drawn by the authors: if an observed rotation curve of a dark-matter-dominated dwarf implies a soliton with spin above Ωmax for its inferred central density, this model would be ruled out for that halo.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this paper delivers a genuinely new result—rotating self-gravitating solitons in the Thomas-Fermi regime of self-interacting ULDM are not large-ℓ eigenstates but solid-body rotators built from a uniform lattice of quantized vortices. The profile (60), the maximum radius Rmax ≈ 1.59 R0, and the maximum rotation rate Ωmax ≈ 1.34 sqrt(ρ0) are new and likely to be cited. The asymptotic matching in the appendices (Painlevé II boundary layers for the soliton edge and vortex core) is careful and technically impressive.

The numerical simulations from stochastic initial conditions are a real plus. The vortex lattice genuinely forms, the density profile matches (60) with a flat center, and the vortex density is uniform as predicted. The authors are honest that the profile comparison uses Ω measured from the same simulation; this makes the agreement a consistency check rather than a prediction. The scaling with ϵ and α gives confidence that the mechanism is right.

The soft spots are in proportion. The stability proof in Sec. IV B is the weakest link. It diagonalizes K = 4πΔ^{-1}+λ on a finite disk with boundary conditions at R > RΩ, assuming the density vanishes identically beyond R. But App. A shows the density has an exponential tail, so the hard boundary is an artificial truncation. The eigenvalue equation does not contain the background density at all; if you enlarge the domain to admit perturbations that penetrate the tail, the lowest Bessel eigenvalue falls below the Jeans threshold and the quadratic form goes negative. The proof therefore establishes stability only within a restricted perturbation class, not for all Ω < Ωmax as claimed. This is a technical gap, not a demonstrated instability—the simulations run many dynamical times without trouble—but the claim about stability should be softened or the tail modes addressed, e.g., by including quantum pressure in the Hessian for the tail region.

The 3D generalization is speculative, but the paper says so. No code or data is released, which would help if this becomes a reference result.

Who is this for? People working on self-interacting ULDM solitons, vortex formation in dark matter, and rotation curves in these models. It deserves a serious referee—the core mechanism is plausible, the analysis is mostly clean, and the bounds are useful—but the referee should press on the stability argument and ask for a genuine prediction of Ω.

Referee Report

3 major / 4 minor

Summary. The paper studies rotating solitons in two-dimensional ultralight dark matter with repulsive quartic self-interactions, described by a Gross-Pitaevskii equation in the Thomas-Fermi regime. The authors derive a variational continuum-limit configuration at fixed mass and angular momentum, obtaining a solid-body rotation profile, a deformed Bessel density profile, and a uniform vortex lattice; they also derive upper bounds on the rotation rate and radius (Eqs. (64), (65)) and argue, via a second-variation analysis, that all such configurations are dynamically stable. Numerical simulations from stochastic initial conditions with injected angular momentum show the formation of a rotating soliton with a regular lattice of single-charge vortices, and the measured density, velocity, vortex count, and angular momentum profiles are compared with the analytical expressions.

Significance. If the formation and stability claims hold, the paper provides a concrete and physically interesting picture for rotating solitons in self-interacting ultralight dark matter, connecting the vortex-lattice physics of rotating Bose-Einstein condensates with cosmological soliton dynamics. The analytical parts are substantial: the Thomas-Fermi profiles, the matched asymptotic expansions for the soliton boundary layer and vortex core, and the energy decomposition of vortex excess energy are worked out in detail and are internally consistent. The numerical simulations are clean and the diagnostics (density, phase winding, velocity field, vortex trajectories) are well chosen to exhibit the claimed phenomenology. The paper also makes falsifiable predictions, notably the scaling Ωmax ≃ 1.343√ρ0 and the radius bound Rmax ≃ 1.593 R0, which are stated in a parameter-free form. However, the dynamical-stability proof has a technical gap, and several numerical comparisons are partly circular because the rotation rate is measured from the same simulation that is then compared with the formulas.

major comments (3)

[Sec. IV B, Eqs. (68)-(70)] The stability proof for all Ω < Ωmax relies on a finite-disk truncation: eigenvectors are Bessel functions with hard-wall conditions δρ(R)=0 (ℓ ≥ 1) or δM(R)=0 (ℓ = 0) at a radius R > RΩ, justified by the statement that the density 'identically vanishes' beyond RΩ. This is inconsistent with App. A, Eq. (A11), which shows that the soliton has an exponential WKB tail that never vanishes identically. The eigenvalue equation (69) contains only κ, ℓ, and the domain size, not the background density; therefore a perturbation that extends into the tail can effectively see a larger R, pushing the lowest eigenvalue κ = (x_ℓ1/R)^2 below the Jeans threshold 4π/λ and reversing the sign of ν. The hard-wall boundary conditions also exclude free-surface modes, including ℓ=0 compression modes and ℓ=2 bar-like deformations, which are the modes most likely to destabilize a rotating self-gravitating body. Consequently, the claim that every Ω < Ωmax soliton is a dynamically stable minimum is not established by the present argument; the proof needs a treatment of the exponential tail or a direct variational test of tail and free-surface modes.
[Sec. V C, Figs. 2, 3, and 5] The numerical confirmation is partly circular. In Fig. 3, Ω is obtained by a least-squares fit to the simulated transverse velocity profile, and this same Ω is then inserted into Eqs. (60), (63), and (73) to produce the red curves in Figs. 2 and 5. This demonstrates that the simulated configuration is internally consistent with the ansatz of a solid-body-rotating Thomas-Fermi soliton, but it does not independently confirm the predictive content of the derivation. A stronger test would predict Ω from the initial conditions, for example through the mass-shell angular-momentum estimate used in Sec. V C 3, and then compare the resulting density, vortex number, and angular-momentum profiles with the simulation output. Alternatively, the measured Lz and Nv should be compared with relations that do not share the fitted Ω. The abstract and conclusion currently state that the numerical results agree with the analytical derivations, which overstates the strength of the available evidence.
[Sec. III B, Eq. (38)] The sentence following Eq. (38) states that the uniform-lattice energy is 'about a quarter of the single vortex energy (35), for Nv = |σ|'. For large Nv and ln(R0/ξ) ≫ 1, Eq. (38) is dominated by the term Nv ρ0 ε² ln(R0/ξ), which is smaller than the single-vortex energy Nv² ρ0 ε² ln(R0/ξ) by a factor of order 1/Nv, not by a factor of four. The qualitative conclusion that a high-spin vortex splits into unit vortices is not affected, but the quantitative statement should be corrected or clarified.

minor comments (4)

[Sec. V C 5] The text says 'at the later times, 84 < t < 84' but the intended interval is evidently 84 < t < 85; this typo appears in the discussion of the late-time vortex trajectories.
[App. B] The matched asymptotic vortex profile is derived under the assumption |σ| ≫ 1, and the authors note that for |σ| = 1 the normalization and the transition shape are known only up to a factor of order unity. Since all vortices in the simulations have |σ| = 1, the quantitative vortex-profile predictions of Sec. III A 2 are not directly verified analytically for the simulated case; this limitation should be acknowledged where Eq. (35) and the related energy estimates are used.
[Sec. V A 3] The anisotropic initial condition in Eq. (99) uses the discontinuous function sign(L) at L = 0, which produces a phase-space distribution that is discontinuous across the L = 0 plane. This is a valid modeling choice for injecting net angular momentum, but its effect on the initial vortex population and on the subsequent relaxation could be commented on, since sign(L) is not smooth.
[Sec. VI] The statement that 'most of these properties' extend to 3D with vortex rings is a conjecture rather than a result of this paper; the distinction should be made explicit, since the 2D logarithmic Green function and the stability analysis do not automatically carry over to 3D point-vortex or vortex-ring systems.

Circularity Check

1 steps flagged · score 6.0 of 10

Analytic derivation is self-contained; numerical 'predictions' of vortex number and angular momentum use the measured rotation rate Ω as input, reducing those checks to consistency tests.

fitted input called prediction [Sec. V C 1 and Fig. 5 caption; Eqs. (60), (63), (73)]
"Upper panel: number of vortices Nv(< r) within radius r (black dashed line). The red solid line is the prediction (73), with the value of Ω obtained from the lower panel in Fig. 3. Lower panel: angular momentum Lz(< r) within radius r (black dashed line). The red solid line is the prediction (63)."

The 'predictions' are evaluated using the rotation rate Ω that was itself obtained by a least-squares fit to the simulated velocity field (best-fit Ω ≃ 1.3, Fig. 3). Equation (73), Nv(<r) = r^2 Ω / ϵ, and Eq. (63) for Lz(<r) are then rearrangements of the measured solid-body rotation and measured density profile, rather than independent determinations of the vortex number or angular momentum. The agreement is a consistency check of the continuum ansatz and of the quantization of circulation, but it does not test the theory's ability to predict the absolute vortex content or angular momentum from the initial conditions.

full rationale

The analytical chain in Secs. II–IV is self-contained: the Thomas-Fermi soliton profile, the vortex ansatz (based on standard external references), the variational continuum limit leading to solid-body rotation, the finite-mass bounds Ωmax and Rmax, and the Bessel stability analysis are derived from the stated energy functional and equations of motion. The finite-disk truncation in Sec. IV B is a technical approximation about perturbation boundary conditions, and the exponential tail discussed in App. A is a correctness concern rather than a circular reduction. The self-citations to Refs. [41,83,88] are methodological (initial-condition construction and Vlasov interpretation), not load-bearing uniqueness claims. The genuine circularity is confined to the numerical comparisons: Ω is fitted from the simulation and then used to evaluate the 'predictions' for the density profile, vortex count, and angular momentum within the soliton. Those comparisons verify internal consistency of the model rather than an independent prediction of Ω or Nv. Because the central existence, stability, and Ωmax/Rmax results do not reduce to that fit, the score is moderate rather than maximal.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard Gross-Pitaevskii physics, the Thomas-Fermi approximation, and the vortex-lattice ansatz. The initial-condition model and shell-ordering assumption are specific to this paper. No invented entities are required.

free parameters (5)

Omega (rotation rate) = ~1.3 for (epsilon=0.01, alpha=1); ~0.65 for alpha=0.5
Fitted from the slope of the transverse velocity in Fig. 3, then inserted into the theoretical density profile (60), vortex count (73), and angular momentum (63).
rho0 (central density) = order unity, measured from the simulated density at the center
Together with Omega, determines the rotating-soliton profile (60); not predicted from initial conditions.
epsilon = 0.03, 0.01, 0.005
Dimensionless ratio of de Broglie wavelength to system size; chosen as simulation parameters, not derived.
alpha = 1.0, 0.5, 0.0
Anisotropy parameter of the initial distribution (99), controlling the initial angular momentum.
lambda (self-interaction coupling) = chosen so that R0 = 0.5 in Eq. (23)
Sets the static soliton radius; an input model parameter rather than a fitted outcome.

assumptions (6)

domain assumption The nonrelativistic ULDM system obeys the Gross-Pitaevskii/Schroedinger-Poisson system (7)-(8) with quartic self-interaction.
Used throughout; standard for scalar field dark matter with repulsive lambda4.
domain assumption Thomas-Fermi approximation: the quantum pressure term is negligible inside the soliton, epsilon << 1.
Invoked in Secs. II E and IV A to obtain profiles (23) and (60).
domain assumption The system is treated as 2D with the logarithmic Poisson kernel; 3D behavior with vortex rings is assumed, not derived.
Simulations and main derivations are in 2D (Sec. V); the abstract and conclusion extend to 3D by analogy.
domain assumption The ansatz (42) of a product of well-separated single-vortex profiles is valid; the continuum limit of a uniform vortex density is taken.
Used in Secs. III C and IV C; requires healing length xi << vortex spacing d, justified for epsilon -> 0 by Eq. (77).
ad hoc to paper Initial conditions use the anisotropic distribution f(E,L)=f0(E)[1+alpha sign(L)] with random phases.
A one-parameter family chosen in Sec. V A 3 to give the target halo density and nonzero net angular momentum.
ad hoc to paper Radial shell ordering is approximately preserved during collapse when estimating the soliton's angular momentum.
Used in Sec. V C 3 to estimate Lz of the soliton from the initial mass shell; approximate.

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Cite this review

Pith. "Pith review of Vortices and rotating solitons in ultralight dark matter." pith.science (2026). https://pith.science/paper/AL2ZTB5Q

@misc{pith2026250102297,
  author       = {Pith},
  title        = {Pith review of: Vortices and rotating solitons in ultralight dark matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AL2ZTB5Q}},
  note         = {Machine review of arXiv:2501.02297}
}

read the original abstract

The dynamics of ultralight dark matter with non-negligible self-interactions are determined by a nonlinear Schr\"odinger equation rather than by the Vlasov equation of collisionless particles. This leads to wave-like effects, such as interferences, the formation of solitons, and a velocity field that is locally curl-free, implying that vorticity is carried by singularities associated with vortices. Using analytical derivations and numerical simulations in 2D, we study the evolution of such a system from stochastic initial conditions with nonzero angular momentum. Focusing on the Thomas-Fermi regime, where the de Broglie wavelength of the system is smaller than its size, we show that a rotating soliton forms in a few dynamical times. The rotation is not associated with a large orbital quantum number of the wave function. Instead, it is generated by a regular lattice of vortices that gives rise to a solid-body rotation in the continuum limit. Such rotating solitons have a maximal radius and rotation rate for a given central density, while the vortices follow the matter flow on circular orbits. We show that this configuration is a stable minimum of the energy at fixed angular momentum and we check that the numerical results agree with the analytical derivations. We expect most of these properties to extend to the 3D case where point vortices would be replaced by vortex rings.

Figures

Figures reproduced from arXiv: 2501.02297 by the authors.

**Figure 2.** FIG. 2. [ [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3. [ [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: FIG. 4. [ [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. [ [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. [ [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: FIG. 11. Superposition of many snapshots of the maps of [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]

**Figure 13.** Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

**Figure 14.** Figure 14: FIG. 14. Angular momentum [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

**Figure 16.** Figure 16: FIG. 16. Superposition of many snapshots of the maps of [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]

**Figure 15.** Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]

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Works this paper leans on

115 extracted references · 30 canonical work pages · cited by 2 Pith papers

[1]

Thus, we start with stochastic initial conditions associated with a col- lisionless virialized halo in the semiclassical limit

Expansion over eigenfunctions As in [41, 83], we start our simulations without a cen- tral soliton since we are interested in the formation of the solitons and their generic properties. Thus, we start with stochastic initial conditions associated with a col- lisionless virialized halo in the semiclassical limit. We choose a target classical density profil...
[2]

Thus, we obtain the expressions ∂ρ ∂t + ∇ ·(ρ⃗ v) = ρ NX j=1 ⃗∇f 2(⃗ r− ⃗ rj) f 2(⃗ r− ⃗ rj) · h ⃗ v− ˙⃗ rj i (44) ∂⃗ v ∂t + (⃗ v· ⃗∇)⃗ v= NX j=1 ([⃗ v− ˙⃗ rj] · ⃗∇)⃗ vj

Equations of motion Following [75] we substitute the ansatz (42) into the hydrodynamic form of the equations of motion. Thus, we obtain the expressions ∂ρ ∂t + ∇ ·(ρ⃗ v) = ρ NX j=1 ⃗∇f 2(⃗ r− ⃗ rj) f 2(⃗ r− ⃗ rj) · h ⃗ v− ˙⃗ rj i (44) ∂⃗ v ∂t + (⃗ v· ⃗∇)⃗ v= NX j=1 ([⃗ v− ˙⃗ rj] · ⃗∇)⃗ vj. (45) The terms in the right-hand sides are dominated by the neighb...
[3]

Thus, we write the wave function as ψ(⃗ r, t) = √ρeis NY j=1 eiσj θj , (47) where again θj(⃗ r) = \(⃗ ex, ⃗ r− ⃗ rj), ⃗∇θj = ⃗ ez × ⃗∇ ln |⃗ r− ⃗ rj|

Effective action A more general and elegant approach is to substitute our ansatz into the action (19). Thus, we write the wave function as ψ(⃗ r, t) = √ρeis NY j=1 eiσj θj , (47) where again θj(⃗ r) = \(⃗ ex, ⃗ r− ⃗ rj), ⃗∇θj = ⃗ ez × ⃗∇ ln |⃗ r− ⃗ rj|. (48) Here ρ(⃗ r, t) and s(⃗ r, t) are smooth functions and we ne- glect the width of the vortices and t...
[4]

(82) Then, we take for the initial wave function [41, 84, 85] a sum with random coefficients anℓ over the eigenmodes ˆψnℓ(⃗ r) of the Schr¨ odinger equation defined by this target gravitational potential Φ N , ψ(⃗ r) = X nℓ anℓ ˆψnℓ(⃗ r), ˆψnℓ(⃗ r) = Rn|ℓ|(r)eiℓθ, (83) where the radial parts satisfy the radial Schr¨ odinger equation − ϵ2 2 1 r d dr r d dr...
[5]

(96) Then, we obtain ρ = 2π Z 0 ΦN dE f0(E), (97) which can be inverted to give the isotropic distribution function ΦN 0 < E <0 : f0(E) = z2 0 8π2R2

Isotropic initial conditions For isotropic initial conditions the classical phase-space distribution does not depend on the angular momentum, f (E, L) = f0(E). (96) Then, we obtain ρ = 2π Z 0 ΦN dE f0(E), (97) which can be inverted to give the isotropic distribution function ΦN 0 < E <0 : f0(E) = z2 0 8π2R2 . (98) Thus, the distribution function is a cons...
[6]

In particular, the mean angular rotation becomes nonzero if f (E, L) depends on the sign of L

Anisotropic initial conditions For anisotropic initial conditions the phase-space dis- tribution depends on L. In particular, the mean angular rotation becomes nonzero if f (E, L) depends on the sign of L. In this paper we take the simple choice f (E, L) = f0(E)[1 + α sign(L)], −1 ≤ α ≤ 1, (99) where f0(E) is the isotropic distribution (98). This re- cove...
[7]

For the initial wave function we take R = 1 and M = 1 for the target halo radius and mass in Eqs.(80)-(81)

Simulation parameters In this paper we consider the cases ϵ = 0.005, 0.01 and 0.03, as we focus on the semi-classical regime, and we mostly illustrate our results with the intermediate case ϵ = 0 .01. For the initial wave function we take R = 1 and M = 1 for the target halo radius and mass in Eqs.(80)-(81). For the anisotropy parameter of Eq.(99) we consi...
[8]

2 how the system evolves from the initial condition (1)

Mass, density and velocity profiles For the case ϵ = 0.01 and α = 1, we show in Fig. 2 how the system evolves from the initial condition (1). As in the 3D isotropic simulations presented in [41], in a few dynamical times a soliton quickly forms at the center of the system. As seen in the left panel, where we show the mass MTF,0 within the radius R0 of Eq....

Show all 115 references

[9]

granules

2D maps We show in Fig. 4 the 2D maps of the system at time t = 500, when the system has relaxed to a rotating cen- tral soliton with an outer virialized halo. We can clearly see in panel (a) the central high-density soliton, with a circular shape, surrounded by a low-density ...
[10]

5 the number of vortices (weighted by their spin σ) and the angular momentum within radius r

Radial angular momentum profile We show in Fig. 5 the number of vortices (weighted by their spin σ) and the angular momentum within radius r. The red solid lines are the analytical predictions (73) and (63), plotted within the radius RΩ. We can check the good agreement between...
[11]

6 the evolution with time of the ro- tation of the system

Evolution of the rotation rate We show in Fig. 6 the evolution with time of the ro- tation of the system. We can see that after the quick relaxation of the system and the formation of the central soliton, in a few dynamical times, the rotation rate (mea- sured from the slope o...
[12]

7 a superposition of snapshots of the locations of the vortices, at times 10 < t < 11 and 84 < t <85

Trajectories of the vortices We show in Fig. 7 a superposition of snapshots of the locations of the vortices, at times 10 < t < 11 and 84 < t <85. The filled circles are the positions at the initial time while the filled squares are the positions at the final time. We can clea...
[13]

This gives ΦN + ΦI = µ, (A5) with the solution r ≥ 0, R 0 − r ≫ ϵ2/3 : ˆψ(r) = p ρ0J0(z0r/R0), (A6) as in Eqs.(22)-(23)

Bulk of the soliton As we focus on the limit ϵ → 0, the bulk of the soliton, where the density is of order unity, is described by the Thomas-Fermi regime where we can neglect the left-hand side in Eq.(A2). This gives ΦN + ΦI = µ, (A5) with the solution r ≥ 0, R 0 − r ≫ ϵ2/3 : ...
[14]

Indeed, for r → ∞we have ΦI → 0 while ΦN keeps growing as in Eq.(A4)

Exponential tail At large distance beyond R0, the density becomes van- ishingly small and the Thomas-Fermi approximation no longer applies. Indeed, for r → ∞we have ΦI → 0 while ΦN keeps growing as in Eq.(A4). Making the change of variable ˆψ = u/√r, the Schr¨ odinger equation...
[15]

In this region the density shows a sharp bend while the gravitational potential re- mains smooth

Boundary layer Around R0 we have a transition between the Thomas- Fermi and WKB regimes. In this region the density shows a sharp bend while the gravitational potential re- mains smooth. From Eq.(A7) we write r ≃ R0 : Φ N − µ ≃ 2M R0 (r − R0). (A12) We can also neglect the fir...
[16]

This Thomas-Fermi expression gives ϵ2/3 ≪ R0 − r ≪ 1 : ˆψ(r) = r 2M λR0 p R0 − r

Asymptotic matchings We first check the matching of the intermediate solu- tion (A17) with the Thomas-Fermi solution (A6). This Thomas-Fermi expression gives ϵ2/3 ≪ R0 − r ≪ 1 : ˆψ(r) = r 2M λR0 p R0 − r. (A18) Using the asymptotic behavior (A16), we can check that this agrees...
[17]

A 1, in the limit ϵ → 0 the bulk of the vortex is described by the Thomas-Fermi regime where we can neglect the spatial derivatives in Eq.(B3)

Bulk of the vortex As for the static soliton in App. A 1, in the limit ϵ → 0 the bulk of the vortex is described by the Thomas-Fermi regime where we can neglect the spatial derivatives in Eq.(B3). However, because we can have |σ| ≫1, we keep the orbital-barrier term in the lef...
[18]

More generally, the density is infinitesimally small below rc from the expression (B6)

V ortex core Close to the center of the vortex the density vanishes as ρ ∝ r2|σ| from Eq.(29). More generally, the density is infinitesimally small below rc from the expression (B6). Therefore, inside the vortex core we can neglect Φ I in the right-hand side in Eq.(B3). Keepin...
[19]

As compared with the boundary layer at R0 found for the static soliton, described in App

V ortex boundary layer At the boundary between the outer Thomas-Fermi regime (B6) and the inner core regime (B8), the self- interaction ΦI , the orbital barrier σ2/r2 and the spatial derivatives are of the same order in the Schr¨ odinger equa- tion (B3). As compared with the b...
[20]

The latter gives close to the core radius |σ|−2/3rc ≪ r − rc ≪ rc : ˆψ = p 2ρ0 r r − rc rc

Asymptotic matchings We first check the outer matching of the boundary- layer solution (B11) with the generalized Thomas-Fermi solution (B6). The latter gives close to the core radius |σ|−2/3rc ≪ r − rc ≪ rc : ˆψ = p 2ρ0 r r − rc rc . (B12) Using Eq.(A16), we can check that th...
[21]

Energy difference For an axisymmetric profile with a vanishing winding number, such as the static soliton (21), the energy (20) reads E0[ρ] = Z drπr " ϵ2 d√ρ dr 2 + ρΦN + λρ2 # . (C1) Then, the static soliton profile ρsol defined by the radial Schr¨ odinger equation (21) is al...
[22]

(C6) We have seen in Eq.(30) that the core radius of the vortex is rc = |σ|ξ ∝ ϵ

Scaling of the E0 terms We first focus on the E0 terms, ∆E0 = E0[ρσ] − E0[ρsol]. (C6) We have seen in Eq.(30) that the core radius of the vortex is rc = |σ|ξ ∝ ϵ. Therefore, choosing a fixed large factor Λ ≫ 1 (e.g., Λ = 10 or 100) that does not depend on ϵ, we have Λ ≫ 1, r ≥...
[23]

From the soliton density profile obtained in App

Scaling of the Eσ term Finally, we estimate the angular kinetic contribu- tion (C4). From the soliton density profile obtained in App. A, with its sharp cutoff beyond radius R0, and the radius rc of the vortex core, we obtain Eσ[ρσ] ≃ ϵ2σ2πρ0 ln(R0/rc) ∝ ϵ2 ln(1/ϵ). (C17) As i...
[24]

Jungman, M

G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rept. 267, 195 (1996), hep-ph/9506380

1996 arXiv
[25]

Drees, R

M. Drees, R. Godbole, and P. Roy, Theory and Phe- nomenology of Sparticles (WORLD SCIENTIFIC, 2005), https://www.worldscientific.com/doi/pdf/10.1142/4001, URL https://www.worldscientific.com/doi/abs/10. 1142/4001

2005 doi
[26]

Steigman and M

G. Steigman and M. S. Turner, Nucl. Phys. B 253, 375 (1985)

1985
[27]

R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977)

1977
[28]

Weinberg, Phys

S. Weinberg, Phys. Rev. Lett. 40, 223 (1978)

1978
[29]

Wilczek, Phys

F. Wilczek, Phys. Rev. Lett. 40, 279 (1978)

1978
[30]

Preskill, M

J. Preskill, M. B. Wise, and F. Wilczek, Phys. Lett. B 120, 127 (1983)

1983
[31]

L. F. Abbott and P. Sikivie, Phys. Lett. B 120, 133 (1983)

1983
[32]

Svrcek and E

P. Svrcek and E. Witten, JHEP 06, 051 (2006), hep- th/0605206

2006
[33]

Arvanitaki, S

A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell, Phys. Rev. D 81, 123530 (2010), 0905.4720

2010 arXiv
[34]

Halverson, C

J. Halverson, C. Long, and P. Nath, Phys. Rev. D 96, 056025 (2017), 1703.07779

2017 arXiv
[35]

T. C. Bachlechner, K. Eckerle, O. Janssen, and M. Kle- ban, JCAP 09, 062 (2019), 1810.02822

2019 arXiv
[36]

D. H. Weinberg, J. S. Bullock, F. Governato, R. Kuzio de Naray, and A. H. G. Peter, Proc. Nat. Acad. Sci. 112, 12249 (2015), 1306.0913

2015 arXiv
[37]

Del Popolo and M

A. Del Popolo and M. Le Delliou, Galaxies 5, 17 (2017), 1606.07790

2017 arXiv
[38]

Nakama, J

T. Nakama, J. Chluba, and M. Kamionkowski, Phys. Rev. D 95, 121302 (2017), 1703.10559. 26

2017 arXiv
[39]

Salucci, Astron

P. Salucci, Astron. Astrophys. Rev. 27, 2 (2019), 1811.08843

2019 arXiv
[40]

Di Luzio, M

L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, Phys. Rept. 870, 1 (2020), 2003.01100

2020 arXiv
[41]

Sikivie and Q

P. Sikivie and Q. Yang, Phys. Rev. Lett. 103, 111301 (2009), 0901.1106

2009 arXiv
[42]

L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Phys. Rev. D 95, 043541 (2017), 1610.08297

2017 arXiv
[43]

Fan, Phys

J. Fan, Phys. Dark Univ. 14, 84 (2016), 1603.06580

2016 arXiv
[44]

W. Hu, R. Barkana, and A. Gruzinov, Phys. Rev. Lett. 85, 1158 (2000), astro-ph/0003365

2000 arXiv
[45]

Schive, T

H.-Y. Schive, T. Chiueh, and T. Broadhurst, Nature Phys. 10, 496 (2014), 1406.6586

2014 arXiv
[46]

Nori and M

M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 478, 3935 (2018), 1801.08144

2018 arXiv
[47]

Irˇ siˇ c, M

V. Irˇ siˇ c, M. Viel, M. G. Haehnelt, J. S. Bolton, and G. D. Becker, Phys. Rev. Lett. 119, 031302 (2017), 1703.04683

2017 arXiv
[48]

Armengaud, N

E. Armengaud, N. Palanque-Delabrouille, C. Y` eche, D. J. E. Marsh, and J. Baur, Mon. Not. Roy. Astron. Soc. 471, 4606 (2017), 1703.09126

2017 arXiv
[49]

S. Lu, A. Ilyas, X.-H. Ma, B. Wang, D. Zhang, and Y.-F. Cai, JCAP 01, 086 (2025), 2407.19521

2025 arXiv
[50]

Khmelnitsky and V

A. Khmelnitsky and V. Rubakov, JCAP 02, 019 (2014), 1309.5888

2014 arXiv
[51]

P. Brax, P. Valageas, C. Burrage, and J. A. R. Cembra- nos, Phys. Rev. D 110, 083515 (2024), 2402.04819

2024 arXiv
[52]

Y.-F. Cai, G. Dom` enech, A. Ganz, J. Jiang, C. Lin, and B. Wang, JCAP 10, 027 (2024), 2311.18546

2024 arXiv
[53]

Chowdhury, A

D. Chowdhury, A. Hait, S. Mohanty, and S. Prakash, Phys. Rev. D 110, 083023 (2024), 2311.10148

2024 arXiv
[54]

D. Blas, S. Gasparotto, and R. Vicente (2024), 2410.07330

2024 arXiv
[55]

Rindler-Daller and P

T. Rindler-Daller and P. R. Shapiro, Mon. Not. Roy. As- tron. Soc. 422, 135 (2012), 1106.1256

2012 arXiv
[56]

Chavanis, Phys

P.-H. Chavanis, Phys. Rev. D 84, 043531 (2011), 1103.2050

2011 arXiv
[57]

Zhang (2024), 2406.05031

H.-Y. Zhang (2024), 2406.05031

2024
[58]

Goodman, New Astron

J. Goodman, New Astron. 5, 103 (2000), astro- ph/0003018

2000
[59]

P. J. E. Peebles, The Astrophysical Journal 534, L127 (2000), URL https://dx.doi.org/10.1086/312677

2000 doi
[60]

P. Mocz, M. Vogelsberger, V. H. Robles, J. Zavala, M. Boylan-Kolchin, A. Fialkov, and L. Hernquist, Mon. Not. Roy. Astron. Soc. 471, 4559 (2017), 1705.05845

2017 arXiv
[61]

Veltmaat, J

J. Veltmaat, J. C. Niemeyer, and B. Schwabe, Phys. Rev. D 98, 043509 (2018), 1804.09647

2018 arXiv
[62]

A. E. Mirasola, N. Musoke, M. C. Neyrinck, C. Prescod- Weinstein, and J. L. Zagorac (2024), 2410.02663

2024 arXiv
[63]

Dawoodbhoy, P

T. Dawoodbhoy, P. R. Shapiro, and T. Rindler- Daller, Mon. Not. Roy. Astron. Soc. 506, 2418 (2021), 2104.07043

2021 arXiv
[64]

R. G. Garc ´ ıa, P. Brax, and P. Valageas, Phys. Rev. D 109, 043516 (2024), 2304.10221

2024 arXiv
[65]

Chen and H.-Y

J. Chen and H.-Y. Zhang, JCAP 10, 005 (2024), 2407.09265

2024 arXiv
[66]

Barnes and G

J. Barnes and G. Efstathiou, Astrophys. J. 319, 575 (1987)

1987
[67]

J. S. Bullock, A. Dekel, T. S. Kolatt, A. V. Kravtsov, A. A. Klypin, C. Porciani, and J. R. Primack, Astrophys. J. 555, 240 (2001), astro-ph/0011001

2001 arXiv
[68]

A. V. Macci` o, A. A. Dutton, F. C. van den Bosch, B. Moore, D. Potter, and J. Stadel, Mon. Not. Roy. As- tron. Soc. 378, 55 (2007), astro-ph/0608157

2007 arXiv
[69]

P. J. E. Peebles, Astrophys. J. 155, 393 (1969)

1969
[70]

A. G. Doroshkevich, Astrofizika 6, 581 (1970)

1970
[71]

S. D. M. White, Astrophys. J. 286, 38 (1984)

1984
[72]

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ket- terle, Science 292, 476 (2001), ISSN 00368075, 10959203, URL http://www.jstor.org/stable/3082780

2001
[73]

A. L. Fetter and A. A. Svidzinsky, Journal of Physics: Condensed Matter 13, R135 (2001), URL https://dx. doi.org/10.1088/0953-8984/13/12/201

2001 doi
[74]

Pitaevskii and S

L. Pitaevskii and S. Stringari, Bose-Einstein Conden- sation, International Series of Monographs on Physics (Clarendon Press, 2003), ISBN 9780198507192, URL https://books.google.fr/books?id=rIobbOxC4j4C

2003
[75]

C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University Press, 2008), 2nd ed

2008
[76]

L. Hui, A. Joyce, M. J. Landry, and X. Li, JCAP 01, 011 (2021), 2004.01188

2021 arXiv
[77]

I.-K. Liu, N. P. Proukakis, and G. Rigopoulos, Mon. Not. Roy. Astron. Soc. 521, 3625 (2023), 2211.02565

2023 arXiv
[78]

A. S. Dmitriev, D. G. Levkov, A. G. Panin, E. K. Push- naya, and I. I. Tkachev, Phys. Rev. D 104, 023504 (2021), 2104.00962

2021 arXiv
[79]

Glennon, A

N. Glennon, A. E. Mirasola, N. Musoke, M. C. Neyrinck, and C. Prescod-Weinstein, JCAP 07, 004 (2023), 2301.13220

2023 arXiv
[80]

M. P. Silverman and R. L. Mallett, Gen. Rel. Grav. 34, 633 (2002)

2002
[81]

Kain and H

B. Kain and H. Y. Ling, Phys. Rev. D 82, 064042 (2010), 1004.4692

2010 arXiv
[82]

Alexander, C

S. Alexander, C. Capanelli, E. G. M. Ferreira, and E. Mc- Donough, Phys. Lett. B 833, 137298 (2022), 2111.03061

2022 arXiv
[83]

C. G. Boehmer and T. Harko, JCAP 06, 025 (2007), 0705.4158

2007 arXiv
[84]

P. Brax, J. A. R. Cembranos, and P. Valageas, Phys. Rev. D 100, 023526 (2019), 1906.00730

2019 arXiv
[85]

Madelung, Z

E. Madelung, Z. Phys. 40, 322 (1927)

1927
[86]

P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), 1103.2054

2011 arXiv
[87]

Harko, Mon

T. Harko, Mon. Not. Roy. Astron. Soc. 413, 3095 (2011), 1101.3655

2011 arXiv
[88]

M. N. Brook and P. Coles (2009), 0902.0605

2009 arXiv
[89]

Landau, E

L. Landau, E. Lifshitz, and L. Pitaevskii, Statisti- cal Physics: Theory of the Condensed State , Course of theoretical physics (Elsevier Science, 1980), ISBN 9780750626361, URL https://books.google.co.jp/ books?id=NaB7oAkon9MC

1980
[90]

Rindler-Daller, Physica A: Statistical Mechanics and its Applications 387, 1851 (2008), ISSN 0378- 4371, URL https://www.sciencedirect.com/science/ article/pii/S0378437107012174

T. Rindler-Daller, Physica A: Statistical Mechanics and its Applications 387, 1851 (2008), ISSN 0378- 4371, URL https://www.sciencedirect.com/science/ article/pii/S0378437107012174

2008
[91]

D. V. Skryabin, Phys. Rev. A 63, 013602 (2000), URL https://link.aps.org/doi/10.1103/PhysRevA. 63.013602

2000 doi
[92]

Kawaguchi and T

Y. Kawaguchi and T. Ohmi, Phys. Rev. A 70, 043610 (2004), URL https://link.aps.org/doi/10. 1103/PhysRevA.70.043610

2004
[93]

Lundh and H

E. Lundh and H. M. Nilsen, Phys. Rev. A 74, 063620 (2006), URL https://link.aps.org/doi/10. 1103/PhysRevA.74.063620

2006
[94]

Takeuchi, M

H. Takeuchi, M. Kobayashi, and K. Kasamatsu, Jour- nal of the Physical Society of Japan 87, 023601 (2018), https://doi.org/10.7566/JPSJ.87.023601

2018 doi
[95]

Giacomelli and I

L. Giacomelli and I. Carusotto, Phys. Rev. Res. 2, 033139 (2020), URL https://link.aps.org/doi/10. 27 1103/PhysRevResearch.2.033139

2020
[96]

Van Alphen, H

W. Van Alphen, H. Takeuchi, and J. Tempere, Phys. Rev. A 109, 043317 (2024), URL https://link.aps. org/doi/10.1103/PhysRevA.109.043317

2024 doi
[97]

A. L. Fetter, Phys. Rev. 151, 100 (1966), URL https: //link.aps.org/doi/10.1103/PhysRev.151.100

1966 doi
[99]

Lund, Physics Letters A 159, 245 (1991), ISSN 0375- 9601, URL https://www.sciencedirect.com/science/ article/pii/037596019190518D

F. Lund, Physics Letters A 159, 245 (1991), ISSN 0375- 9601, URL https://www.sciencedirect.com/science/ article/pii/037596019190518D

1991
[100]

G. K. Batchelor, An Introduction to Fluid Dynamics , Cambridge Mathematical Library (Cambridge University Press, 2000)

2000
[101]

Feynman, in Chapter II Application of Quantum Mechanics to Liquid Helium , edited by C

R. Feynman, in Chapter II Application of Quantum Mechanics to Liquid Helium , edited by C. Gorter (El- sevier, 1955), vol. 1 of Progress in Low Temperature Physics, pp. 17–53, URL https://www.sciencedirect. com/science/article/pii/S0079641708600773

1955
[102]

A. L. Fetter, Rev. Mod. Phys. 81, 647 (2009), URL https://link.aps.org/doi/10.1103/RevModPhys.81. 647

2009 doi
[103]

M. P. Hertzberg and E. D. Schiappacasse, JCAP 08, 028 (2018), 1804.07255

2018 arXiv
[104]

F. S. Guzm´ an and L. A. Ure˜ na L´ opez, Phys. Rev. D101, 081302 (2020), 1912.10585

2020 arXiv
[105]

Banik and P

N. Banik and P. Sikivie, Phys. Rev. D 88, 123517 (2013), 1307.3547

2013 arXiv
[106]

Galazo Garc ´ ıa, P

R. Galazo Garc ´ ıa, P. Brax, and P. Valageas (2024), 2412.02519

2024 arXiv
[107]

Lin, H.-Y

S.-C. Lin, H.-Y. Schive, S.-K. Wong, and T. Chiueh, Phys. Rev. D 97, 103523 (2018), 1801.02320

2018 arXiv
[108]

T. D. Yavetz, X. Li, and L. Hui, Phys. Rev. D 105, 023512 (2022), 2109.06125

2022 arXiv
[109]

L. M. Widrow and N. Kaiser, Astrophys. J. Lett. 416, L71 (1993)

1993
[110]

P. Mocz, L. Lancaster, A. Fialkov, F. Becerra, and P.-H. Chavanis, Phys. Rev. D 97, 083519 (2018), 1801.03507

2018 arXiv
[111]

Galazo Garc ´ ıa, P

R. Galazo Garc ´ ıa, P. Brax, and P. Valageas, Phys. Rev. D 105, 123528 (2022), 2203.05995

2022 arXiv
[112]

R. Liu, W. Hu, and H. Xiao (2024), 2406.12970

2024 arXiv
[113]

Pathria and J

D. Pathria and J. L. Morris, Journal of Computational Physics 87, 108 (1990)

1990
[114]

Zhang and M

Q. Zhang and M. I. Hayee, Journal of Lightwave Tech- nology 26, 302 (2008)

2008
[115]

Edwards, E

F. Edwards, E. Kendall, S. Hotchkiss, and R. Easther, JCAP 2018, 027 (2018), 1807.04037

2018 arXiv
[116]

Olver, D

F. Olver, D. Lozier, R. Boisvert, and C. Clark, The NIST Handbook of Mathematical Functions (Cambridge Uni- versity Press, New York, NY, 2010)

2010

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Thus, we start with stochastic initial conditions associated with a col- lisionless virialized halo in the semiclassical limit

Expansion over eigenfunctions As in [41, 83], we start our simulations without a cen- tral soliton since we are interested in the formation of the solitons and their generic properties. Thus, we start with stochastic initial conditions associated with a col- lisionless virialized halo in the semiclassical limit. We choose a target classical density profil...

[2] [2]

Thus, we obtain the expressions ∂ρ ∂t + ∇ ·(ρ⃗ v) = ρ NX j=1 ⃗∇f 2(⃗ r− ⃗ rj) f 2(⃗ r− ⃗ rj) · h ⃗ v− ˙⃗ rj i (44) ∂⃗ v ∂t + (⃗ v· ⃗∇)⃗ v= NX j=1 ([⃗ v− ˙⃗ rj] · ⃗∇)⃗ vj

Equations of motion Following [75] we substitute the ansatz (42) into the hydrodynamic form of the equations of motion. Thus, we obtain the expressions ∂ρ ∂t + ∇ ·(ρ⃗ v) = ρ NX j=1 ⃗∇f 2(⃗ r− ⃗ rj) f 2(⃗ r− ⃗ rj) · h ⃗ v− ˙⃗ rj i (44) ∂⃗ v ∂t + (⃗ v· ⃗∇)⃗ v= NX j=1 ([⃗ v− ˙⃗ rj] · ⃗∇)⃗ vj. (45) The terms in the right-hand sides are dominated by the neighb...

[3] [3]

Thus, we write the wave function as ψ(⃗ r, t) = √ρeis NY j=1 eiσj θj , (47) where again θj(⃗ r) = \(⃗ ex, ⃗ r− ⃗ rj), ⃗∇θj = ⃗ ez × ⃗∇ ln |⃗ r− ⃗ rj|

Effective action A more general and elegant approach is to substitute our ansatz into the action (19). Thus, we write the wave function as ψ(⃗ r, t) = √ρeis NY j=1 eiσj θj , (47) where again θj(⃗ r) = \(⃗ ex, ⃗ r− ⃗ rj), ⃗∇θj = ⃗ ez × ⃗∇ ln |⃗ r− ⃗ rj|. (48) Here ρ(⃗ r, t) and s(⃗ r, t) are smooth functions and we ne- glect the width of the vortices and t...

[4] [4]

(82) Then, we take for the initial wave function [41, 84, 85] a sum with random coefficients anℓ over the eigenmodes ˆψnℓ(⃗ r) of the Schr¨ odinger equation defined by this target gravitational potential Φ N , ψ(⃗ r) = X nℓ anℓ ˆψnℓ(⃗ r), ˆψnℓ(⃗ r) = Rn|ℓ|(r)eiℓθ, (83) where the radial parts satisfy the radial Schr¨ odinger equation − ϵ2 2 1 r d dr r d dr...

[5] [5]

(96) Then, we obtain ρ = 2π Z 0 ΦN dE f0(E), (97) which can be inverted to give the isotropic distribution function ΦN 0 < E <0 : f0(E) = z2 0 8π2R2

Isotropic initial conditions For isotropic initial conditions the classical phase-space distribution does not depend on the angular momentum, f (E, L) = f0(E). (96) Then, we obtain ρ = 2π Z 0 ΦN dE f0(E), (97) which can be inverted to give the isotropic distribution function ΦN 0 < E <0 : f0(E) = z2 0 8π2R2 . (98) Thus, the distribution function is a cons...

[6] [6]

In particular, the mean angular rotation becomes nonzero if f (E, L) depends on the sign of L

Anisotropic initial conditions For anisotropic initial conditions the phase-space dis- tribution depends on L. In particular, the mean angular rotation becomes nonzero if f (E, L) depends on the sign of L. In this paper we take the simple choice f (E, L) = f0(E)[1 + α sign(L)], −1 ≤ α ≤ 1, (99) where f0(E) is the isotropic distribution (98). This re- cove...

[7] [7]

For the initial wave function we take R = 1 and M = 1 for the target halo radius and mass in Eqs.(80)-(81)

Simulation parameters In this paper we consider the cases ϵ = 0.005, 0.01 and 0.03, as we focus on the semi-classical regime, and we mostly illustrate our results with the intermediate case ϵ = 0 .01. For the initial wave function we take R = 1 and M = 1 for the target halo radius and mass in Eqs.(80)-(81). For the anisotropy parameter of Eq.(99) we consi...

[8] [8]

2 how the system evolves from the initial condition (1)

Mass, density and velocity profiles For the case ϵ = 0.01 and α = 1, we show in Fig. 2 how the system evolves from the initial condition (1). As in the 3D isotropic simulations presented in [41], in a few dynamical times a soliton quickly forms at the center of the system. As seen in the left panel, where we show the mass MTF,0 within the radius R0 of Eq....