Formation of solitons and their transitions in scalar-field dark matter models with a non-polynomial self-interaction potential

Patrick Valageas; Philippe Brax; Raquel Galazo Garc\'ia

REVIEW 3 major objections 4 minor 1 cited by

Formation of solitons and their transitions in scalar-field dark matter models with a non-polynomial self-interaction potential

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper shows that scalar-field dark matter halos with a saturating self-interaction can form Thomas-Fermi solitons, fuzzy dark matter solitons, and an abrupt transition between the two as the central density crosses the saturation…

desk verdict A solid numerical study of soliton formation with a saturating self-interaction; the TF-to-FDM transition is well supported, but the 'subdominant seeding' headline is under-evidenced by single-realization runs. read the letter →

arxiv 2412.02519 v1 pith:IFUPB5UC submitted 2024-12-03 astro-ph.CO

Raquel Galazo García , Philippe Brax , Patrick Valageas This is my paper · ORCID

classification astro-ph.CO

keywords scalar-fielddarkmattersolitonsThomas-Fermiregimefuzzyself-interactionsaxionmonodromySchrödinger-Poissonsystemgravitationalcondensation

topics Dark Matter

open problems 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 scalar-field dark matter halos whose repulsive self-interaction rises linearly with density at low density but saturates once the density exceeds a threshold $\rho_c$, as in axion-monodromy inspired potentials. It argues that the same halo can settle into a Thomas-Fermi soliton, where self-interaction pressure balances gravity, or a fuzzy dark matter soliton, where quantum pressure balances gravity, and that when the central density crosses $\rho_c$ the Thomas-Fermi soliton becomes unstable and collapses into the fuzzy one. It also argues that self-interactions too weak to support any soliton can still play a critical role: they give the central region a small early density boost that seeds a fuzzy soliton much earlier than it would form without them. These behaviours are captured by a simple Gaussian ansatz, which reduces the soliton problem to finding minima of an energy curve as a function of central density. This matters because the same particle model could then produce different cores in different halos, altering predictions for galactic rotation curves and small-scale structure formation.

What carries the argument

The central objects are the dimensionless saturating self-interaction potential $\Phi_I(\rho) = \lambda\rho$ below the threshold density $\rho_c$ and $\Phi_I(\rho)=\lambda\rho_c$ above it, together with the Gaussian ansatz $\rho(r)=\rho_0 e^{-r^2/R^2}$ inserted into the total energy functional at fixed mass. The ansatz turns the search for solitons into a one-dimensional minimization problem: local minima of $E_{\rm tot}(\rho_0)$ correspond to the two soliton types, the depth and presence of those minima tell whether a transition happens, and the energy gap between them determines how much mass falls into the new soliton.

What would settle it

Re-run the $R_{\rm TF}=0.5$, $\rho_c=0.5$ and $R_{\rm TF}=0.1$, $\rho_c=0.5$ simulations with many independent random-phase initial conditions and compare the distributions of fuzzy soliton formation times; if the distributions overlap substantially, the seeding role attributed to the weak self-interaction is not established.

Watch

Extended reading notes

Core claim

Starting from a virialized, wave-like halo in the semiclassical regime, the formation and fate of the central soliton is controlled by the relation between its central density and the saturation density $\rho_c$ of the self-interaction. If the central density stays below $\rho_c$, a Thomas-Fermi soliton of fixed radius $R_{\rm TF}$ forms and is stable. If the self-interactions saturate before they can support such a soliton, the halo eventually condenses into a fuzzy dark matter soliton supported by quantum pressure. If a Thomas-Fermi soliton grows until its central density reaches $\rho_c$, the self-interaction pressure vanishes and the soliton collapses into a much smaller, denser fuzzy soliton. The paper further claims that even when self-interactions are always subdominant, they can seed the later fuzzy soliton by boosting central density contrasts at early times, and that all of these regimes are predictable from the energy landscape of a Gaussian density profile.

Load-bearing premise

The numerical evidence that always-subdominant self-interactions seed the later fuzzy soliton compares a single random-phase realization per parameter set, so the difference in formation times could in principle be a large stochastic fluctuation.

Editorial extensions

If this is right

A single scalar-field model can produce halos whose central cores are of different types, Thomas-Fermi or fuzzy, depending on the halo's mass and formation history rather than one universal soliton profile.
When the Thomas-Fermi soliton becomes unstable, its collapse into a fuzzy soliton is fast (around $t=7$ for $R_{\rm TF}=0.5$, $\rho_c=3$) and ejects most of the original core mass when the two energy levels are far apart.
Self-interactions that never dominate the dynamics can still act as a catalyst, shortening fuzzy soliton formation from $t\sim3000$ to $t\sim100$ for the $\rho_c=0.5$ configurations studied.
The Gaussian energy analysis predicts for each parameter set a critical soliton mass above which the Thomas-Fermi minimum disappears, so the transition threshold can be computed before running a simulation.

Reading between the lines

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

A direct testable extension is to repeat the $\rho_c=0.5$, $R_{\rm TF}=0.5$ run with an ensemble of independent random-phase realizations; if the early boost is causal, fuzzy soliton formation times should cluster near $t\sim100$ rather than spread toward the no-interaction condensation time $t\sim3000$.
The same two-minimum energy structure should persist for smooth versions of the saturating potential, so the Thomas-Fermi to fuzzy collapse is likely a generic feature of saturating self-interactions rather than an artifact of the piecewise-linear model.
Astrophysically, the seeding effect implies that a self-interaction too weak to alter halo-averaged dynamics could still leave an observable imprint through the earlier appearance of compact central cores; observations tied to the presence or timing of a core could constrain the saturation density more tightly than hydrostatic arguments alone.

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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

Colleague — read this one for the saturating self-interaction. The paper studies a non-polynomial axion-monodromy-like potential that is quartic at low density but saturates above rho_c, and shows numerically and with a Gaussian ansatz that a halo can form either a TF soliton or an FDM soliton, and that a TF soliton can collapse into an FDM soliton when it exceeds the saturation density. That transition is the strongest part: the collapse at t~7 in Fig 4 matches the energy analysis (critical mass ~0.45, density ~5) and the two cases (RTF=0.5 vs 0.1) give different mass capture fractions that the energy argument explains. The Gaussian ansatz is a genuinely useful tool here; it is not fitted to the simulations, and the agreement is qualitative but credible.

The soft spot is the second headline claim: that even when self-interactions are always subdominant they play a critical role by seeding the later FDM soliton. The evidence is a comparison of two runs, one per parameter set (Fig 3 vs Fig 9). Initial density fluctuations are O(1) on the de Broglie scale (Eq 53), so the time at which a random peak grows into a soliton could easily vary by more than the factor of 30 reported. The paper does not provide an ensemble, convergence tests, or a variance estimate for the condensation time. The Gaussian ansatz supports the existence of the TF branch and the early boost, but it does not quantify the magnitude or variance of the seeding effect. So I would treat that specific claim as suggestive, not established. The other results — the TF-to-FDM transition, the energy-barrier explanation, and the caution about the hydrodynamical picture — do hold up.

The citation pattern is fine; the method follows their prior work and standard references. The simplified potential (15) drops the Bessel oscillations, which is acceptable at this level.

Who is this for? People working on soliton cores in scalar-field DM, especially axion-monodromy models. It is a within-subfield advance, not a breakthrough. It deserves a serious referee: the science is sound in its core, and the seeding claim can be fixed by adding a few ensembles or softening the language. If I were refereeing, I would ask for that before acceptance.

Referee Report

3 major / 4 minor

Summary. This paper studies the formation and evolution of solitons in a scalar-field dark matter model with a repulsive self-interaction that saturates at high density, motivated by axion monodromy potentials. The authors use a simplified piecewise potential (15) and simulate the Gross-Pitaevskii–Poisson system in the semi-classical regime (epsilon = 0.01) for two values of the Thomas-Fermi radius (RTF = 0.5 and 0.1) and several saturation densities rho_c. They identify three regimes: a stable Thomas-Fermi soliton when rho_c is large, a directly formed FDM soliton when rho_c is small, and a TF-to-FDM transition when the soliton density exceeds rho_c. A Gaussian energy ansatz reproduces the existence and approximate locations of the TF and FDM energy minima and the disappearance of the TF minimum at a critical mass. The paper further claims that even always-subdominant self-interactions can accelerate FDM soliton formation by boosting early density fluctuations, based on the comparison of Fig. 3 (RTF = 0.5, rho_c = 0.5) with Fig. 9 (RTF = 0.1, rho_c = 0.5).

Significance. The paper is clearly written and provides a useful contribution to the phenomenology of scalar-field dark matter with saturating self-interactions. Its main strengths are the explicit mapping from a monodromy-type potential to a tractable nonrelativistic model, the systematic exploration of the (RTF, rho_c) parameter space, and the Gaussian ansatz, which gives a simple analytic account of the TF and FDM branches and of the transition between them. The TF-to-FDM transition at intermediate rho_c is well supported: the disappearance of the low-density energy minimum in Fig. 7 (M ~ 0.45, rho_0 ~ 5) coincides with the simulated collapse near rho_max ~ 6 in Fig. 4. The paper is also honest about its assumptions, such as neglecting the Bessel oscillations after Eq. (15) and the energy-matching ansatz in Sec. V B 3. However, the second headline claim concerning the 'critical role' of subdominant self-interactions rests on a single realization per parameter set, and the numerical simulations lack convergence tests and error estimates.

major comments (3)

[Secs. V A 2, VI A 2, Figs. 3 and 9, Eqs. (50)-(53), (56)] The claim that subdominant self-interactions play a critical role in seeding a later FDM soliton rests on comparing one realization with RTF = 0.5, rho_c = 0.5 (soliton at t ~ 100) with one realization with RTF = 0.1, rho_c = 0.5 (soliton at t ~ 3000). The initial conditions have order-unity density fluctuations on the de Broglie scale, as stated in Eq. (53), so the time at which a rare density peak grows into a soliton is itself a stochastic quantity. The fact that all runs share the same random-phase realization (Sec. IV A) controls for differences between parameter sets, but it does not establish that the observed boost is typical; a single common realization could be one where the effect is unusually strong or weak. The quoted gravitational condensation time t_gr ~ 3300 in Eq. (56) is a mean kinetic-theory estimate, not a measure of the realization-to-realization scatter. I recommend running an ensemble of realizations for these two parameter sets and reporting the distribution (or at least the mean and standard deviation) of FDM soliton formation times and masses, so that the reader can assess whether the t ~ 100 versus t ~ 3000 difference exceeds the stochastic scatter.
[Sec. IV B and all simulation results] No numerical resolution or convergence information is provided. The manuscript does not state the grid size, box size, time step, or the number of eigenmodes used in the initial conditions of Eq. (48), nor does it report tests with different resolutions. This matters because the FDM soliton radius is of order epsilon^2/M ~ 0.01 (Eq. (44)) for the masses found in the simulations, so the spatial grid must resolve scales comparable to epsilon = 0.01, while density contrasts reach values of order 10^5. Without convergence tests, the quantitative values used for validation—for example the transition time t ~ 7 in Fig. 4, the final MFDM values, and the comparison with the Gaussian-ansatz critical mass—could depend on resolution. Please provide the numerical parameters and at least one convergence test for each regime (TF-stable, TF-to-FDM transition, and FDM-seeded).
[Secs. V B 3 and VI B 3, Figs. 7 and 13] The predicted final mass of the FDM soliton after the TF collapse relies on the explicit assumption Etot,FDM ~ Etot,TF, with the leftover TF matter assigned zero energy. This assumption is acknowledged in the text, but it is load-bearing for the claim that the Gaussian ansatz quantitatively explains the mass partition between the TF and FDM solitons (the 10% versus 60% cases). The agreement is only to within a factor of about 1.6 (MFDM ~ 0.08 predicted versus ~ 0.05 simulated in Sec. V B 3), so the test is not very stringent. I ask the authors to test this assumption directly from the simulation data, for instance by measuring the energy of the material that ends up in the FDM soliton and the energy of the ejected material at the transition time (t ~ 8 in Fig. 4), or to present the energy-matching estimate more cautiously as a heuristic with an explicit statement of its uncertainty.

minor comments (4)

[Sec. II B, Eq. (15)] The paper explicitly neglects the decaying oscillations of the Bessel function in Eq. (11) when introducing the simplified potential (15). Since this is the main modeling step connecting the results to the axion monodromy potential, a brief discussion of how the oscillations could affect the transition threshold, or one test run with the full Phi_I, would strengthen the robustness of the conclusions.
[Sec. V B 2] The text states that 'the self-interactions still prevent the existence of a FDM soliton for M < 0.012' but does not explain how this threshold is obtained from the Gaussian-ansatz energy curves. Please clarify the criterion used to define existence of a soliton branch in these figures.
[Eq. (56)] The estimate t_gr ~ 3300 uses rho ~ 0.5 and v ~ 1, but the provenance of these values is not given. Please state explicitly how these values follow from the initial profile (46)-(47).
[General] There are several typographical and formatting errors, for example 'wich' in Sec. V B 1 and some broken equation layouts in the text. A careful proofread would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Gaussian-ansatz predictions are not fitted to the simulation outputs, and the self-citations are methodological rather than load-bearing.

full rationale

The paper's central claims are supported by two independent lines of evidence that are not circular. First, the numerical simulations solve the Schrodinger-Poisson system with the specified potential (19), and the observed TF and FDM solitons are identified from the resulting density and potential profiles, not from any pre-imposed soliton solution. Second, the Gaussian ansatz of Sec. III C provides energy curves Etot(rho0) from Eqs. (36)-(38), with no free parameters fitted to the simulation outcomes; the minima of these curves are then compared with the simulated soliton densities and masses, and the comparisons are approximate but not constructed to match. The transition estimates in Secs. V B 3 and VI B 3 use explicit assumptions such as Etot,FDM about Etot,TF, which are stated assumptions rather than fits. The 'critical role' seeding claim in Sec. VI A 2 is an interpretation of a deterministic comparison between two runs that share the same initial realization (as stated in Sec. IV A: 'This initial condition, which is common to all the runs performed in this paper'), so it does not reduce to a definition or a fitted parameter. The paper cites the authors' prior work [44] and [33,37] for the initial-condition construction and the Gaussian-ansatz method, respectively, but these are methodological references and the present results rest on the simulations and the external kinetic-theory estimate from [45]. No uniqueness theorem is imported from the authors' prior work, and no prediction is equivalent to its input by construction. The single-realization nature of the seeding comparison is a legitimate statistical robustness concern, but it is not a circularity. Overall, only minor, non-load-bearing self-citations are present, so the paper is essentially self-contained and non-circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

All model inputs are scenario parameters (epsilon, RTF, rho_c) rather than fitted constants. The main added assumptions are the simplified potential (15) and the representativeness of single realizations. No new particles, forces, or conserved quantities are introduced.

free parameters (3)

epsilon = 0.01
Chosen to place the simulations in the semi-classical regime (de Broglie wavelength much smaller than system); it sets the quantum pressure scale and controls all FDM scalings.
RTF = 0.5 and 0.1
Equivalently the low-density self-interaction strength lambda via Eq. (30); chosen to represent strong and weak self-interactions compared with the halo radius.
rho_c = 100, 3, 0.5, 500, 80
Saturation density threshold in Eq. (15); chosen to probe large, intermediate, and small saturation values and to trigger or suppress TF-to-FDM transitions.

assumptions (5)

domain assumption Schrodinger-Poisson equations (17)-(18) correctly describe nonrelativistic scalar-field dark matter on galactic scales.
Derived from the Klein-Gordon equation in the weak-field nonrelativistic limit, following refs [11,12,33]; standard in the ultralight DM literature.
ad hoc to paper The simplified piecewise potential (15) captures the essential physics of the full monodromy cosine potential (4).
Eq. (15) discards the decaying Bessel oscillations present in Eq. (11) without a quantitative check; the saturation plateau is the key feature the paper studies.
domain assumption The random-phase eigenmode initial condition (Eqs. 48-50) statistically represents a virialized classical halo in the limit epsilon to 0.
The construction is standard (refs [38,39]) and the mean density converges to the Lane-Emden profile (46), but individual realizations fluctuate with <rho^2>_c = <rho>, Eq. (53).
ad hoc to paper A single random-phase realization per parameter set is representative of the ensemble.
No ensemble averaging or statistical error bars are provided; comparisons in Figs 3 and 9 use one run each.
domain assumption The Gaussian density profile (33) is a sufficient variational ansatz for soliton minima.
It recovers TF and FDM scalings up to order-unity factors, as shown in Sec III C, but is not exact.

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Cite this review

Pith. "Pith review of Formation of solitons and their transitions in scalar-field dark matter models with a non-polynomial self-interaction potential." pith.science (2026). https://pith.science/paper/IFUPB5UC

@misc{pith2026241202519,
  author       = {Pith},
  title        = {Pith review of: Formation of solitons and their transitions in scalar-field dark matter models with a non-polynomial self-interaction potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IFUPB5UC}},
  note         = {Machine review of arXiv:2412.02519}
}

read the original abstract

We study the formation of solitons inside scalar-field dark matter halos with a non-polynomial self-interaction potential. We consider a self-interaction potential that is quartic in the scalar field in the low-density regime but saturates at large densities. This mimics the behaviour of axion monodromy potentials. We concentrate on the semi-classical regime, where the de Broglie wavelength is much smaller than the size of the system. We find that depending on the strength and scale of the self-interactions, the system can form solitons of the Thomas-Fermi type (dominated by self-interactions) or of the Fuzzy Dark Matter type (dominated by the quantum pressure). The system can also display transitions from a Thomas-Fermi soliton to a Fuzzy Dark Matter soliton as the former becomes unstable. We show that these behaviours can be understood from a simple Gaussian ansatz. We find that even in cases where the self-interactions are always subdominant they can play a critical role, by providing a small density boost that is enough to generate the seed for the formation of a Fuzzy Dark Matter soliton at much later times. We also point out that the intuition derived from a hydrodynamical picture can be misleading in regimes where wave effects are important.

Figures

Figures reproduced from arXiv: 2412.02519 by the authors.

**Figure 2.** FIG. 2. [ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. [ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. Evolution of a halo with [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. [ [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 7.** Figure 7: FIG. 7. [ [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. Evolution of a halo with [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: FIG. 9. Evolution of a halo with [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: FIG. 10. Evolution of a halo with [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: FIG. 11. [ [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 13.** Figure 13: FIG. 13. [ [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]

**Figure 12.** Figure 12: FIG. 12. [ [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

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Reference graph

Works this paper leans on

65 extracted references · 22 canonical work pages · cited by 1 Pith paper

[1]

Low-density regime In the Thomas-Fermi regime where we can neglect the quantum pressure because ϵ ≪ 1 and gravity is balanced by the repulsive self-interactions, the hydrostatic equilib- rium is given by TF regime : Φ N + ΦI = µ. (28) At low density ρ < ρc, where ΦI = λρ, the soliton density profile reads [23, 33] ρ0 < ρc : ρTF(r) = ρ0 sin(πr/RTF) πr/RTF ...

work page
[2]

The Thomas- Fermi regime no longer exists and the soliton is deter- mined by the balance between gravity and the quantum pressure

High-density regime At high density, ρ ≫ ρc, the potential ΦI is flat, which corresponds to vanishing self-interactions. The Thomas- Fermi regime no longer exists and the soliton is deter- mined by the balance between gravity and the quantum pressure. This gives FDM regime : Φ Q + ΦN = µ − λρc, (31) with the scaling ρ0 ≫ ρc : RFDM ∼ ϵ2 M , ρ 0 ∼ M 4 ϵ6 , ...

work page
[3]

(44) We recover the scaling (32)

High-density regime In the high density regime (32) we write ρ0 ≫ ρc : Etot ≃ EK +EN = ϵ23πM 1/3ρ2/3 0 4 − M 5/3ρ1/3 0√ 2 , (43) and the saddle-point condition (40) gives ρ0 = √ 2 ϵ23π !3 M 4, R = r π 2 3ϵ2 M . (44) We recover the scaling (32). For a virial velocity v2 = M/R, this gives for the dimensionless de Broglie wave length (22) λdB = π 2 1/4 2π √ ...

work page
[4]

(33) This bypasses the numerical computation of the soliton profiles and should provide the correct scalings and be- haviors

Energy functional To study the transitions between low and high density solitons we can use a simple analytical approach where we use a Gaussian ansatz for the density radial profile [23, 24, 37], ρ(r) = ρ0 e−(r/R)2 , ψ = √ρ, with ρ0 = M π3/2R3 . (33) This bypasses the numerical computation of the soliton profiles and should provide the correct scalings a...

work page
[5]

(42) Up to factors of order unity, we recover the exact result (30)

Low-density regime In the low-density Thomas-Fermi regime (29) we write ρ0 < ρc : Etot ≃ EN + EI = − M 5/3ρ1/3 0√ 2 + λM ρ0 4 √ 2 , (41) and the saddle-point condition (40) gives ρ0 = 4 3λ 3/2 M, R = r 3λ 4π . (42) Up to factors of order unity, we recover the exact result (30)

work page
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Arcadi, M

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Large density threshold, ρc = 100 We first set ρc = 100 in Eq. (15). This high den- sity threshold means that the central density will never reach the threshold ρc and the self-interaction potential always takes the form Φ I = λρ, as for a quartic self- interaction λ4ϕ4/4. Thus, as seen in Fig. 2 we recover the results obtained in Fig. 3 of [44], where we...

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Small density threshold, ρc = 0.5 We now consider the opposite case, ρc = 0.5, in Fig. 3. This low density threshold means that the self-interaction potential is mostly constant and small and it is unable to form a TF soliton, as we can see in the figure. Nev- ertheless, we can see in the second row, by comparison with the initial condition in Fig. 1, tha...

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[9]

Intermediate density threshold, ρc = 3 We now discuss the intermediate case, ρc = 3, shown in Fig. 4. At early times the dynamics are identical to those of the high-density threshold case shown in Fig. 2, with the fast formation by t ∼ 6 of a soliton of radius RTF = 0 .5 suppo...
[10]

Large density threshold, ρc = 100 0 2 4 0 0.015 0.010 0.005 0.000 0.005 0.010 0.015 0.020 E EK EN EI Etot M = 0.08 10 1 101 103 105 107 0 2 1 0 1 2 3 4 E EK EN EI Etot M = 0.08 0 5 10 15 20 0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 E EK EN EI Etot M = 0.6 10 1 101 103 105 107 109 0 20...
[11]

Small density threshold, ρc = 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0 0.0004 0.0002 0.0000 0.0002 0.0004 0.0006 E EK EN EI Etot M = 0.012 10 1 100 101 102 103 104 0 0.002 0.000 0.002 0.004 0.006 E EK EN EI Etot M = 0.012 0 1 2 3 4 5 0 0.004 0.002 0.000 0.002 0.004 E EK EN EI Etot M = 0....
[12]

Intermediate density threshold, ρc = 3 0 2 4 0 0.015 0.010 0.005 0.000 0.005 0.010 0.015 0.020 E EK EN EI Etot M = 0.08 10 1 101 103 105 0 0.2 0.1 0.0 0.1 0.2 E EK EN EI Etot M = 0.08 0 5 10 15 20 0 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 E EK EN EI Etot M = 0.45 10 1 101 103 105 ...
[13]

V A 1, we first set the threshold ρc = 500 to a very high value so that the self-interaction potential always takes the form Φ I = λρ

Large density threshold, ρc = 500 As in Sec. V A 1, we first set the threshold ρc = 500 to a very high value so that the self-interaction potential always takes the form Φ I = λρ. Therefore, the system behaves as in [44]. In contrast with the case RTF = 0.5 shown in Fig. 2, th...
[14]

9 the case of a small thresh- old, ρc = 0.5

Small density threshold, ρc = 0.5 We now consider in Fig. 9 the case of a small thresh- old, ρc = 0.5. As noticed in Sec. V A 2, as compared with the case shown in Fig. 3 which has the same threshold ρc = 0 .5 but a larger TF radius RTF = 0 .5, the self- interactions no longer...
[15]

Intermediate density threshold, ρc = 80 We now discuss the intermediate case, ρc = 80, shown in Fig. 10. This is somewhat similar to the other inter- mediate case shown in Fig. 4, in the sense that there is a transition from a TF soliton to a FDM soliton. At early times, the d...
[16]

Large density threshold, ρc = 500 We show the case ρc = 500 in Fig. 11. We can see that even to form a relatively low mass TF soli- 14 0 50 100 150 200 t 100 101 102 max 0 50 100 150 200 t 10 2 10 1 100 M Mtot MTF 0 50 100 150 200 t 0.6 0.4 0.2 0.0 0.2 0.4 0.6 E EK EN EI Etot ...
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Small density threshold, ρc = 0.5 We show the case ρc = 0 .5 in Fig. 12. We can see that for a small mass MFDM = 0 .001, of the order of the central mass in the simulation in Fig. 9, a FDM soli- ton would have a negligible density ρ0 ∼ 0.01 and en- ergy |E| ∼10−7. This means s...
[18]

13 the intermediate case ρc = 80

Intermediate density threshold, ρc = 80 We show in Fig. 13 the intermediate case ρc = 80. We can see that until M ≲ 0.037 there is only one energy minimun, at a moderate density. This is the TF solution (42), which disappears at M ≃ 0.11 to leave only the new high-density FDM ...
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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Low-density regime In the Thomas-Fermi regime where we can neglect the quantum pressure because ϵ ≪ 1 and gravity is balanced by the repulsive self-interactions, the hydrostatic equilib- rium is given by TF regime : Φ N + ΦI = µ. (28) At low density ρ < ρc, where ΦI = λρ, the soliton density profile reads [23, 33] ρ0 < ρc : ρTF(r) = ρ0 sin(πr/RTF) πr/RTF ...

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[2] [2]

The Thomas- Fermi regime no longer exists and the soliton is deter- mined by the balance between gravity and the quantum pressure

High-density regime At high density, ρ ≫ ρc, the potential ΦI is flat, which corresponds to vanishing self-interactions. The Thomas- Fermi regime no longer exists and the soliton is deter- mined by the balance between gravity and the quantum pressure. This gives FDM regime : Φ Q + ΦN = µ − λρc, (31) with the scaling ρ0 ≫ ρc : RFDM ∼ ϵ2 M , ρ 0 ∼ M 4 ϵ6 , ...

work page

[3] [3]

(44) We recover the scaling (32)

High-density regime In the high density regime (32) we write ρ0 ≫ ρc : Etot ≃ EK +EN = ϵ23πM 1/3ρ2/3 0 4 − M 5/3ρ1/3 0√ 2 , (43) and the saddle-point condition (40) gives ρ0 = √ 2 ϵ23π !3 M 4, R = r π 2 3ϵ2 M . (44) We recover the scaling (32). For a virial velocity v2 = M/R, this gives for the dimensionless de Broglie wave length (22) λdB = π 2 1/4 2π √ ...

work page

[4] [4]

(33) This bypasses the numerical computation of the soliton profiles and should provide the correct scalings and be- haviors

Energy functional To study the transitions between low and high density solitons we can use a simple analytical approach where we use a Gaussian ansatz for the density radial profile [23, 24, 37], ρ(r) = ρ0 e−(r/R)2 , ψ = √ρ, with ρ0 = M π3/2R3 . (33) This bypasses the numerical computation of the soliton profiles and should provide the correct scalings a...

work page

[5] [5]

(42) Up to factors of order unity, we recover the exact result (30)

Low-density regime In the low-density Thomas-Fermi regime (29) we write ρ0 < ρc : Etot ≃ EN + EI = − M 5/3ρ1/3 0√ 2 + λM ρ0 4 √ 2 , (41) and the saddle-point condition (40) gives ρ0 = 4 3λ 3/2 M, R = r 3λ 4π . (42) Up to factors of order unity, we recover the exact result (30)

work page

[6] [6]

Arcadi, M

G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mam- brini, M. Pierre, S. Profumo, and F. S. Queiroz, Eur. Phys. J. C 78, 203 (2018), 1703.07364

arXiv 2018

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Large density threshold, ρc = 100 We first set ρc = 100 in Eq. (15). This high den- sity threshold means that the central density will never reach the threshold ρc and the self-interaction potential always takes the form Φ I = λρ, as for a quartic self- interaction λ4ϕ4/4. Thus, as seen in Fig. 2 we recover the results obtained in Fig. 3 of [44], where we...

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[8] [8]

Small density threshold, ρc = 0.5 We now consider the opposite case, ρc = 0.5, in Fig. 3. This low density threshold means that the self-interaction potential is mostly constant and small and it is unable to form a TF soliton, as we can see in the figure. Nev- ertheless, we can see in the second row, by comparison with the initial condition in Fig. 1, tha...

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