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Can dark-matter Q-balls grow to the mass gap masses?

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Dark-matter Q-balls can grow to about five solar masses near the galactic center, but their radii then reach roughly 10^9 km, so in this model they are diffuse clouds and cannot explain LIGO and Virgo's mass-gap events.

desk verdict The growth law in Eq. (41)-(43) overcounts the ambient dark-matter density and produces an artificial t^4 growth; the paper's central mass estimate is unsupported, but the qualitative conclusion may still survive. read the letter →

arxiv 2412.08803 v2 pith:JHYVMTKO submitted 2024-12-11 astro-ph.CO

Alexander Libanov This is my paper · ORCID
classification astro-ph.CO
keywords Q-ballsdarkmatternontopologicalsolitonsmassgapgravitationalwavesFriedberg-Lee-Sirlinmodelfirst-orderphasetransitionLIGO/Virgo
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 asks whether dark-matter Q-balls, nontopological solitons born in a first-order phase transition in the early Universe, can merge into the mass-gap objects between neutron stars and light black holes that LIGO and Virgo detect. In the flat expanding Universe, mergers are so diluted by cosmic expansion that Q-balls barely grow. Once Q-balls are confined inside a galaxy, the selected Q-ball follows a growth law that, for benchmark parameters, reaches about 5 solar masses near the galactic center after 13 Gyr. The catch is that its radius is then about $10^{9}$ km, far larger than its Schwarzschild radius, so the object is a diffuse dark-matter cloud rather than a compact star. The paper therefore concludes that Q-balls can grow to mass-gap masses, but the presented model cannot explain the unusual gravitational-wave events.

What carries the argument

The central object is the Friedberg-Lee-Sirlin Q-ball, a spherical nontopological soliton in which a complex field $\chi$ carries a conserved charge $Q$ and a real field $\phi$ forms a potential well; for large $Q$ its mass and radius scale as $m_Q = \frac{4\sqrt{2}\pi}{3} v Q^{3/4}$ and $R_Q = (Q/4)^{1/4}/v$. The argument is carried by the galaxy merger law (41), simplified to $\dot{Q} = Q u_\star \sigma(Q) n(Q)$, whose solution $Q(t,r) = \left[\frac{3u_\star\rho(r)}{64\sqrt{2}v^3} t + Q_\star^{1/4}\right]^4$ produces the $t^4$ growth. Substituting this into the mass formula yields the final expression (46) that gives the $\sim 5\,M_\odot$ estimate and, through (50), the $\sim 10^9$ km radius.

What would settle it

Recompute the charge growth in (41) using the birth distribution (18) for the ambient Q-ball population, so that $n(kQ_\star)$ is not set to the full local dark-matter density divided by $m_Q(kQ_\star)$ for every $k$, and check whether the selected Q-ball still reaches $1\,M_\odot$ by 13 Gyr; if it does not, the paper's mass estimate collapses.

Watch

Extended reading notes

Core claim

The central claim is that within the Friedberg-Lee-Sirlin Q-ball model, a selected Q-ball in a galactic dark-matter halo can grow to masses of order the solar mass, yet its size becomes of order the Solar system, making it unlike the compact objects that LIGO and Virgo observe. Concretely, the mass formula (46) with the benchmark parameters (48) gives $m_Q \approx 5\,M_\odot$ at distance $r = 0.05$ kpc from the galactic center at $t = 13$ Gyr, with radius $R_Q \approx 10^9$ km from (50). Because this radius is far above the Schwarzschild radius for one solar mass, the paper's answer to the title question is a qualified yes: Q-balls can reach mass-gap masses, but the presented scenario cannot explain unusual gravitational-wave events such as GW190814, GW200105, and GW200115. The paper also estimates a population of about $4\times 10^9$ stellar-mass Q-balls within 0.17 kpc of the galactic center and about $10^{24}$ Q-balls in the Milky Way halo.

Load-bearing premise

The load-bearing premise is that the ambient dark matter around the growing Q-ball contains every charge multiple $kQ_\star$ with the full local dark-matter density, so the largest-charge contribution is always available to be absorbed; if that population is finite or follows the charge distribution in (18), the $t^4$ growth is not justified.

Editorial extensions

If this is right

  • Free cosmological Q-balls in the flat expanding Universe almost never merge, so their masses remain essentially fixed after the phase transition.
  • Inside a galactic halo, the selected Q-ball's mass grows toward about 5 solar masses at $r = 0.05$ kpc after 13 Gyr for the benchmark parameters.
  • The corresponding radius is about $10^9$ km, far larger than the Schwarzschild radius for one solar mass.
  • Stellar-mass Q-balls in this model are therefore diffuse clouds, not compact objects, and cannot explain LIGO/Virgo mass-gap events like GW190814.
  • The model predicts about $4\times 10^9$ stellar-mass Q-balls within 0.17 kpc of the galactic center and about $10^{24}$ Q-balls in the Milky Way halo.

Reading between the lines

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

  • A direct corollary the paper does not foreground is a compactness bound: for any allowed parameter set, $R_Q$ stays orders of magnitude above the Schwarzschild radius, so within this model Q-balls cannot mimic black holes in gravitational-wave templates.
  • The same merger equation applied to denser environments, such as the innermost parsec or early protogalactic halos, would give even larger masses, but the radius grows in lockstep, so the diffuse-cloud conclusion is robust.
  • The overcounting assumption behind the growth law can be tested with N-body simulations of Q-ball accretion in which the ambient charge spectrum is finite and follows the birth distribution; such simulations would settle whether the $t^4$ law is physical.
  • If roughly $10^9$-km stellar-mass Q-balls exist in the Galactic center, they would produce astrometric or microlensing signatures distinct from point masses; the paper's comparison with microlensing observations is a first step toward searching for them.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies Friedberg-Lee-Sirlin Q-balls as a dark-matter candidate and asks whether they can grow through mergers to masses in the ~3-5 M_sun mass gap. It derives the cosmological Q-ball charge from a first-order phase transition, obtains a charge distribution (18), and develops two growth models: one in the expanding universe (Section IV.A) and one in galactic potentials (Section IV.B). The galactic model yields a t^4 growth law (45) and, with the parameter set (48), masses up to ~5 M_sun and radii ~10^9 km at r=0.05 kpc. The paper concludes that although Q-balls can reach stellar masses, their radii are far larger than their Schwarzschild radii, so they resemble dark-matter clouds and cannot explain LIGO/Virgo mass-gap events; it also estimates the present population of such Q-balls.

Significance. If the growth calculation were correct, the paper would provide a concrete demonstration that even in a minimal Q-ball model, stellar-mass solitons with Solar-system-sized radii are possible, ruling out this scenario for LIGO/Virgo events and providing population estimates. The paper is transparent: it states its assumptions, admits uncertainties in Eqs. (8)-(9), and gives closed-form expressions that can be checked. However, the central accretion equation (41) contains a serious overcounting of the ambient dark-matter density, and the reduction to Eq. (43) is not justified. The quantitative masses, radii, and populations are therefore unsupported. The paper's final negative conclusion about LIGO/Virgo may survive in spirit, but the present derivation does not establish it.

major comments (3)
  1. [Section IV.B, Eqs. (41)-(43)] Equation (41) sums over charge species kQ_star with n(kQ_star)=rho(r)/m_Q(kQ_star) from Eq. (39). Since the same rho(r) is used for every k, the total mass density implied by the ambient population is sum_k m_Q(kQ_star)n(kQ_star)=K rho(r), with K=Q(t)/Q_star~10^18 for the final parameters of Section V. This exceeds the available dark-matter density by many orders of magnitude. The reduction to the single k=Q(t) term in Eq. (43) retains the same normalization for the largest charge species and is not derived from the initial distribution (18), which is dominated by charges near Q_star. The t^4 growth law and the resulting masses in Eqs. (46), (49), (52), and (53) are therefore not established.
  2. [Section IV.B and Section V] A consistent treatment should use the actual charge distribution (18) normalized to rho(r), rather than assigning the full density to each k. If the ambient population is dominated by Q_star-charge Q-balls, the accretion rate scales as dQ/dt proportional to Q^{1/2}, giving Q proportional to t^2 and m_Q proportional to t^{3/2}; with the parameter set (48) the mass at 13 Gyr would be orders of magnitude below 1 M_sun. The paper does not address this alternative, and the claim in Section VI that Q-balls 'are able to gain the necessary masses' is not robust.
  3. [Section V, Eq. (47) and parameter set (48)] The example parameters sit at the boundaries of the allowed region (v at the lower bound from Eq. (21), eta_chi=1, u=1, Tc=v), which is acceptable for an existence argument. However, the paper's stronger statement in Section VI that 'with any physically meaningful set of free parameters' the radii are of order the Solar system is not demonstrated, since no scan over the full parameter space is presented. The dependence of the radius (50) on v and the mass condition (47) should be analyzed to justify that all allowed parameter sets giving m_Q greater than or similar to M_sun produce R_Q much greater than R_Schwarzschild.
minor comments (5)
  1. [Eq. (6)] Equation (6) defines Qmin = mQ/mchi, which is self-referential because mQ depends on Q; it should be solved for Q_min using m_Q from Eq. (5) and m_chi from Eq. (2).
  2. [Eq. (7)] The notation 'nQQ/s = eta_chi' is ambiguous; it should be written as n_Q Q / s or similar to clarify that n_Q is the number density of Q-balls.
  3. [Eq. (41)] The sum over k in N with kQ_star <= Q(t) has an upper limit that grows as Q grows; the notation should indicate this explicitly, for example k = 1, ..., floor(Q/Q_star).
  4. [Section VI] The phrase 'with any physically meaningful set of free parameters' is too strong; the analysis only demonstrates the example (48). Consider rephrasing to 'for the parameter set considered here'.
  5. [References] The reference list contains duplicates and formatting inconsistencies: [12] repeats [4], and several entries have nonstandard formatting (e.g., [13] and [14]).

Circularity Check

2 steps flagged · score 6.0 of 10

The ~5 M_sun mass claim is built into the assumed ambient Q-ball population in Eq. (43), and the quoted parameter set is selected to satisfy the target mass inequality (47), so the positive mass-growth result is partially forced by construction.

  1. self definitional [Section IV.B, Eqs. (39)-(45), especially Eq. (43)]
    "the concentration is n(r) = ρ(r)/m_Q(Q) ... ˙Q = P_k kQ⋆u⋆σ(kQ⋆)n(kQ⋆), t ∈ [0; 13] Gyr, k ∈ N, Q(0) = Q⋆ ... it can be obtained ˙Q ∼ (kQ⋆)^3/4. From the view of (42), it can be seen that the main contribution to the charge changing in (41) is made by the term with maximal k which corresponds to the situation when kQ⋆ = Q(t). According to this, (41) can be simplified for estimates, ( ˙Q = Qu⋆σ(Q)n(Q) ..."

    Eq. (39) defines the ambient number density by dividing the full local dark-matter density by m_Q(Q). Eq. (41) applies this same full density separately to every species kQ⋆, which already overcounts the available mass by K = Q/Q⋆. The step to Eq. (43) does not evaluate that sum; it keeps only the term with kQ⋆ = Q(t), i.e., it assumes at every instant that ambient Q-balls of exactly the selected ball's current charge exist with number density ρ/m_Q(Q). Since m_Q(Q) is the selected ball's own mass formula, the growth law dQ/dt ∝ Q^{3/4}, the solution (45), and the final mass (46) are rearrangements of this assumed self-similar reservoir rather than consequences of the initial charge distribution (18), which is dominated by charges near Q⋆.

  2. fitted input called prediction [Section V, Eqs. (47)-(49) and the parameter set (48)]
    "it is necessary to satisfy the following limitation: mQ(v,u,ηχ,u⋆,Tc,r)|_{t=13 Gyr} ≳ 1M⊙. ... As an example of one of these sets of free parameters, the following values are presented ... v ≈ 10^-7 GeV, u = 1, ηχ = 1, u⋆ = 0.0007, Tc ≈ 10^-7 GeV ... The mass limit (47) is satisfied. ... mQ|_{r=0.05 kpc} ≈ 5 M⊙."

    The parameter set (48) is not derived from independent data; it is selected precisely because it makes the target inequality (47) hold, with v at its lower limit (21), ηχ = 1, u = 1, and Tc = v. Evaluating formula (46) at this hand-picked point and reporting m_Q ≈ 5 M_sun is a restatement of the selection criterion, not an independent prediction of the model. The mass-gap mass was imposed as a requirement on the parameter choice, so the positive conclusion that Q-balls 'can grow to the mass gap masses' is partly forced by that choice; the large radius (51) is a separate consequence of the same low-v parameters.

full rationale

The paper's central growth derivation is not self-contained. In Section IV.B, the ambient Q-ball population is represented by Eq. (39), n(r)=ρ(r)/m_Q(Q), which ties every ambient ball's mass to the selected ball's current mass. The sum in Eq. (41) then assigns the full local dark-matter density simultaneously to every charge kQ⋆, overcounting the available mass by a factor of Q/Q⋆; the reduction to Eq. (43) keeps only the k=Q(t) term, effectively assuming at every instant a reservoir of Q-balls with charge Q(t) and density ρ/m_Q(Q). That assumption builds the self-similar t^4 growth and the ~M_sun result into the inputs. Additionally, the quoted 'most successful set' (48) is chosen after imposing the target inequality (47), m_Q ≥ 1 M⊙, so the reported m_Q ≈ 5 M_sun at r=0.05 kpc is a demonstration that the authors could find parameters satisfying their own criterion, not an independent prediction. The radius conclusion (R_Q ~ 10^9 km) follows algebraically from the same low-v parameters, so it is not independently circular, but the positive mass-growth claim is substantially circular. The flat-Universe negative result in Eq. (35) is independent of these issues and remains a non-circular statement.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a chain of assumptions: the FLS Q-ball model, the first-order phase-transition production mechanism, the large-Q mass-radius scaling, the Burkert profile in galaxies, the inelastic merger cross-section, and an undepleted accretion population. Most are taken from prior work; the undepleted-population assumption is introduced in this paper and is the weakest link. The most successful parameters are chosen to hit the mass target, so they are free parameters rather than derived quantities.

free parameters (7)
  • v (Lagrangian scale) = v ~ 10^-7 GeV in the most successful set; allowed range about 10^-7 GeV to 5.6 GeV from Eqs. (21)-(24)
    Set at the lower bound to maximize the final Q-ball mass and satisfy the mass condition (47).
  • eta_chi (chi-field asymmetry) = eta_chi = 1
    Chosen at its upper limit, 0 < eta_chi <= 1, to maximize Q_star and the final mass.
  • u (bubble wall velocity) = u = 1
    Chosen at its physical upper limit, 0 < u < 1, to maximize accretion and satisfy constraints.
  • Tc (phase transition temperature) = Tc ~ 10^-7 GeV = v
    Assumed equal to v and chosen so the requirement v ~ Tc is well satisfied and the phase transition occurs in the photon epoch.
  • u_star (orbital velocity in galaxy) = u_star = 220 km/s ~ 0.0007
    Milky Way stellar orbital velocity, not fitted by the paper, but it enters the growth law linearly.
  • rho_b (Burkert central density) = rho_b = 2.34 x 10^4 GeV/cm^3
    Normalized so the Milky Way halo mass is about 10^12 solar masses; the growth rate is proportional to rho(r).
  • collision inelasticity factor = 1/2
    Multiplicative factor in the cross-section (29), taken from Refs. [46,47]; the growth rate is proportional to it.
assumptions (6)
  • domain assumption Dark matter consists of Q-balls of the Friedberg-Lee-Sirlin model described by Lagrangian (1).
    The entire analysis is specific to this model; the paper notes that other Q-ball models may give different results.
  • domain assumption Q-balls are formed in a first-order phase transition with the maximal charge Q_star given by Eq. (9), following Ref. [36].
    The initial mass and growth start from Q_star; the formula depends on bubble-wall velocity u, nucleation parameters, and asymmetry eta_chi.
  • domain assumption Large-Q thin-wall expressions for radius and mass, Eqs. (4) and (5), remain valid throughout the growth, including for solar-mass Q-balls.
    The merger calculation uses these scaling laws even when the selected Q-ball has grown by many orders of magnitude beyond the Q_star seed.
  • domain assumption In galaxies, cosmic expansion can be neglected and Q-balls follow the Burkert density profile (37) with Milky Way parameters.
    This removes the scale-factor suppression from Eq. (28) and sets the local number density via Eq. (39).
  • domain assumption Q-ball collisions are inelastic with probability about 50% and the geometric cross-section is sigma = pi R_Q^2 / 2, independent of relative velocity and charge asymmetry.
    Based on MSSM Q-ball collision studies in Refs. [46,47]; not established for Friedberg-Lee-Sirlin Q-balls at galactic velocities.
  • ad hoc to paper The selected Q-ball accretes from an undepleted population in which every charge species kQ_star is present at density rho(r)/m_Q(kQ_star).
    This is the assumption behind Eq. (41); it overcounts the number density and is the paper's main derivation gap.

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Pith. "Pith review of Can dark-matter Q-balls grow to the mass gap masses?." pith.science (2026). https://pith.science/paper/JHYVMTKO

@misc{pith2026241208803,
  author       = {Pith},
  title        = {Pith review of: Can dark-matter Q-balls grow to the mass gap masses?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JHYVMTKO}},
  note         = {Machine review of arXiv:2412.08803}
}
read the original abstract

Within the framework of general relativity, it can be shown that gravitational waves are radiated with the merger of massive compact objects. Such gravitational wave signals are observed on Earth on various detectors, in particular, on Laser Interferometer Gravitational Wave Observatory (LIGO) and Virgo. During the operation of these detectors, many events have been detected. Those events are associated with the merger of massive compact objects, however, the nature of some merging objects has not yet been reliably established. This work considers nontopological solitons of dark matter -- Q-balls, as candidates for the role of massive compact objects. In this work one of the simplest models of Q-balls, the mechanism of their birth during a phase transition in the early Universe and the mechanism of their mass gaining during the evolution of the Universe, which is based on their mutual merger, are considered. As a result, it is analyzed whether Q-balls of dark matter can grow to the mass gap masses and be candidates for the role of massive compact objects.

Figures

Figures reproduced from arXiv: 2412.08803 by the authors.

Figure 1
Figure 1. FIG. 1. A schematic representation of the I-st order phase [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Dependence of cosmological constraints on the pa [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 5
Figure 5. FIG. 5. The evolution of selected Q-ball radius ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figures from the paper (1 more)
Figure 6
Figure 6. Figure 6: FIG. 6. The radius profile of the selected Q-ball ( [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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