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REVIEW 4 major objections 4 minor 64 references

Black Holes from Fermi Ball Collapse

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Saturated Fermi balls in a quartic-coupled dark sector collapse into black holes.

desk verdict A genuine new scaling branch for saturated Fermi balls, but the PBH mechanism hinges on an unsolved inter-ball force; worth refereeing, not yet a closed case. read the letter →

arxiv 2411.17074 v2 pith:BF72TQ7R submitted 2024-11-26 astro-ph.CO hep-ph

Yifan Lu , Zachary S. C. Picker , Stefano Profumo , Alexander Kusenko This is my paper · ORCID
classification astro-ph.COhep-ph
keywords Fermiballsnon-topologicalsolitonsprimordialblackholesdarksectorquarticscalarpotentialYukawainteractionsaturationasymmetricmatter
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

This paper argues that a minimal dark sector—a heavy fermion plus a light scalar with a quartic $\lambda\varphi^4$ self-interaction—can produce primordial black holes through a purely particle-physics route. The quartic term makes Fermi balls, non-topological soliton bound states of the dark fermions, reach saturation at much smaller fermion numbers than in the pure Yukawa case. Once saturated, their radius grows only as $N^{1/3}$ while their mass grows as $N$, so sufficiently heavy balls fall inside their own Schwarzschild radius and collapse. The same term keeps the exterior Yukawa force long-ranged, which the paper argues lets the balls keep merging and accreting until collapse. If right, this gives an economical dark-sector mechanism for primordial black hole formation without fine-tuning initial conditions.

What carries the argument

The load-bearing object is the saturated Fermi ball in the strong-coupling branch. Saturation is the state where the ball radius is comparable to the effective scalar interaction length inside the ball; in this branch the effective scalar mass is set by $\sqrt{\lambda}\varphi$, so saturation happens at $N_{\rm sat} \sim 1/(y\sqrt{\lambda})$. The exterior scalar field obeys a nonlinear equation whose long-range part is approximated by an effective charge $n(r)=r\varphi$, bounded between $N_{\rm sat}$ and $N_{\rm sat} R/R_{\rm sat}$. This effective-charge picture supplies the inter-ball force bounds (Eq. 40) that carry the merger-growth argument, and the $R_s/R \sim N^{2/3}$ scaling carries the collapse argument.

What would settle it

A dedicated numerical solution of the two-ball scalar boundary-value problem would settle the mechanism: if the computed inter-ball force at formation-relevant separations is weaker than gravity for all balls below the collapse threshold, merger-driven growth stalls before black hole formation. A search for primordial black holes in the mass range predicted by Eqs. (47) and (49) that finds none at the expected abundance would also constrain the scenario.

Watch

Extended reading notes

Core claim

In the strong-coupling branch, defined by $1 \gtrsim \lambda \gg (m_\varphi/y m_\psi)^2$ and small $m_\varphi$, the Fermi ball interior is non-relativistic and the scalar VEV is small; solving the mean-field energy minimization gives $R \simeq (4/5)(9\pi/2g_f)^{1/3} N^{1/3}/(\alpha^{1/3} m_\psi)$ with $\alpha = g_f y^4/(6\pi^2\lambda)$ and $M \simeq m_\psi N$. Because $R_s/R \sim N^{2/3}$, every sufficiently heavy saturated Fermi ball must collapse into a black hole. The paper further estimates the long-range force between two saturated balls from the exterior scalar profile, giving upper and lower bounds $F_{\rm low}$ and $F_{\rm high}$, and uses these to argue that growth by mergers can continue. This combination—early saturation plus long-range attraction—is the paper's central resolution of the previous tradeoff between black-hole-capable density and merger-capable interactions.

Load-bearing premise

The growth-to-collapse story depends on two saturated Fermi balls actually attracting each other strongly enough over cosmological distances to merge; the paper gives only upper and lower bound estimates for this force and leaves the exact two-ball solution to future work.

Editorial extensions

If this is right

  • Fermi balls formed by this mechanism with fermion number above a critical value inevitably collapse, because $R_s/R\sim N^{2/3}$ ensures the Schwarzschild radius eventually exceeds the ball radius.
  • The quartic term removes the sub-saturation $R\sim N^{2/3}$ plateau, so even balls whose radius is smaller than the Yukawa interaction range are already saturated.
  • The estimated long-range force bounds imply that merger-driven growth is possible, and the Bullet-cluster self-interaction constraint only excludes a small corner of the parameter space.
  • The instantaneous and non-instantaneous formation scenarios give analytic compact-object mass estimates, so the model makes definite predictions for the primordial black hole mass function in terms of $m_\varphi$, $m_\psi$, and $f_{\rm DM}$.

Reading between the lines

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

  • If the exact two-ball force turns out to lie near the lower bound $F_{\rm low}$, then the merger channel is efficient only for the lightest Fermi balls; heavier collapses would require accretion-driven growth, shifting the mass function predicted in Fig. 3.
  • A dedicated numerical solver for the nonlinear exterior scalar equation, of the type used in chameleon-screening studies, could replace the bounding estimates with a definite force law and sharpen the collapse criterion.
  • The same $\lambda\varphi^4$ saturation mechanism may also leave a population of stable, never-collapsing Fermi balls; these would be a macroscopic dark-matter component whose self-interaction signatures differ from point-like particle dark matter.
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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

4 major / 4 minor

Summary. The paper studies Fermi-ball solitons in a dark-sector model with a heavy fermion and a light scalar, focusing on the case where the scalar potential is dominated by a quartic term λφ⁴. In the strong-coupling branch it derives closed-form saturation scalings R∼N^{1/3} and M∼mψN (Sec. II C, Eq. (26)), and argues that because Rs/R∼N^{2/3}, sufficiently massive saturated Fermi balls must eventually fall inside their Schwarzschild radius and collapse to black holes. It then estimates the long-range force between two saturated balls (Sec. III, Eqs. (40)-(41)) and combines this with an earlier cosmological structure-formation mechanism (Refs. [20,37]) to give formation masses in instantaneous and non-instantaneous scenarios, with results summarized in Fig. 3. The central advertised outcome is that a minimal renormalizable dark sector can naturally produce primordial black holes.

Significance. If established, the result would be significant: it would connect a very simple two-field dark sector to primordial black hole formation without inflation, phase transitions, or fine-tuned initial conditions. The paper's strongest contribution is the interior scaling derivation in Sec. II C: the analytic R∼N^{1/3}, M∼mψN relations are internally consistent and are supported by the numerical solutions shown in Fig. 2, and the derivation does not involve fitting to data. The authors also deserve credit for clearly flagging the limitations of their exterior-solution treatment and the two-ball force estimate. The weakest link is quantitative: the growth of saturated balls to the collapse mass relies on an order-of-magnitude force estimate, and the final collapse statement is based on a flat-space Schwarzschild-radius comparison rather than a general-relativistic structure calculation. The manuscript therefore demonstrates a plausible and attractive mechanism, but the black-hole formation rate and even the existence of a guaranteed growth channel are not yet established with the same rigor as the interior scaling.

major comments (4)
  1. [Sec. III B, Eqs. (40)-(41); Sec. IV A] The central growth step, namely that saturated Fermi balls can merge until they reach the collapse mass, is not established. The two-ball force is only bracketed: Eq. (40) gives lower and upper estimates whose ratio grows roughly as y^{1/3} λ^{1/6} N^{1/3} for equal balls, and the exact two-ball solution is explicitly deferred to future work. Moreover, Eq. (41) shows that even the upper-bound force can fall below Newtonian gravity for reasonably large masses, and Sec. IV A states that essentially all free fermions are absorbed during formation, so late-time accretion is not a backstop. Since the abstract's claim that the balls 'can therefore grow by mergers' is the bridge from the reliable soliton scaling to black hole formation, the paper needs either a credible force calculation or a demonstrated alternative growth channel before the PBH claim can be accepted.
  2. [Sec. II C, Eq. (26); Sec. IV] The collapse criterion is a Newtonian comparison Rs/R∼N^{2/3} based on the flat-space mean-field energy functional (3). The paper does not solve the Tolman-Oppenheimer-Volkoff equations or otherwise show that the mean-field ball remains a valid description up to the compactness at which collapse is unavoidable. Because the title and abstract claim actual black hole formation, the transition from a pressure-supported saturated ball to a black hole should be modeled or at least bounded: one needs to know where the flat-space scaling breaks down and to verify that R<Rs is reached before that point. As written, the statement that the balls 'will naturally eventually' collapse is a plausible inference rather than a demonstrated result.
  3. [Sec. IV, Fig. 3] The black-dot region labeled 'immediate black hole formation' is a central quantitative output, but the criterion used to draw it is never written down. The text after Eq. (49) only says that saturated balls collapse when they are heavy enough; no formula for the critical fermion number N_crit (obtained by setting Rs=R using Eq. (26)) is given, and no explicit comparison of the formation masses in Eqs. (47) and (49) with that critical mass is shown. Without this information the figure cannot be reproduced or checked, and the distinction between the 'immediate black hole' and 'stable Fermi ball' regions remains opaque.
  4. [Sec. IV A] The text says that 'essentially all of the fermions' are absorbed during Fermi ball formation, yet the abstract and the formation discussion mention accretion of ambient dark fermions as a growth mechanism. These statements are in tension: if no free fermions remain, accretion cannot contribute to late-time growth, and mergers are the only channel. Please state clearly which growth channel is assumed in each scenario and whether the conclusion depends on it.
minor comments (4)
  1. [Throughout] There are several typographical errors, including 'Specificially' in Sec. II A, 'Largangian' in Sec. II C, and 'non-instaneous' in Sec. IV. Please correct them.
  2. [Sec. III A, Eq. (37)] The condition for negligible nonlinear screening is stated as n/(r² d²n/dr²) = 1/(λ n²) ≳ 1, which is not transparent. It would help to rewrite the criterion directly in terms of λ(yN)² or N/N_sat, so that the reader can see where the bound N≲N_sat comes from.
  3. [Sec. III A, Eq. (38)] The effective charge range N_sat ≲ N_eff ≲ N_sat R/R_sat is presented as an expectation rather than a controlled inequality. Please state explicitly that these are heuristic bounds with no rigorous error estimate, since they feed directly into the force bounds in Eq. (40).
  4. [Sec. IV B, Eq. (50)] The Bullet Cluster constraint is applied with σ≃π m_φ^{-2}. It would be useful to state whether this is intended as an upper bound on the geometric cross-section and to note that the actual cross-section for extended objects may be smaller, since this affects the interpretation of the gray region in Fig. 3.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Fermi-ball scaling and collapse criterion are derived from the Lagrangian, not fitted or self-referential; the several self-citations enter only in the cosmological formation context and are not load-bearing.

full rationale

The central derivation is self-contained. Eqs. (14)-(16) minimize the mean-field energy functional with V(phi)=lambda phi^4/4, and the strong-coupling branch (Eqs. (20)-(26)) follows from solving the resulting algebraic equations in the k_F << m_* limit; the 'ansatz m_* -> m_psi' is not an input equivalent to the output, because it is justified by the small-alpha expansion and is checked against the numerical solution in Fig. 2. The collapse statement Rs/R ~ N^{2/3} is a consequence of the derived scalings R ~ N^{1/3}, M ~ m_psi N (Eq. (26)), not an assumption. No parameter is fitted to the predicted PBH mass or collapse condition. The cosmological formation mechanism in Sec. IV is imported from Refs. [20,37,59] (which include some current authors), but these are prior published results, including N-body simulations [59], and they are not used to derive the mass-radius relations; thus the self-citations are background support rather than a circular load-bearing chain. The paper itself flags the weak link, the inter-ball force: Sec. III A states the exterior scalar solution becomes 'rapidly unreliable,' Sec. III B leaves the exact two-ball solution to future work and gives only bounding estimates, and Eq. (41) shows the upper-bound force can fall below gravity for large balls. That is an acknowledged incompleteness in a physical input, not a reduction of the prediction to its own input. Accordingly no circular step is exhibited.

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

The central scaling rests on a long chain of imported assumptions: mean-field Thomas-Fermi treatment, neglect of self-gravity in the structure equations, an unsolved exterior scalar profile, and formation physics taken from the authors' prior work. The only truly free cosmological parameter is the asymmetry η; the Lagrangian parameters are scanned inputs.

free parameters (5)
  • asymmetry parameter η = free (Sec. IV A)
    Sets the dark fermion relic abundance fDM and the compact object mass in Eqs. (46)-(49); chosen freely, not determined by data.
  • Yukawa coupling y = scanned; example y=5e-2
    Controls the attractive force and enters the strong-coupling parameter α∝y^4; chosen by hand.
  • quartic coupling λ = scanned; example λ=1e-2
    Required for renormalizability; the strong-coupling branch requires 1≳λ≫(mφ/(y mψ))²; chosen by hand.
  • fermion mass mψ = scanned, roughly 1e5 to 1e12 GeV
    Sets the mass scale of the Fermi ball and the saturation parameters; input parameter.
  • scalar mass mφ = scanned, down to 1e-25 GeV
    Sets the Yukawa range and the formation temperature; must be small for the strong-coupling branch (Eq. 22); input parameter.
assumptions (6)
  • domain assumption Mean-field/Thomas-Fermi approximation: the scalar is replaced by its classical VEV and fermions by a local Fermi gas.
    Used throughout Sec. II as the basis for the energy functional (Eq. 3) and the scalar equation of motion (Eq. 6).
  • domain assumption Zero temperature and conserved fermion number N.
    Assumed in the Fermi ball construction of Sec. II; fixes the relation between N and kF in Eq. (4).
  • standard math Existence of Fermi ball solutions via the ball-shooting argument for an effective potential.
    Borrowed from Coleman's Q-ball existence proof (Ref. [49]) and applied in Sec. II A.
  • domain assumption Self-gravity is neglected in the structure equations; collapse is inferred from comparing the flat-space radius to the Schwarzschild radius.
    The energy functional (Eq. 3) contains no gravitational term, yet the black hole formation criterion in Sec. IV compares R to Rs.
  • ad hoc to paper In the strong-coupling branch, fermions are non-relativistic (m*→mψ) and the scalar mass term is negligible under Eq. (22).
    Required to obtain the analytic solution Eq. (25); the paper verifies the resulting scaling numerically but not the full condition space.
  • domain assumption Yukawa-force-driven halos form as in Ref. [20] and formation absorbs essentially all free fermions.
    Imported from prior work and used in Sec. IV to estimate compact object masses and the absence of a residual accretion population.

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Cite this review

Pith. "Pith review of Black Holes from Fermi Ball Collapse." pith.science (2026). https://pith.science/paper/BF72TQ7R

@misc{pith2026241117074,
  author       = {Pith},
  title        = {Pith review of: Black Holes from Fermi Ball Collapse},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BF72TQ7R}},
  note         = {Machine review of arXiv:2411.17074}
}
abstract

Fermi balls are non-topological solitons that can naturally form in an early universe containing a dark sector with heavy fermions and an attractive interaction mediated by a light scalar field. We compute the Fermi ball mass and radius scaling relations when the potential of the scalar field $\varphi$ has a non-negligible quartic coupling $\lambda\varphi^4$. The resulting Fermi balls reach `saturation' very rapidly, even when their radius is much smaller than the effective Yukawa force range. These objects can therefore grow by mergers or by accretion of ambient dark fermions, until they become so dense that they fall within their Schwarzschild radius and collapse to black holes. This setup, therefore, provides an example of a rather natural and economical dark sector scenario for the formation of primordial black holes.

Figures

Figures reproduced from arXiv: 2411.17074 by the authors.

Figure 1
Figure 1. FIG. 1: The shape of the effective potential and the scalar field profile are demonstrated for several choices of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The scaling of the Fermi ball radius (left panel) and binding energy (right panel) with the number of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Mass contours for the compact objects formed in the non-instantaneous and instantaneous scenarios [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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Works this paper leans on

64 extracted references · 27 canonical work pages

  1. [1]

    M. Dine, W. Fischler and M. Srednicki, A Simple Solution to the Strong CP Problem with a Harmless Axion, Phys. Lett. B 104 (1981) 199

    1981

  2. [2]

    Kim, Weak Interaction Singlet and Strong CP Invariance, Phys

    J.E. Kim, Weak Interaction Singlet and Strong CP Invariance, Phys. Rev. Lett. 43 (1979) 103

    work page 1979

  3. [3]

    Shifman, A.I

    M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Can Confinement Ensure Natural CP Invariance of Strong Interactions?, Nucl. Phys. B 166 (1980) 493

    work page 1980

  4. [4]

    Zhitnitsky, On Possible Suppression of the Axion Hadron Interactions

    A.R. Zhitnitsky, On Possible Suppression of the Axion Hadron Interactions. (In Russian) , Sov. J. Nucl. Phys. 31 (1980) 260

    1980

  5. [5]

    Ellis, J.S

    J.R. Ellis, J.S. Hagelin, D.V. Nanopoulos, K.A. Olive and M. Srednicki, Supersymmetric Relics from the Big Bang, Nucl. Phys. B 238 (1984) 453

    work page 1984

  6. [6]

    Jungman, M

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

    arXiv 1996

  7. [7]

    Nussinov, TECHNOCOSMOLOGY: COULD A TECHNIBARYON EXCESS PROVIDE A ’NATURAL’ MISSING MASS CANDIDATE? , Phys

    S. Nussinov, TECHNOCOSMOLOGY: COULD A TECHNIBARYON EXCESS PROVIDE A ’NATURAL’ MISSING MASS CANDIDATE? , Phys. Lett. B 165 (1985) 55

    work page 1985

  8. [8]

    Barr, R.S

    S.M. Barr, R.S. Chivukula and E. Farhi, Electroweak Fermion Number Violation and the Production of Stable Particles in the Early Universe , Phys. Lett. B 241 (1990) 387

    work page 1990

Show all 64 references
  1. [9]

    Kaplan, M.A

    D.E. Kaplan, M.A. Luty and K.M. Zurek, Asymmetric Dark Matter , Phys. Rev. D 79 (2009) 115016 [0901.4117]

    2009 arXiv

  2. [10]

    Davoudiasl and R.N

    H. Davoudiasl and R.N. Mohapatra, On Relating the Genesis of Cosmic Baryons and Dark Matter , New J. Phys. 14 (2012) 095011 [ 1203.1247]

    2012 arXiv

  3. [11]

    Petraki and R.R

    K. Petraki and R.R. Volkas, Review of asymmetric dark matter, Int. J. Mod. Phys. A 28 (2013) 1330028 [1305.4939]

    2013 arXiv

  4. [12]

    Zurek, Asymmetric Dark Matter: Theories, Signatures, and Constraints , Phys

    K.M. Zurek, Asymmetric Dark Matter: Theories, Signatures, and Constraints , Phys. Rept. 537 (2014) 91 [1308.0338]

    2014 arXiv

  5. [13]

    Zel’dovich and I.D

    Y.B. Zel’dovich and I.D. Novikov, The hypothesis of cores retarded during expansion and the hot cosmological model, Sov. Astron. 10 (1966) 602

    1966

  6. [14]

    Hawking, Gravitationally collapsed objects of very low mass, Mon

    S. Hawking, Gravitationally collapsed objects of very low mass, Mon. Not. Roy. Astron. Soc. 152 (1971) 75

    1971

  7. [15]

    Carr and S

    B.J. Carr and S. Hawking, Black holes in the early Universe, Mon. Not. Roy. Astron. Soc. 168 (1974) 399

    1974

  8. [16]

    Chapline, Cosmological effects of primordial black holes, Nature 253 (1975) 251

    G.F. Chapline, Cosmological effects of primordial black holes, Nature 253 (1975) 251

    1975

  9. [17]

    Green, Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function, Phys

    A.M. Green, Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function, Phys. Rev. D 94 (2016) 063530 [ 1609.01143]

    2016 arXiv

  10. [18]

    Cotner and A

    E. Cotner and A. Kusenko, Primordial black holes from supersymmetry in the early universe , Phys. Rev. Lett. 119 (2017) 031103 [ 1612.02529]

    2017 arXiv

  11. [19]

    Cotner, A

    E. Cotner, A. Kusenko, M. Sasaki and V. Takhistov, Analytic Description of Primordial Black Hole Formation from Scalar Field Fragmentation , JCAP 10 (2019) 077 [ 1907.10613]

    2019 arXiv

  12. [20]

    Flores and A

    M.M. Flores and A. Kusenko, Primordial Black Holes from Long-Range Scalar Forces and Scalar Radiative Cooling, Phys. Rev. Lett. 126 (2021) 041101 [2008.12456]

    2021 arXiv

  13. [21]

    B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Constraints on primordial black holes , Rept. Prog. Phys. 84 (2021) 116902 [ 2002.12778]

    2021 arXiv

  14. [22]

    Hardy, R

    E. Hardy, R. Lasenby, J. March-Russell and S.M. West, Big Bang Synthesis of Nuclear Dark Matter , JHEP 06 (2015) 011 [ 1411.3739]

    2015 arXiv

  15. [23]

    Wise and Y

    M.B. Wise and Y. Zhang, Stable Bound States of Asymmetric Dark Matter , Phys. Rev. D 90 (2014) 055030 [1407.4121]

    2014 arXiv

  16. [24]

    Wise and Y

    M.B. Wise and Y. Zhang, Yukawa Bound States of a Large Number of Fermions, JHEP 02 (2015) 023 [1411.1772]

    2015 arXiv

  17. [25]

    Gresham, H.K

    M.I. Gresham, H.K. Lou and K.M. Zurek, Early Universe synthesis of asymmetric dark matter nuggets , Phys. Rev. D 97 (2018) 036003 [ 1707.02316]

    2018 arXiv

  18. [26]

    Kawana and K.-P

    K. Kawana and K.-P. Xie, Primordial black holes from a cosmic phase transition: The collapse of Fermi-balls , Phys. Lett. B 824 (2022) 136791 [ 2106.00111]

    2022 arXiv

  19. [27]

    Huang and K.-P

    P. Huang and K.-P. Xie, Primordial black holes from an electroweak phase transition, Phys. Rev. D 105 (2022) 115033 [2201.07243]

    2022 arXiv

  20. [28]

    Kawana, P

    K. Kawana, P. Lu and K.-P. Xie, First-order phase transition and fate of false vacuum remnants , JCAP 10 (2022) 030 [ 2206.09923]

    2022 arXiv

  21. [29]

    P. Lu, K. Kawana and A. Kusenko, Late-forming primordial black holes: Beyond the CMB era , Phys. Rev. D 107 (2023) 103037 [ 2210.16462]

    2023 arXiv

  22. [30]

    Chang, D

    J.H. Chang, D. Egana-Ugrinovic, R. Essig and C. Kouvaris, Structure Formation and Exotic Compact Objects in a Dissipative Dark Sector , JCAP 03 (2019) 036 [1812.07000]

    2019 arXiv

  23. [31]

    Bramante, C.V

    J. Bramante, C.V. Cappiello, M.D. Diamond, J.L. Kim, Q. Liu and A.C. Vincent, A Dissipative Dark Cosmology: From Early Matter Dominance to Delayed Compact Objects, 2405.04575

    arXiv

  24. [32]

    Gradwohl and J.A

    B.-A. Gradwohl and J.A. Frieman, Dark matter, long range forces, and large scale structure , Astrophys. J. 398 (1992) 407

    1992

  25. [33]

    Gubser and P.J.E

    S.S. Gubser and P.J.E. Peebles, Structure formation in a string inspired modification of the cold dark matter model, Phys. Rev. D 70 (2004) 123510 [hep-th/0402225]

    2004 arXiv

  26. [34]

    Nusser, S.S

    A. Nusser, S.S. Gubser and P.J.E. Peebles, Structure formation with a long-range scalar dark matter interaction, Phys. Rev. D 71 (2005) 083505 [astro-ph/0412586]

    2005 arXiv

  27. [35]

    Amendola, J

    L. Amendola, J. Rubio and C. Wetterich, Primordial black holes from fifth forces , Phys. Rev. D 97 (2018) 081302 [1711.09915]

    2018 arXiv

  28. [36]

    Savastano, L

    S. Savastano, L. Amendola, J. Rubio and C. Wetterich, Primordial dark matter halos from fifth forces , Phys. Rev. D 100 (2019) 083518 [ 1906.05300]

    2019 arXiv

  29. [37]

    Flores, Y

    M.M. Flores, Y. Lu and A. Kusenko, Structure formation after reheating: Supermassive primordial black holes and Fermi ball dark matter , Phys. Rev. D 108 (2023) 123511 [ 2308.09094]. 13

    2023 arXiv

  30. [38]

    Dom` enech and M

    G. Dom` enech and M. Sasaki,Cosmology of strongly interacting fermions in the early universe , JCAP 06 (2021) 030 [ 2104.05271]

    2021 arXiv

  31. [39]

    Lee and Y

    T.D. Lee and Y. Pang, Fermion Soliton Stars and Black Holes, Phys. Rev. D 35 (1987) 3678

    1987

  32. [40]

    Lee and Y

    T.D. Lee and Y. Pang, Nontopological solitons, Phys. Rept. 221 (1992) 251

    1992

  33. [41]

    Gresham, H.K

    M.I. Gresham, H.K. Lou and K.M. Zurek, Nuclear Structure of Bound States of Asymmetric Dark Matter , Phys. Rev. D 96 (2017) 096012 [ 1707.02313]

    2017 arXiv

  34. [42]

    Xie, Revisiting the fermion-field nontopological solitons, 2405.01227

    K.-P. Xie, Revisiting the fermion-field nontopological solitons, 2405.01227

    arXiv

  35. [43]

    Del Grosso and P

    L. Del Grosso and P. Pani, Fermion soliton stars with asymmetric vacua , Phys. Rev. D 108 (2023) 064042 [2308.15921]

    2023 arXiv

  36. [44]

    Del Grosso, G

    L. Del Grosso, G. Franciolini, P. Pani and A. Urbano, Fermion soliton stars , Phys. Rev. D 108 (2023) 044024 [2301.08709]

    2023 arXiv

  37. [45]

    Del Grosso, P

    L. Del Grosso, P. Pani and A. Urbano, Compact objects in and beyond the standard model from nonperturbative vacuum scalarization, Phys. Rev. D 109 (2024) 095006 [2401.06716]

    2024 arXiv

  38. [46]

    Walecka, A Theory of highly condensed matter , Annals Phys

    J.D. Walecka, A Theory of highly condensed matter , Annals Phys. 83 (1974) 491

    1974

  39. [47]

    Serot and J.D

    B.D. Serot and J.D. Walecka, Relativistic nuclear many body theory, Rec. Prog. Many Body Theor. 3 (1992) 49

    1992

  40. [48]

    Spruch, Pedagogic notes on Thomas-Fermi theory (and on some improvements) : atoms, stars, and the stability of bulk matter , Rev

    L. Spruch, Pedagogic notes on Thomas-Fermi theory (and on some improvements) : atoms, stars, and the stability of bulk matter , Rev. Mod. Phys. 63 (1991) 151

    1991

  41. [49]

    Coleman, Q-balls, Nucl

    S.R. Coleman, Q-balls, Nucl. Phys. B 262 (1985) 263

    1985

  42. [50]

    Fubini, A New Approach to Conformal Invariant Field Theories, Nuovo Cim

    S. Fubini, A New Approach to Conformal Invariant Field Theories, Nuovo Cim. A 34 (1976) 521

    1976

  43. [51]

    Linde, Decay of the False Vacuum at Finite Temperature, Nucl

    A.D. Linde, Decay of the False Vacuum at Finite Temperature, Nucl. Phys. B 216 (1983) 421

    1983

  44. [52]

    Burrage, B

    C. Burrage, B. Elder and P. Millington, Particle level screening of scalar forces in 1+1 dimensions , Phys. Rev. D 99 (2019) 024045 [ 1810.01890]

    2019 arXiv

  45. [53]

    Burrage, B

    C. Burrage, B. Elder, P. Millington, D. Saadeh and B. Thrussell, Fifth-force screening around extremely compact sources, JCAP 08 (2021) 052 [ 2104.14564]

    2021 arXiv

  46. [54]

    Tamosiunas, C

    A. Tamosiunas, C. Briddon, C. Burrage, W. Cui and A. Moss, Chameleon screening depends on the shape and structure of NFW halos , JCAP 04 (2022) 047 [2108.10364]

    2022 arXiv

  47. [55]

    Berezhiani and J

    L. Berezhiani and J. Khoury, Emergent long-range interactions in Bose-Einstein Condensates , Phys. Rev. D 99 (2019) 076003 [ 1812.09332]

    2019 arXiv

  48. [56]

    Picker and A

    Z.S.C. Picker and A. Kusenko, Constraints on late-forming exploding black holes , Phys. Rev. D 108 (2023) 023012 [ 2305.13429]

    2023 arXiv

  49. [57]

    Picker and A

    Z.S.C. Picker and A. Kusenko, Explaining the GeV excess with exploding black holes , Phys. Lett. B 845 (2023) 138175 [ 2305.13434]

    2023 arXiv

  50. [58]

    Chakraborty, P.K

    A. Chakraborty, P.K. Chanda, K.L. Pandey and S. Das, Formation and Abundance of Late-forming Primordial Black Holes as Dark Matter , The Astrophysical Journal 932 (2022) 119

    2022

  51. [59]

    Dom` enech, D

    G. Dom` enech, D. Inman, A. Kusenko and M. Sasaki, Halo formation from Yukawa forces in the very early Universe, Phys. Rev. D 108 (2023) 103543 [2304.13053]

    2023 arXiv

  52. [60]

    Markevitch, A.H

    M. Markevitch, A.H. Gonzalez, D. Clowe, A. Vikhlinin, L. David, W. Forman et al., Direct constraints on the dark matter self-interaction cross-section from the merging galaxy cluster 1E0657-56 , Astrophys. J. 606 (2004) 819 [ astro-ph/0309303]

    2004 arXiv

  53. [61]

    Shen, P.F

    X. Shen, P.F. Hopkins, L. Necib, F. Jiang, M. Boylan-Kolchin and A. Wetzel, Dissipative Dark Matter on FIRE. II. Observational Signatures and Constraints from Local Dwarf Galaxies , Astrophys. J. 966 (2024) 131 [ 2206.05327]

    2024 arXiv

  54. [62]

    Harris, K.J

    C.R. Harris, K.J. Millman, S.J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau et al., Array programming with NumPy, Nature (London) 585 (2020) 357 [2006.10256]

    2020 arXiv

  55. [63]

    Virtanen, R

    P. Virtanen, R. Gommers, T.E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau et al., SciPy 1.0: fundamental algorithms for scientific computing in Python, Nature Methods 17 (2020) 261 [ 1907.10121]

    2020 arXiv

  56. [64]

    Hunter, Matplotlib: A 2d graphics environment , Computing in Science and Engineering 9 (2007) 90

    J.D. Hunter, Matplotlib: A 2d graphics environment , Computing in Science and Engineering 9 (2007) 90

    2007

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Reviewed August 12, 2026 · model on record in the stance chip above.