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Primordial Black Holes from Cosmic Domain Walls

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that primordial black holes can form from the collapse of spherical domain-wall bubbles nucleated during inflation, giving a spike-like mass function that can place PBHs around $10^{20}$ g as all dark matter or around…

desk verdict A genuine new mechanism — time-dependent DW tension produces a narrow PBH mass spike — but the all-DM and LIGO windows rest on an unchecked radiation-domination assumption and need a clearer reheating treatment. read the letter →

arxiv 1908.02662 v1 pith:7MIJ42D3 submitted 2019-08-07 astro-ph.CO gr-qchep-th

Jing Liu , Zong-Kuan Guo , Rong-Gen Cai This is my paper · ORCID
classification astro-ph.COgr-qchep-th
keywords primordialblackholesdomainwallsdarkmatterinflationquantumtunnelingmassfunctiongravitationalwavesLIGO
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

This paper argues that primordial black holes (PBHs) can form from the collapse of spherical domain-wall bubbles that nucleate during inflation, and that this channel avoids the usual uncertainties of PBH formation from overdense density fluctuations. The key move is a two-field inflationary potential in which the domain-wall tension changes with time, so quantum nucleation is overwhelmingly concentrated near the moment when the Euclidean action $S_E$ is minimal. Because the resulting PBH mass depends on the nucleation time, the mass function has a narrow, spike-like peak. For one parameter choice the spike sits near $10^{20}$ g, where PBHs could constitute all of the dark matter; for another it sits near $10^{34}$ g, matching the LIGO binary-black-hole merger rate. If the claim holds, the model offers a production mechanism with a sharply peaked mass function and no accompanying stochastic gravitational-wave background.

What carries the argument

The central machinery is the Euclidean action of a nucleating domain wall, $S_E(t)=2\pi^2\sigma(t)H^{-3}(t)$, together with the nucleation rate $\lambda(t)=H^4(t)A e^{-S_E(t)}$. Because the wall tension $\sigma(t)$ varies through the two-field potential $V(\varphi,\chi)=\lambda_\chi[\chi^2-\alpha^2(\varphi-\varphi_c)^2-m^2]^2/4+f(\varphi)$, the action has a minimum at $\varphi=\varphi_c$, concentrating nucleation in a short time interval. The mass–time relation $M=5.6\times8\pi R^2(t_e)H(t_e)M_p^2$ then converts that narrow nucleation window into a spike-like mass function $f(M)$.

What would settle it

Recompute the mass function $f(M)$ from Eq. (22) with an inflaton-dominated matter era between inflation and radiation domination; if the $10^{20}$ g spike shifts or broadens so that evaporation and microlensing bounds exclude it, the all-dark-matter claim is refuted.

Watch

Extended reading notes

Core claim

Domain walls form because the effective potential $V(\varphi,\chi)$ has two degenerate vacua in the $\chi$ direction, with the vacuum separation controlled by $(\varphi-\varphi_c)^2$. During inflation the field $\varphi$ rolls, so the wall tension $\sigma(t)$ and the Euclidean action $S_E(t)=2\pi^2\sigma(t)H^{-3}(t)$ vary; nucleation is exponentially suppressed except near $\varphi=\varphi_c$, where $S_E$ is minimal. The number density of nucleated walls is $\lambda(t)=H^4(t) A e^{-S_E(t)}$, and the final PBH mass is approximated by $M=5.6\times 8\pi R^2(t_e)H(t_e)M_p^2$, where $R(t_e)$ is the wall radius at the end of inflation. Combining these gives the mass function $f(M)$ with a spike-like peak. The authors compute three parameter sets: peak at $M\sim10^{17}$ g, at $M\sim10^{20}$ g where PBHs could be all dark matter, and at $M\sim10^{34}$ g to explain LIGO merger events. They stress that the spike shape is independent of the detailed dynamics away from $t_*$.

Load-bearing premise

The calculation assumes the universe is radiation-dominated from the end of inflation until matter-radiation equality, and it takes the simulated final-mass formula as given; if a standard matter-dominated reheating phase intervenes, the mass–formation-time relation and the spike-shaped mass function would change.

Editorial extensions

If this is right

  • PBHs with masses around $10^{20}$ g can make up all of the dark matter, avoiding the threshold uncertainties of the usual overdense-collapse mechanism.
  • PBHs with masses around $10^{34}$ g can explain the binary-black-hole merger rate reported by LIGO.
  • Because the nucleated walls are spherically symmetric, Birkhoff's theorem implies their collapse emits no stochastic gravitational-wave background, so the usual gravitational-wave constraints on PBH abundance do not apply.
  • The mass function has a spike-like structure that can in principle be centered at any scale of cosmological interest by choosing when $S_E$ reaches its minimum.
  • The semiclassical nucleation regime requires $S_E>1$ for PBHs heavier than $10^{15}$ g, so PBH observations can constrain the Euclidean action during inflation.

Reading between the lines

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

  • The paper leaves implicit that the same control—the moment when $\varphi$ crosses $\varphi_c$—can place the spike at intermediate masses, such as the $10^{17}$ g window probed by current evaporation and microlensing bounds, if a viable parameter set exists.
  • If standard reheating includes an inflaton-dominated matter era, the relation between PBH mass and formation time in Eq. (18) changes; recomputing $f(M)$ under that early matter phase is a direct test of whether the spike survives and where it lands.
  • The no-gravitational-wave prediction is checkable: a future stochastic-background detection in the LISA or Taiji band whose amplitude tracks the claimed PBH abundance would count against the Birkhoff-based argument.
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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. This paper proposes a two-field inflationary model in which the tension of domain walls of the χ field changes as the inflaton φ rolls, so that the Euclidean action S_E(t)=2π²σ(t)H^{-3}(t) passes through a minimum at φ=φ_c. Spherical domain-wall bubbles nucleated near this minimum are produced in a short time interval, and their radius at the end of inflation maps to a PBH mass through Eq. (18). The authors derive a PBH mass function f(M) with a spike-like shape and present three parameter sets whose peaks fall at M~10^17, 10^20, and 10^34 g; the last two are claimed to explain all dark matter and the LIGO binary-black-hole merger rate, respectively. The technical core is the identification of the time dependence of the nucleation rate as the source of the narrow mass function, together with the use of published numerical collapse formulas for the PBH mass.

Significance. The proposed mechanism is genuinely different from the usual overdensity-threshold route to PBHs and, if the quantitative formulas are correct, would give a narrow mass function with reduced sensitivity to the threshold ambiguity. The paper also correctly notes that spherically symmetric collapse does not generate a stochastic gravitational-wave background, avoiding a class of constraints that apply to scalar-curvature PBH models. However, the advertised all-dark-matter and LIGO windows are not parameter-free predictions: the peak mass and abundance are controlled by φ_c, m, λ_χ, α, and the unspecified reheating history, and the current manuscript does not supply a complete or dimensionally consistent set of formulas for f(M). The conceptual result is worth publishing after the technical issues are fixed, but the quantitative claims are not yet supported.

major comments (4)
  1. [Sec. IV, Eq. (18)] The mass function and all claimed windows assume the universe is radiation-dominated from the end of inflation to matter-radiation equality, but standard reheating generically includes an inflaton-dominated, effectively matter-dominated phase of uncertain duration. During that phase the supercritical PBH mass is M_f,MD ≈ 4π R^3(t_e)H^2(t_e)M_p^2 (Sec. II), not the radiation-era formula in Eq. (18). For parameter set 2, R(t_e)H(t_e) ≳ 10^8, so the same nucleation time gives a mass of order 10^27 g rather than 10^20 g, and the wall radius at the onset of radiation domination is also larger. Since the inflaton decay rate is never specified, the mapping from nucleation time to PBH mass and abundance in Eqs. (20)–(22) is not determined. The authors should either specify a reheating scenario and recompute f(M), or restrict the claims to the case of instantaneous reheating and state the resulting conditional nature of the all-dark-matter and LIGO windows.
  2. [Sec. IV, Eqs. (21)–(22)] As displayed, Eq. (21) is not the derivative of Eq. (18) with respect to t_*. Differentiating M = 5.6×8π R^2(t_e)H(t_e)M_p^2 with R(t_e)=H^{-1}(t_*)a(t_e)/a(t_*) gives, in the slow-roll limit, |dM/dt_*| ≈ 2 M H(t_*), not the printed expression containing √(K M H(t_e) a(t_e)/a(t_*) M_p). The printed right-hand side has mass dimension 3/2 in Planck units while the left-hand side has mass dimension 2. Equation (22) then also has the wrong dimension for the dimensionless fraction f(M). Because Fig. 4 is computed with this Jacobian, the plotted mass function and the resulting all-dark-matter and LIGO constraints are not reproducible. The authors need to correct the Jacobian |dt_*/dM|, derive the corresponding f(M), and regenerate all figures and bounds.
  3. [Sec. III, Eq. (9) and Fig. 4] The model parameters used to generate the figures are incomplete. The coupling λ_φ in the inflaton potential f(φ)=λ_φ p φ^p is never assigned a numerical value, despite being fixed by the CMB normalization quoted in Sec. III. Without it the time axis in Figs. 2 and 3, the Hubble scale H(t), and the mass normalization in Fig. 4 cannot be reproduced. The paper should state the full parameter set, including λ_φ and any reheating parameters, used for each curve.
  4. [Sec. IV and Table I] The peak positions and amplitudes in Fig. 4 are controlled by the free parameters φ_c and m, together with λ_χ, α, and the unspecified λ_φ, and the paper provides no independent constraint that fixes these parameters. Therefore the agreement of parameter set 2 with the all-dark-matter bound and parameter set 3 with the LIGO merger rate is a demonstration of parameter flexibility rather than a falsifiable prediction. This should be stated explicitly in Sec. IV and the Conclusion, alongside the acknowledged exponential sensitivity to S_E.
minor comments (4)
  1. [Sec. III] The field-dependent tension σ(t) of the domain walls is never written explicitly; from Eq. (7) it is σ(t)=(4/3)√(λχ/2)[α²(φ(t)−φ_c)²+m²]^{3/2}, and stating this would make the minimum of S_E in Fig. 3 transparent.
  2. [Eq. (22)] The prefactor A from the nucleation rate in Eq. (14) is omitted in Eq. (22); if A is not exactly unity the normalization of f(M) must be recomputed.
  3. [Abstract] The phrase 'the mass function of PBHs in general has a spike-like structure' is too broad; the spike occurs only when S_E has a minimum, which is a model-dependent condition.
  4. [Fig. 4 caption] The caption contains typographical errors ('mas functions') and the figure would benefit from explicit mention of the normalization and of which constraint curves are plotted.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the spike-shaped PBH mass function follows from the model's explicit construction and external collapse numerics; the quoted mass windows are parameter examples, not fitted predictions presented as derived facts.

full rationale

The paper's central derivation is self-contained and does not reduce to its inputs by construction. The nucleation rate (Eq. 14) and Euclidean action (Eq. 13) are standard semiclassical results cited to external work [55], and the PBH mass formula (Eq. 18) is taken from numerical simulations in Ref. [31]; neither is a restatement of the two-field potential (Eq. 7). The spike-like mass function follows because the potential is deliberately designed so that the χ field's mass, and hence the DW tension σ(t) and S_E(t), reach a minimum at φ = φ_c; the model then correctly derives a narrow nucleation window from that construction. The quantitative peaks (10^20 g, 10^34 g) are presented as examples for the parameter sets in Table I, and the paper explicitly says 'In principle this model can provide DW radius concentrated upon any scale of cosmological interest,' so the peak position is a tunable model output rather than a fitted quantity disguised as a prediction. The stated assumption that 'the universe is radiation-dominated from the end of inflation to the matter-radiation equality' is an external modeling choice that affects the mass-abundance mapping; it could be challenged as a robustness limitation, but it is not a circular step. The authors' self-citations concern gravitational-wave constraints and scalar perturbation spectra, not the load-bearing PBH-from-DW mechanism, which rests on external references [31,33,55]. No circularity is found.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The central quantitative claims depend on hand-chosen parameters lambda_chi, alpha, phi_c, and m, plus the external collapse formulas from [31] and the RD-era assumption. The model introduces a new scalar field chi without independent evidence.

free parameters (5)
  • lambda_chi = 0.3
    Chosen by hand; sets the domain wall tension scale through the chi potential.
  • alpha = 3 x 10^-5
    Chosen by hand; controls how sharply the wall tension and Euclidean action change around phi_c.
  • phi_c = 3.74, 4.17, 5.50 M_p for sets 1-3
    Free parameter that sets the nucleation time and therefore the peak PBH mass.
  • m = 3.24, 3.16, 2.98 x 10^-5 M_p for sets 1-3
    Free parameter that sets the minimum Euclidean action and thus the peak PBH abundance.
  • lambda_phi = not stated
    Normalization of the power-law inflaton potential f(phi) = lambda_phi phi^{2/5}; fixed implicitly by the scalar amplitude P_R ~ 2 x 10^-9, but never given, making the absolute PBH abundance hard to reproduce.
assumptions (5)
  • domain assumption Thin-wall approximation and the planar wall metric of Refs. [51,52] describe the DW spacetime, and the collapse mass formulas of Ref. [31] with C = 0.62 (RD) and C = 0.15 (MD) apply to the spherical bubbles.
    The PBH mass formulas in Sec. II and Eq. (18) rely directly on these external results.
  • domain assumption The nucleation rate lambda = H^4 A e^{-S_E} with S_E = 2 pi^2 sigma H^{-3} from Refs. [55,56] is valid for semiclassical tunneling with sigma approximately H^3.
    Eqs. (13)-(14) are the basis for the time distribution of nucleated DWs and hence the spike.
  • domain assumption The adiabatic approximation for chi tracking the phi-dependent minimum is valid, as estimated in Eqs. (10)-(12).
    The time-dependent tension sigma(t) is read off from the classical background value of chi.
  • domain assumption The universe is radiation-dominated from the end of inflation to matter-radiation equality.
    Used in Sec. IV when applying the RD collapse and mass formulas to the PBH mass function; standard reheating may include an intermediate matter-dominated phase.
  • domain assumption Power-law inflation with f(phi) = lambda_phi phi^{2/5}, initial phi_i = 6.25 M_p, and end phi_e = M_p gives N = 50, n_s = 0.976, and r = 0.03.
    This background sets the Hubble scale and the relation between phi and time, entering the S_E(t) and f(M) calculations.
invented entities (1)
  • The chi scalar field with potential V(phi,chi) of Eq. (7)
    purpose: Makes the domain wall tension time-dependent so that S_E has a sharp minimum and DWs nucleate in a narrow time interval.
    No direct detection handle for chi is provided; the only testable consequences are the PBH mass function and CMB observables of the full model.

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Pith. "Pith review of Primordial Black Holes from Cosmic Domain Walls." pith.science (2026). https://pith.science/paper/7MIJ42D3

@misc{pith2026190802662,
  author       = {Pith},
  title        = {Pith review of: Primordial Black Holes from Cosmic Domain Walls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7MIJ42D3}},
  note         = {Machine review of arXiv:1908.02662}
}
abstract

We investigate the formation of primordial black holes (PBHs) from the collapse of spherically symmetric domain wall bubbles, which spontaneously nucleate via quantum tunneling during inflation. Since the tension of domain walls changes with time and so domain walls nucleate in a short time interval, the mass function of PBHs in general has a spike-like structure. In contrast to models in which PBHs produced from overdense regions, our model avoids the uncertainties of PBHs production mechanism. PBHs from domain walls with mass around $10^{20}\mathrm{g}$ may constitute all dark matter, those with mass around $10^{34}\mathrm{g}$ can explain the merger events of binary black holes detected by LIGO.

Figures

Figures reproduced from arXiv: 1908.02662 by the authors.

Figure 1
Figure 1. FIG. 1: Potential [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Evolutions of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. At the time φ = φc, SE reaches the minimum, which depends on m. During inflation SE is larger than one, so the semiclassical approximation is valid. DWs nucleated at the end of inflation has the smallest radius, 1/H(te), which collapse into PBHs with smallest mass shortly after inflation ends. Following the results in Sec. II, the minimum mass is Mmin = 4πσCH(te) −2 . (16) In our three parameter sets, σ(te) ∼ 10−5M3… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The mas functions of PBHs for the parameter set [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Forward citations

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