Topological dark matter in the literature

Topological dark matter is the established proposal that dark matter consists of topological defects, such as magnetic monopoles, left behind when a symmetry of a hidden sector breaks in the early universe. The defects are stable because their field configuration cannot unwind, and how many survive is set by how many form at the transition and how many then annihilate in pairs. Murayama and Shu found that such defects could be the dark matter at masses of 1–10 PeV, noting that Kibble’s simple count underestimates how many form; Liveoak, Maharana and Wells showed in 2026 that hidden monopoles overclose the universe if the symmetry breaks above about 100 PeV, unless a later epoch dilutes them. This low-scale scenario is what remains of the program’s retired soliton candidate once its abundance is counted correctly. Nothing in it depends on Born reciprocity, so the program claims compatibility with it and nothing more.

Status: Established — literature, live-searched [28S, 29S] (#11 §12) and checked at the level of the abstracts in round 3 (rulings 3 and 10). The program’s relation to it: “The framework is merely compatible with known PeV-scale topological dark matter (prior art)” (CLAUDE.md). Claiming the scenario as the program’s own was rejected as misattribution (round-3 Decision 2, route (a)). The general physics below is [T], written from general knowledge.

What topological dark matter is#

Defects as relics [T]#

When a gauge symmetry breaks as the universe cools, the field that breaks it chooses among equivalent vacuum states. If the space of vacua has the right topology, field configurations exist that wrap it and cannot be undone by any continuous change: domain walls, strings or monopoles, depending on the topology. For a hidden SU(2) broken to U(1) by an adjoint scalar the vacua form a sphere, and the defects are ’t Hooft–Polyakov monopoles (the ’t Hooft–Polyakov monopole; the hidden SU(2) → U(1) proposal). A monopole cannot decay, because its winding number is conserved; only a monopole and an antimonopole can annihilate. If the hidden sector does not couple to ours, its monopoles interact with ordinary matter only through gravity, which is what dark matter requires.

The density such monopoles leave is a relic abundance, as for any stable species (relic abundance mechanisms), but it is set by two processes specific to defects [T].

Formation#

The Kibble mechanism gives the least number that can form. Regions farther apart than the horizon cannot coordinate their choice of vacuum, so at least about one defect forms per horizon volume. Realistic transitions form more. Near a continuous transition, the slow response of the field near the critical point (Kibble–Zurek dynamics) freezes in a correlation length shorter than the horizon; in a first-order transition, defects form where bubbles of the new phase meet. The soliton manuscript’s error lay here: it treated the one-per-horizon count as an upper end and normalized it too low (Obituary: the Kibble normalization).

Annihilation#

A monopole and an antimonopole attract through the magnetic field of the unbroken U(1), can capture each other into a bound pair and annihilate. In the hot plasma of the early universe, scattering off charged particles slows their approach, which helps capture, until the plasma thins out [T]. The classic analysis is Preskill’s (1979), for the monopoles of grand-unified theories: as #14 §3 cites it, “The classic relic analysis [19] found monopole-antimonopole annihilation inefficient for superheavy monopoles in our plasma.” Its abstract, as the round-3 referee quotes it: “density of such monopoles in the early universe is estimated to be unacceptably large” (review/round3/referee/audit-notes.md, Step 1). That overabundance is the monopole problem. Annihilation can reduce the abundance only from above, toward a value set by the capture rate; a population that starts below that value is left as it formed.

The two works #11 cites#

Murayama and Shu (2010) [28S]#

H. Murayama, J. Shu, “Topological dark matter,” Phys. Lett. B 686, 162 (2010), arXiv:0905.1720 (#11 ref [28], live-searched). The round-3 referee quotes its abstract: “Kibble mechanism drastically underestimates the production of topological defects … dark matter of mass 1-10 PeV” (Step 4(c) item 1). Ruling 3 relies on the same abstract: Kibble “drastically underestimates the production of topological defects”. #11 §12 cites it for the statement that “Kibble–Zurek and bubble-nucleation production only add to” the one-per-horizon count.

Liveoak, Maharana and Wells (2026) [29S]#

D. Liveoak, A. Maharana, J. D. Wells, “Dark monopoles, bounds on hidden sectors, and cosmological implications,” arXiv:2607.20843 (2026) (#11 ref [29], live-searched). The referee quotes the abstract: “for symmetry-breaking scales >~ 100 PeV, the monopole abundance is unacceptably high” (Step 4(c) item 1). The adjudication reads it as: “hidden monopoles overclose for breaking scales ≳ 100 PeV unless diluted, e.g. by early matter domination” (ruling 3).

The referee also used the paper’s body (Steps 2 and 4(a)):

  • its Section III sets one defect per Hubble volume as the “minimal number density”, the floor of the Kibble count;
  • its eq. 13 gives Kibble–Zurek production, “n ~ lambda H T_c^2”;
  • its eqs. 16, 18 and 28 give diffusive and radiative capture of monopole–antimonopole pairs, and its eq. 25 an analytic estimate of the final abundance.

The referee applied these to #14 v3’s parameters and found the benchmark overproduced whichever way the annihilation was computed (overproduction at the benchmark). Those figures were accepted, not re-run: the adjudicator “did not re-run the referee’s Kibble–Zurek integration” (ruling 3).

The scenario as the program states it#

#11 §12, in the obituary of the soliton candidate:

Hidden-sector monopoles avoid overclosing the universe only for symmetry-breaking scales ≲ 100 PeV, or with a period of early matter domination [29S].

Below that scale, annihilation in the hidden plasma sets the abundance. That is the established topological-dark-matter scenario [28S, 29S], and nothing in it depends on Born reciprocity.

Read with the formation and annihilation physics above: at low symmetry-breaking scales the monopoles form abundantly and annihilate in the hidden plasma, and the capture rate sets how many remain. Above about 100 PeV the surviving abundance is too large, and only a later epoch that adds entropy, such as the decay of a field that dominated the universe for a while, can dilute it (inflation, reheating and entropy dilution).

Strong-coupling annihilation is a solved case#

#14 had treated the annihilation of its monopoles as open. Its acceptance criterion 3 asked for “the strong-coupling relic freeze-out calculation of Sec. 3, in the hidden plasma” (freeze-out of defects), because the monopoles’ mutual coupling is strong, α_m ≈ 34 (the coupling α_d ≈ 34). The referee found that this is the ordinary case: “Acceptance criterion 3 is not open: the “strong-coupling” regime alpha_m = 34 is exactly the Dirac-monopole case of Preskill 1979 (h = 2 pi/e); its modern hidden-sector version (LMW 2026) excludes v >~ 100 PeV for hidden monopoles” (Step 4(a) item 3; LMW is Liveoak–Maharana–Wells). The reason is Dirac quantization: any monopole whose magnetic charge is the minimal one relative to a weak electric charge is strongly coupled, so Preskill’s monopoles were already in this regime. Ruling 10, upheld at the level of the abstracts: “Annihilation at α_m ≈ 34 is Preskill’s Dirac case, already done for hidden sectors”. The handoff’s chat-only “√N floor” addressed the same annihilation question; round-3 Decision 3 dropped it because its premise, Kibble as a ceiling, is inverted (ruling 3) and no candidate was left for it to serve (Obituary: the √N topological floor).

Other prior art#

The referee listed two further papers on the same model, both uncited by #14 (Step 4(c) item 10): Baek, Ko and Park, JCAP 10 (2014) 067 [1311.1035], and Khoze and Ro, JHEP 10 (2014) 061 [1406.2291], “both: hidden SU(2)->U(1) monopoles + W’ + dark photon”. The adjudication accepted the list as a to-do (“Uncited prior art: Murayama–Shu; LMW; Baek–Ko–Park; Khoze–Ro”). #11 now cites Murayama–Shu and Liveoak–Maharana–Wells, but not Baek–Ko–Park or Khoze–Ro.

#14 itself cited one study of the same model: Hiramatsu, Ibe, Suzuki and Yamaguchi, JHEP 12 (2021) 122, arXiv:2109.12771 (#14 ref [12]), of which the v3 worklog says they “study exactly this case with kinetic mixing” (item R6). Their result on mixing is on kinetic mixing between sectors.

How the program relates to it#

The referee concluded that #14’s proposal is “the standard hidden ’t Hooft-Polyakov-monopole scenario (Baek-Ko-Park 2013; Khoze-Ro 2014; Murayama-Shu 2009; LMW 2026) with one coupling choice (alpha_d = alpha), and it stands or falls on that literature, independently of Born reciprocity” (Step 4(d)), and that the only salvage is “a different model (v <~ 1e8 GeV, or explicit dilution), which already exists in the literature” (Step 5). The adjudication’s verdict sets out the same two routes: “a superheavy monopole that needs a named dilution mechanism, or a model at v ≲ 100 PeV that already exists in the literature.”

Round-3 Decision 2 rejected adopting the low-scale model: “Route (a), v ≲ 100 PeV, works, but it is Murayama–Shu’s and Liveoak–Maharana–Wells’s model. Nothing in it depends on Born reciprocity (the referee and the pre-registration agree), so claiming it would add no testable content and would misattribute it.” The candidate was retired (route (c)), and “The surviving version is noted as prior art (route a)” (round-3 decisions). #11 §12’s verdict: “The framework is compatible with the prior-art PeV-scale scenario and adds nothing testable to it.” This is the round-3 instance of the ‘framework does no work’ critique.

What was checked#

The two works #11 cites carry [S]: they were found by live search, not recalled from memory. Round 3 checked them at the level of their abstracts: ruling 3 quotes both abstracts, and ruling 10 is upheld “(abstract level)”. The referee’s audit verified its literature list on INSPIRE and arXiv (Step 4(c)). Beyond the referee’s use of the Liveoak–Maharana–Wells equations, no document records a full-text reading of either paper, and the referee’s integration with those capture rates was not re-run by the adjudicator.

Used by#

References#

As #11 lists them, live-searched:

  • [28][S] H. Murayama, J. Shu, “Topological dark matter,” Phys. Lett. B 686, 162 (2010), arXiv:0905.1720.
  • [29][S] D. Liveoak, A. Maharana, J. D. Wells, “Dark monopoles, bounds on hidden sectors, and cosmological implications,” arXiv:2607.20843 (2026).

Named by the round-3 referee, verified on INSPIRE and arXiv in its audit, and not in #11’s list:

  • Baek, Ko and Park, JCAP 10 (2014) 067, arXiv:1311.1035.
  • Khoze and Ro, JHEP 10 (2014) 061, arXiv:1406.2291.

The referee gives these by surname only.

From #14’s reference list, which carries no marks; the round-3 referee found that every one of its 23 references resolves on INSPIRE (Step 1):

  • [19] J. Preskill, Phys. Rev. Lett. 43, 1365 (1979).
  • [12] T. Hiramatsu, M. Ibe, M. Suzuki, S. Yamaguchi, JHEP 12 (2021) 122, arXiv:2109.12771.

Sources#

  • CLAUDE.md ‘Since the handoff’ — “merely compatible”
  • papers/broken-born-checkpoint-v2.md §12 — the scenario and the [28S, 29S] tags; refs [28]–[29] — the citations
  • review/round3-adjudication.md ‘Verdict’ — the two routes; ‘Rulings’ 3 and 10 — the literature readings; ‘Decisions’ 2 — routes (a) and (c); ‘Decisions’ 3 — the √N floor dropped
  • review/round3/referee/audit-notes.md Step 1 — Preskill’s abstract; Steps 2 and 4(a) — the Liveoak–Maharana–Wells equations and acceptance criterion 3; Step 4(c) — the quoted abstracts and the uncited prior art; Step 4(d) and Step 5 — the referee’s conclusions
  • review/v3-worklog.md ‘Round-2 pending list’ — Hiramatsu et al. (R6)
  • notes/knot-dark-matter-standalone.md §3 — the classic relic analysis; §6 — acceptance criterion 3; ‘References’ — [12], [19]