Open: polarization-genesis and the 5.4 : 1 ratio (OP7)

Polarization-genesis is the program’s name for a mechanism it does not yet have: the process that would set how much dark matter the early universe made compared with ordinary matter, whose densities stand in the observed ratio Ω_χ : Ω_b = 5.4 : 1. For particle χ the lab notebook argues that the standard route is closed: a light sector that interacts strongly with itself and touches ordinary matter only through gravity cannot get its abundance from thermal freeze-out, so the abundance has to come from an asymmetry, as the abundance of ordinary baryons does. The notebook sketched a mechanism in which heavy fields decay into both kinds of matter, with the split set by an angle of the vacuum’s polarization, and called it “a shape, not a mechanism”. The problem is open and applies to both remaining dark-matter candidates.

Status: Open [O] — #11 §14 item 7, “Polarization-genesis: Ω_χ : Ω_b = 5.4 : 1 [O]”. The notebook’s sketch is tagged “[O, sketched]” (notes/broken-born-theory.md §8). It qualifies both the χ-particle candidate and the X-condensate candidate. Nothing is calculated and nothing is machine-checked. Listed among open problems; not on the current queue in the root CLAUDE.md.

The question#

Dark matter outweighs ordinary (baryonic) matter by a fixed factor. Checkpoint #11 writes the ratio Ω_χ : Ω_b = 5.4 : 1 (§14 item 7), the lab notebook Ω_χ/Ω_b = 5.4 (#1 §5.3), and the handoff “Ω_DM:Ω_b = 5.4:1” (HANDOFF.md §2), where DM is dark matter. Ω_b is the density of baryons and Ω_χ that of the dark matter, each as a fraction of the critical density. The documents treat the value as an observational input and give no reference for it.

The question is what physics fixes the ratio. #11 §12 states its scope: “The 5.4 : 1 abundance question (polarization-genesis) applies to both” candidates. The name follows baryogenesis, the process that set the excess of baryons over antibaryons; #9 calls the problem “the program’s baryogenesis analog [O]” (#9 §10 item 6).

Why the standard route fails for particle χ#

The notebook states the argument in two sentences [O]:

A sub-GeV, strongly self-coupled, gravitationally-only-connected sector cannot freeze out thermally without cannibalism problems; the abundance must be set asymmetrically — by polarization-genesis (§8). The observed Ω_χ/Ω_b = 5.4 is an input awaiting a mechanism, exactly as η_B awaited baryogenesis. (#1 §5.3)

On the wiki’s reading, the three adjectives describe χ as the notebook had just fixed it. “Sub-GeV” and “strongly self-coupled” match its benchmark two subsections earlier: a mass m_χ ≈ 70 MeV × α̃^{2/3}(σ₀/m ÷ 30 cm²g⁻¹)^{−1/3} for a coupling α̃ of order one (“At α̃ ∼ 1 the Born approximation is marginal”), fixed by the cross-section per mass σ₀/m_χ ≈ 30 cm²/g that fits dwarf-galaxy cores (#1 §5.1; the scattering benchmark). “Gravitationally-only-connected” is postulate P4: no gauge force crosses from ordinary matter to χ, a statement #11 has taken strictly since v2.1 (#11 §4).

Why that combination rules out freeze-out is standard [T] (relic abundance mechanisms). Thermal freeze-out sets the abundance by annihilation into lighter particles, at a rate fixed by the coupling to them. A sector connected to ours only by gravity cannot annihilate into ordinary particles at any useful rate, and if χ is the lightest state of its own sector it has nothing lighter to annihilate into. What remains are number-changing self-interactions, in which several χ particles scatter into fewer and the lost rest mass heats the survivors: cannibalism, whose outcome depends on the sector’s self-coupling and its own initial temperature rather than on any coupling to ordinary matter. The notebook does not spell out which “cannibalism problems” it means; this is the standard picture.

The notebook’s conclusion is that the abundance “must be set asymmetrically”: by an excess of χ over its antiparticle, as the baryon density is set by an excess of baryons.

The template: baryogenesis [T]#

The baryon density is not a freeze-out relic. It is set by the baryon asymmetry η_B, a small excess of baryons over antibaryons that survived when the rest annihilated [T]. Producing such an excess requires the three Sakharov conditions: baryon-number violation, violation of C (charge conjugation) and of CP (charge conjugation and parity combined), and a departure from thermal equilibrium [T]. Asymmetric dark matter extends the idea to the dark sector, so that a shared origin of the two excesses can fix the ratio of the two densities [T]. The notebook’s comparison, “exactly as η_B awaited baryogenesis”, places the 5.4 : 1 ratio where η_B stood before a mechanism for it existed: an observed number with no derivation.

The notebook’s sketch [O, sketched]#

#1 §8 gives the only proposal in the record, in full:

Why Ω_χ : Ω_b = 5.4 : 1? Sketch: at reheating, the heavy Grassmannian moduli (the nine non-Maslov directions) decay; their branching into x- versus p-polarized quanta is set by the vacuum tilt angle, Ω_χ/Ω_x^(0) ~ tan²θ_pol, after which baryogenesis processes the x-side. A relation between the observed ratio and a Grassmannian angle would be the theory’s second nontrivial number. Status: a shape, not a mechanism — the same maturity baryogenesis had in 1966.

Its steps, with what the record supplies for each:

The decaying fields. The vacuum’s polarization is a point of the Lagrangian Grassmannian Λ(4), the space of position/momentum splittings, which has “dimension d(d+1)/2 = 10 for d = 4” (#1 §2 [T]). One direction is the Maslov circle, whose angle τ is the dark-energy field (τ as the Maslov phase); the other nine are the “non-Maslov directions”. That τ “is the lightest modulus, with the other nine directions heavy, is assumed [P]” (#1 §2).
Decay at reheating. Reheating is the end of inflation, when the energy of the fields that drove it is converted into a hot plasma [T]. In the sketch the nine heavy moduli decay then, into quanta of both polarizations: x-polarized ordinary matter and p-polarized χ.
The split. The branching between the two is “set by the vacuum tilt angle”, giving Ω_χ/Ω_x^(0) ~ tan²θ_pol. The notebook defines θ_pol only by that phrase; its postulate P5 calls our vacuum “x-tilted” (#1 §1). It gives no amplitudes. A ratio of the form tan²θ is what a split by angle produces when the two channels receive amplitudes proportional to sin θ and cos θ, as when polarized light meets a polarizing beam splitter at angle θ and the two output intensities go as cos²θ and sin²θ [T]; that this is the intended reading is the wiki’s inference, not the notebook’s statement.
Baryogenesis on the x-side. Ω_x^(0) is not defined further. The superscript reads as the x-side share before “baryogenesis processes the x-side”, but the notebook does not say how Ω_x^(0) becomes Ω_b, so the sketch does not by itself give Ω_χ : Ω_b.
What it would buy. A relation between the ratio and a Grassmannian angle “would be the theory’s second nontrivial number”. The notebook does not say which number is the first.

The sketch has no decay rates, no value for θ_pol and no number for the ratio, and no later document develops it. Its own verdict, “a shape, not a mechanism”, is the status the wiki records.

An unreconciled dependence#

The sketch needs the nine heavy moduli to exist as fields and to decay. Calc note #5, written later, counts them in the field content of the broken vacuum and expects most of them to disappear: “Λ(d) σ-modes (10: τ plus nine, of which inverse-Higgs constraints likely eliminate most [O])” (notes/debt1a-geometric-core.md §D). Inverse-Higgs constraints are relations that make some would-be Goldstone fields derivatives of others, removing them as independent fields [T]. #5 lists “(4) inverse-Higgs accounting of the nine heavy moduli” among the items “Now precisely open” (§F), and no later note returns to it. No document reconciles the sketch with #5’s expectation (Open: field content of the broken vacuum; conflicts between sources).

Scope: particle χ, wave χ and the X-condensate#

Wave χ. Very light χ on the wave island gets its abundance by misalignment, like an axion. Calc note 3b: “Open problem #5 (cannibalism-safe relic abundance) is moot on the wave island: misalignment is the mechanism, and Ω_χ trades against (m, χᵢ) in the standard way. The 5.4:1 question remains (polarization-genesis), but the production channel is no longer exotic” (‘Bonus’; relic abundance by misalignment on the wave island). The notebook’s addendum v0.2 records “open problems 3 and 5 close (5 via misalignment production on the wave island)”. That closes the production channel on the wave island only; the ratio itself stays open. “Open problem #5” here is item 5 of the notebook’s list, not item 5 of #11 §14, which is the dual-stress enhancement.
Particle χ. The freeze-out argument above applies, and no mechanism is known.
The X-condensate. No document discusses how it would be produced. #11 §12 applies the ratio question to it as to χ.

#11 outranks the notebook, and it keeps the ratio open for both candidates.

History#

The lab notebook (v0.1). §5.3 states the problem and §8 the sketch. Its list of open problems, “in order of severity”, ranks “Polarization-genesis mechanism for 5.4 : 1. [O]” fourth and makes the fifth depend on it: “Relic abundance without cannibalism: needs the asymmetric mechanism of (4). [O]” (#1 §11).
Debt 3 (addendum v0.2). The relic-abundance item closes on the wave island; polarization-genesis does not.
#9. Open problem 6: “Polarization-genesis: the 5.4 : 1 ratio Ω_χ : Ω_b — the program’s baryogenesis analog [O]” (#9 §10).
The handoff. Open item 7, “Polarization-genesis (the Ω_DM:Ω_b = 5.4:1 ratio)” (HANDOFF.md §2).
#11. §14 item 7, and §12’s statement that it applies to both remaining candidates.
Now. The current queue in the root CLAUDE.md (‘Since the handoff’) does not include it.

A date to check#

The notebook’s “the same maturity baryogenesis had in 1966” is not checked against a source. Sakharov’s conditions are usually dated 1967 (from model memory, †); the register entry is on references, and the discrepancy awaits a ruling as review/wiki-build-flags.md item 18. The flags’ audit rated the item unclear: no repository file dates Sakharov’s conditions independently, so under the † rule the usual date stays unverified.

Sources#

papers/broken-born-checkpoint-v2.md §14 item 7 — the open problem; §12 — it applies to both candidates; §8 — the benchmark; §4 — P4, strict since v2.1
notes/broken-born-theory.md §5.3 — why freeze-out fails; §8 — the sketch; §11 — items 4 and 5 of the notebook’s open problems; §2 — the dimension of Λ(4) and the nine heavy directions [P]; §5.1 — the benchmark; §1 — P5, “x-tilted”; ‘Addendum (v0.2)’ — the wave-island closure
notes/debt3b-wave-island.md ‘Bonus: the relic debt closes itself’ — misalignment on the wave island; the ratio remains
notes/debt1a-geometric-core.md §D — the nine moduli, likely eliminated; §F item 4 — “inverse-Higgs accounting of the nine heavy moduli”, open
notes/broken-born-program-v1.md §10 — “the program’s baryogenesis analog”
HANDOFF.md §2 — open item 7
CLAUDE.md ‘Since the handoff’ — the current queue
review/wiki-build-flags.md item 18 — the notebook’s 1966 date for baryogenesis