The scattering benchmark

The scattering benchmark is the set of numbers obtained by requiring the self-scattering of conjugate matter to produce the flat central cores observed in dwarf galaxies. Fitting the cores fixes the cross-section per unit mass at low velocity; with an assumed mass for the carrier of the momentum transfer, that fixes the mass of a χ particle at about 70 MeV for a coupling of order one. The checkpoint keeps only this mass relation. The other benchmark numbers, among them the prediction that galaxy clusters show no self-interaction at any sensitivity, appear only in the lab notebook, rest on a Gaussian gate tail that is now an open question, and have never been checked by machine. The benchmark also fixes the coupling that the program’s Debt-3 calculation used to rule out collective clumping of χ particles.

Status: Alive — [D] (notes/broken-born-theory.md §5.1) — the mass relation is carried by #11 §8, and v1 (#9 §6) states it for particle χ; no benchmark number has a verify/ script, and the numbers found only in the lab notebook still lack the executed check that the iron rule requires. The lab notebook’s predictions above the gate are superseded.

The relation#

The checkpoint, #11, gives the benchmark in one line of §8:

“Benchmark: m_χ ≈ 70 MeV·α̃^{2/3}(σ₀/m ÷ 30 cm²g⁻¹)^{−1/3}.”

m_χ ≈ 70 MeV · α̃^{2/3} · (σ₀/m ÷ 30 cm²g⁻¹)^{−1/3}

Here m_χ is the mass of a χ particle, α̃ the coupling of its self-interaction amplitude, and σ₀/m the low-velocity cross-section per unit mass of the gated scattering law, σ_T(v) = σ₀F(v²/v*²). The relation is scaled to a reference value of σ₀/m of 30 cm²/g: at that value and α̃ = 1 it gives 70 MeV, and other values scale by the powers shown. v1 (#9 §6) gives the same relation “for particle-χ, with the wave-χ alternative in §7”.

Where the input comes from#

Dark matter that does not scatter off itself, cold dark matter, is predicted to pile up in a steep central density peak, a cusp, while many dwarf galaxies show a flat core instead (dwarf galaxies) [T]. Self-scattering carries heat into the cold center of a halo and turns a cusp into a core, provided the cross-section per unit mass is large enough at the velocities found in dwarfs (self-interacting dark matter) [T].

The lab notebook takes the value from that requirement: “Fitting dwarf-scale cores wants σ₀/m_χ ≈ 30 cm²/g at v ≲ v* ≈ 120 km/s” (#1 §5.1). It cites no source for the value. The gate velocity v* is where the cross-section turns down, so dwarfs, at lower velocities, see the full σ₀. The checkpoint’s row 1 places the knee at “v* ~ 10² km/s” (#11 §13), less precisely than the notebook’s 120 km/s.

How the relation follows#

The lab notebook derives the relation in §5.1, tagged [D], from its v0.1 scattering law:

σ_T(v) = σ₀ e^{−v²/v*²},   v* ≡ √2 μ/m_χ,   σ₀ = 16πα̃²m_χ²/m_φ⁴

Here m_φ is the mass of the carrier of the momentum transfer and μ the gate’s momentum width. “With m_φ ~ m_χ this fixes” the mass relation together with

μ = m_χ v*/√2 ≈ 20 keV   (α̃ ≈ 1)

(#1 §5.1). The notebook does not show the algebra. The form of the relation can be read from σ₀ [T]: with m_φ set equal to m_χ, σ₀/m_χ is proportional to α̃²/m_χ³, so fixing σ₀/m fixes m_χ³ in proportion to α̃², which gives the powers 2/3 and −1/3. The prefactor of 70 MeV is the notebook’s number and is not machine-checked. The value of μ follows from the definition of v* at that mass and v* ≈ 120 km/s.

The notebook attaches a caveat to the coupling: “At α̃ ~ 1 the Born approximation is marginal (expect O(1) corrections); the shape claims above survive because they follow from the k- vs q-dependence, not from the coupling strength” (#1 §5.1). The Born approximation is the first-order treatment of scattering, reliable when the interaction is weak [T]; at α̃ ~ 1 the absolute cross-section can be off by a factor of order one, and the notebook’s argument is that the velocity dependence set by the gate survives because it does not depend on the coupling’s size.

The coupling is “spent”#

The benchmark leaves no free strength in the χ-particle sector, and the Debt-3 calculation used that. Calc note #2, the first of the Debt-3 calc notes, writes the self-interaction strength as a single coupling,

g ≡ 4πα̃/m_φ² = √(πσ₀/m²)

(‘Setup’), and then evaluates the mean-field criterion for collective clumping at the benchmark:

“The dwarf-core fit fixes σ₀/m = 30 cm²/g, hence g is spent — no freedom remains. With m_χ = 70 MeV, v* = 120 km/s:” (notes/debt3a-stream-stability.md, ‘Numbers at the scattering benchmark’)

The ratio of interaction energy to the threshold comes out “∼ 10⁻²⁸” in a dwarf core and “∼ 10⁻³¹” in the local halo of the Milky Way: “Fails by 28–31 orders of magnitude.” #11 §9 carries it as “fails by 10²⁸–10³¹ at the scattering benchmark”. See the mean-field criterion and its 10²⁸–10³¹ shortfall. The same calculation, run the other way, gives the classicality theorem: “satisfying the criterion at fixed σ₀/m forces m ≲ 0.1 meV, where N_occ ~ 10²⁰” (#11 §9): “collective momentum-space behavior and classical particles are mutually exclusive”. On the regime in which Debt 3a evaluates that mass, see flag 19 in review/wiki-build-flags.md.

The benchmark is therefore the particle-island benchmark. On the wave island, where χ is light enough to act as a classical wave, the dwarf-core fit is imposed on a different quantity, the gated condensation rate, and the combination it fixes is g²/m³ (Obituary: Scenario A and the ultra-faint-dwarf endpoint).

Assumptions and caveats#

  • An assumption from before the mechanism. The choice m_φ ~ m_χ was made in v0.1, when the gate was a postulate. The v0.3 addendum later identified the carrier as exchange of the momentum graviton X (#1 ‘Addendum (v0.3, overnight)’; the gate mechanism). No later document revisits m_φ or relates it to the mass M_X of X (conflicts between sources).
  • An uncited input. The 30 cm²/g that fixes the scale is stated without a reference (#1 §5.1).
  • The Born approximation is marginal at α̃ ~ 1, as the notebook says.
  • Numbers found only in the lab notebook. μ ≈ 20 keV and the predictions above the gate (below) appear only in #1. The formulas for v* and σ₀ come from #1 and are not restated in #11; #13 §4 uses the definition of v* with μ read as the χ quantum’s fiber width (the invariant-length shell), and a suspected factor of √2 in μ between the two gate conventions is recorded for a ruling in review/wiki-build-flags.md item 17 (two conventions). The project’s rule is that every asserted number gets executed code; for these, no verify/ script exists (the iron rule).

Why the strength needs explaining#

The checkpoint calls this σ₀ “nuclear-scale”, comparable to the cross-sections of nuclear physics [T]. In the same sentence it lists reconciling that value with the gravitational-strength couplings of ordinary matter to X as the “Main open quantitative item” (#11 §8). The benchmark fits σ₀; it does not explain how X exchange produces a cross-section that large. See Open: the dual-stress enhancement (OP5).

Superseded: the v0.1 predictions from the benchmark#

In v0.1 the benchmark fed predictions that the checkpoint no longer makes.

  • Cross-sections above the gate. “Then the theory predicts, with zero remaining freedom: σ_T/m (300 km/s) ≈ 6×10⁻² cm²/g, σ_T/m (1500 km/s) ≈ 30 e^{−156} ≈ 0.” (#1 §5.1)
  • Clusters. “Cluster self-interaction is identically zero at any future sensitivity — a sharper statement than any Yukawa model can make, and a standing kill condition.” (#1 §5.1; scattering through a Yukawa mediator)
  • What two parameters covered. The notebook’s parameter count gave the conjugate sector two parameters, “σ₀/m, v* (2)”, covering “Dwarf cores, LSB/group knee, cluster nulls, stream granularity scale, friction suppression (≥5)” (#1 §10), LSB meaning low surface brightness galaxies.

The cross-section numbers come from the exponential cliff e^{−v²/v*²}, which the v0.4 addendum weakened to a knee whose tail may be Gaussian or rational (the gated scattering law). They hold only if the tail is Gaussian, which is now Open: the gate shape (OP4), and none is machine-checked. The checkpoint keeps the qualitative statement, “zero cluster signal (particle χ)”, in row 1 (#11 §13), with no number attached. Of the five phenomena, two have died: velocity-space granularity (obituary) and the suppression of dynamical friction in dwarf cores (obituary).

Sources#

  • papers/broken-born-checkpoint-v2.md §8 — the mass relation; σ₀ as nuclear-scale; the main open quantitative item; §9 — the 10²⁸–10³¹ shortfall at the benchmark and the classicality theorem; §13 row 1 — v* ~ 10² km/s, zero cluster signal for particle χ
  • notes/broken-born-program-v1.md §6 — the relation for particle χ
  • notes/debt3a-stream-stability.md ‘Setup’ — the coupling g; ‘Numbers at the scattering benchmark’ — “spent” at the benchmark, the 28–31 orders; ‘Corollary (the classicality theorem)’
  • notes/broken-born-theory.md §5.1 — the core-fit input, σ₀ and v*, m_φ ~ m_χ, μ ≈ 20 keV, the Born caveat, the 300 and 1500 km/s numbers, the cluster statement; §10 — the v0.1 parameter count; §3 — the v0.1 amplitude
  • notes/broken-born-theory.md ‘Addendum (v0.3, overnight)’ — the gate’s carrier identified as X exchange
  • papers/conjugate-special-relativity.md §4 — the definition of v* reused
  • review/wiki-build-flags.md items 17, 19 — the suspected √2 in μ between the two gate conventions; the regime of the classicality corollary; both awaiting a ruling
Backward The gated scattering law σ_T(v) Open: the gate shape (OP4) Forward