Row 10: an early step in G_eff

Row 10 describes two possible imprints of the program’s dual, momentum-side gravitational sector on the early universe. The first is a step in the effective strength of gravity, G_eff: the dual sector adds to the square of the effective Planck mass a term that grows toward the past as the sixth inverse power of the cosmic scale factor, a⁻⁶, so it matters only at early times, before the light elements formed in Big Bang nucleosynthesis. The second is negative dark radiation: if the dual space is closed, its curvature acts on our expansion like radiation of negative energy, which lowers the effective number of neutrino species, N_eff. Both follow from the lab notebook’s calculation with the cosmological constraint a·ã = 1. The step is optional and the dark radiation conditional on a closed dual space, and their sizes are free parameters (knobs), so a null result bounds the knobs rather than killing the program; #11 names the cosmic microwave background experiment CMB-S4 as the probe.

Status: Alive, optional and knob-bounded — [D] (the lab notebook’s minisuperspace calculation and dictionary, #1 §6; #11 §10’s dictionary is [D — corrected en route]) — not machine-checked; no document compares either imprint with data.

The row#

Checkpoint #11 §13, row 10:

G is Newton’s constant and G_eff the effective gravitational coupling in the expansion equation. N_eff is the effective number of neutrino species and ΔN_eff its departure from the standard value. CMB-S4 is a proposed ground-based experiment on the cosmic microwave background (CMB) [T]; whether it is still going ahead is unchecked (flag 32 in review/wiki-build-flags.md, a lead from model memory that the flags’ audit left unclear). The kill column is a parenthesis: the row names no result that would end a claim.

The observables [T]#

Two measurements of the early universe bear on the row (the radiation era: thermal history and N_eff):

• The expansion rate during nucleosynthesis. In the first minutes, the ratio of neutrons to protons freezes when the weak reactions that convert them become slower than the expansion. A faster expansion freezes it earlier, leaves more neutrons and so more helium. The primordial helium abundance therefore measures the expansion rate H during Big Bang nucleosynthesis (BBN), and H depends on both the radiation content and the strength of gravity [T] (the expansion rate: radiation and gravity together).
• N_eff. The radiation of the universe after electron–positron annihilation is photons and three neutrino species. N_eff measures the energy density of all light species other than photons, in units of one neutrino species; extra light particles raise it, and the cosmic microwave background measures it through the expansion history before recombination [T].

The standard expectation#

“Standard G, N_eff” (#11 §13): Newton’s constant has the same value in the early universe as now, and N_eff takes the value of the Standard Model’s three neutrinos, a little above 3. The round-3 record uses 3.044 for it (ruling 5) [T].

The Broken-Born requirement#

Both imprints come from the lab notebook’s cosmology, which rests on two results of scale-factor duality.

The constraint a·ã = 1#

The program describes the universe with two scale factors, a for position space and ã for momentum space. Rigidity of the symplectic form fixes the volume of a comoving cell of phase space, so (#1 §6, [D]): “each mode occupies (aã)³ × (comoving cell) = ℏ³”, and a·ã = const ≡ 1, “the statement that ℏ does not run” (the derivation). ã is therefore not an independent variable: wherever it appears, it can be replaced by 1/a. Quotations from #1 on this page are transcribed from its LaTeX.

The a⁻⁶ step in G_eff#

The notebook adds the dual sector’s own gravitational term to the minisuperspace description of the universe, in which gravity is reduced to the scale factor and the lapse (#1 §6, [D]): “With one shared lapse N and dual Planck mass M̃, substituting ã = 1/a into the dual Einstein–Hilbert kinetic term” — the term ã(dã/dt)², which the substitution turns into ȧ²/a⁵ — “gives”

3H²(M_P² + M̃² a⁻⁶) = ρ

where H is the expansion rate and ρ the energy density. Without the M̃ term this is the ordinary Friedmann equation. The steps are on Friedmann cosmology and minisuperspace.

What the term means, in the notebook’s words: “The dual gravitational sector is not a new fluid and not a ghost — ã was never independent — but a stiff-like (a⁻⁶) modification of the effective Planck mass, relevant only at early times. It predicts a step in G_eff before BBN and, conditionally on its one knob, a spectral step in the primordial gravitational-wave background. [D]”

The wiki’s reading of the equation: the bracket is an effective Planck mass squared, and the effective strength of gravity is inversely proportional to it. The dual term grows toward the past as a⁻⁶, so at early enough times it dominates the bracket and gravity is weaker than today; as a grows, M_P² takes over and the ordinary Friedmann equation returns. The passage between the two regimes is the “step”. The notebook gives neither the step’s size nor its time, beyond “before BBN”; M̃ is free. #11 §10 keeps the result in one clause, “the dual Einstein term is an a⁻⁶ step in G_eff” (the dual gravity term).

“Not a ghost” is a minisuperspace statement. The notebook records that “no Boulware–Deser sector arises in minisuperspace” [D] and that “The statement for full inhomogeneous H is open” [O] (#1 §4). That analysis predates the notebook’s v0.4 addendum, after which the momentum-space partner of the metric is a massive spin-2 field (the momentum graviton X).

Negative dark radiation from a closed dual space#

The second imprint comes from the duality dictionary, which translates content of the dual frame into its effect on our expansion. The notebook’s derivation (#1 §6, [D]): a fluid comoving in the dual frame with equation of state w̃ has density ρ̃ ∝ ã^{−3(1+w̃)} and energy E = ρ̃ã³ ∝ ã^{−3w̃} = a^{3w̃}; entering our expansion as ρ_eff = E/a³ ∝ a^{−3(1−w̃)}, it acts as a fluid with

w_eff = −w̃

Spatial curvature behaves like a fluid with w̃ = −1/3 [T], so curvature of the dual space acts on our side as w = +1/3, the equation of state of radiation. The notebook’s table: “Spatial curvature (w̃ = −1/3)” appears as “Dark radiation (w = +1/3; closed dual space ⇒ negative ΔN_eff)” (the table). The sign comes from the curvature: in the Friedmann equation the curvature of a closed space enters with the opposite sign to an energy density, so read as a fluid it has negative energy density [T], and the dictionary carries that sign across. Extra radiation raises or lowers N_eff according to its sign, so a closed dual space gives ΔN_eff < 0. The table singles out the closed case, and no document discusses an open dual space (row 10 on the radiation-era page).

#11 §10 keeps it as “dual curvature ↦ negative dark radiation”, with the dictionary tagged “[D — corrected en route]”: the rule w_eff = −w̃ superseded an earlier heuristic, −2 − w̃, which identified densities across the two frames without the constraint’s volume factor (Superseded: the earlier heuristic rule).

Optional, and bounded by knobs#

The notebook’s parameter table counts the dual geometric sector separately (#1 §10):

The last entry is paraphrased; the notebook abbreviates it. M̃ is the dual Planck mass. k̃ is not defined in the notebook; written beside the dual curvature, it reads as the curvature of the dual space (the wiki’s reading). Both can be zero: with M̃ = 0 there is no step, and with a flat or open dual space no negative dark radiation. “Knob-bounded” means that measurements consistent with the standard values bound M̃ and k̃ rather than refute the program; v0.1 and v1 wrote “Bounds constrain knobs” (#1 §9; #9 §9).

What kills it#

Nothing in the row: “(Knob-bounded; CMB-S4 probes)”. A measurement of standard G and N_eff is consistent with the row at small M̃ and k̃, and the documents name no values of the knobs that the program expects, so a result within the standard values bounds the knobs without counting against the program. What the row looks for is a step in the expansion rate before nucleosynthesis or a negative ΔN_eff.

The data situation#

No program document compares either imprint with a measurement, and none computes the size of the step or of the negative ΔN_eff for any value of the knobs.

Apart from the notebook’s ±0.03 for CMB-S4 (below), the N_eff figures in the record belong to the retired soliton candidate, whose hidden photon would have added positive dark radiation (Obituary: superheavy conjugate solitons). The round-3 referee checked the Planck 2018 interval against its source, Planck’s combination of its own data with baryon-acoustic-oscillation measurements at 95% confidence: “N_eff = 2.99 +0.34 -0.33” (review/round3/referee/audit-notes.md, Step 1, citation [23]; #14 had written “2.99 +- 0.34”, review/wiki-build-flags.md item 22). With the Standard Model value 3.044, ruling 5 put the 95% bound on extra radiation at ΔN_eff < 0.286, which the gate checks (the soliton-obituary script):

PASS  95% bound Delta N_eff < 0.286  [0.286]

These figures bound positive ΔN_eff for a different proposal; no document applies them to row 10.

#11 names CMB-S4 as the probe and gives no sensitivity. #11 §15 names rows 1, 4 and 7 as the program’s near-term falsification exposure; row 10 is not among them.

Superseded: the spectral step and the ±0.03 target#

The lab notebook’s row 10 (#1 §9) required an “a⁻⁶ step in G_eff; optional negative ΔN_eff from closed dual curvature”, together with a conditional step in the spectrum of primordial gravitational waves, and its kill column read “(Bounds constrain knobs; CMB-S4 N_eff at ±0.03 probes the curvature term)”. The ±0.03 figure appears only in #1, and no document or script checks it.

v1 (#9 §9) changed the row in three ways, and #11 kept v1’s form:

• it dropped the spectral step in the gravitational-wave background;
• it dropped the ±0.03 target, reading “(Bounds constrain knobs; CMB-S4 probes)”;
• it moved “optional” from the dark radiation to the step: “Optional a⁻⁶ G_eff step; closed dual curvature ⇒ negative ΔN_eff”. v1’s cosmology section calls the step “an optional early-universe fingerprint” (#9 §8).

#11 changed only the kill column’s first words, from “Bounds constrain knobs” to “Knob-bounded”. No document says why the spectral step and the target were dropped. The mechanism the notebook implies for the spectral step is discussed on the dropped spectral step, and the whole table’s history on the history of the predictions table.

Sources#

• papers/broken-born-checkpoint-v2.md §13 row 10 — the row; §10 — the dictionary [D — corrected en route] and “the dual Einstein term is an a⁻⁶ step in G_eff”; §15 — the near-term rows
• notes/broken-born-theory.md §6 — the constraint, the minisuperspace equation, the a⁻⁶ step and the dictionary table [D]; §4 — the minisuperspace ghost analysis; §9 — v0.1’s row 10; §10 — the knobs M̃ and k̃. Quotations are transcribed from its LaTeX.
• notes/broken-born-program-v1.md §8, §9 — the step as “an optional early-universe fingerprint” and v1’s row 10
• review/round3/referee/audit-notes.md Step 1, citation [23] — the Planck 2018 interval, checked against its source
• review/round3-adjudication.md ‘Rulings’, ruling 5 — the Standard Model value 3.044 and the bound ΔN_eff < 0.286
• verify/soliton_obituary.py, dark radiation — the gate’s check of that bound
• notes/knot-dark-matter-standalone.md §4 — #14’s dark radiation and its Planck interval (withdrawn manuscript)
• review/wiki-build-flags.md item 22 — #14’s symmetric interval against the source’s asymmetric one; item 32 — CMB-S4’s status, unchecked