Scale-factor duality

Scale-factor duality, in this program, is the reciprocal relation a·ã = 1 between the scale factor a of our universe, which multiplies the distances between distant galaxies, and the scale factor ã of its momentum-space description, the conjugate universe: as one grows, the other shrinks by the same factor. The relation follows from holding Planck’s constant fixed, and it is the cosmological case of a fact familiar from signal analysis, that stretching a waveform compresses its spectrum. Read through this relation, energy in the conjugate description acts on our expansion in a translated form, the far-future halt of our expansion is the same event as the conjugate universe’s bounce, and one early idea of the program, that the conjugate universe’s collapse drives our acceleration, did not survive.

Stands: the established physics of Friedmann cosmology and minisuperspace and redshift; the program’s claims the cosmological constraint a·ã = 1, the duality dictionary w_eff = −w̃, the turnaround as the conjugate bounce (its premise, a negative V(τ), unresolved) · Died: conjugate collapse powers our expansion · Open: dual kinetic energy versus τ’s potential · Superseded: the heuristic rule w_eff = −2 − w̃ · Conjectures: the phase-space diameter relation (chat-only)

Map#

Friedmann cosmology and minisuperspace is established physics and comes first: the metric of a uniform expanding universe written with a lapse, gravity reduced to the scale factor and the lapse, the Friedmann equation as the constraint that varying the lapse produces, how each kind of fluid thins as the universe grows (the stiff fluid among them), and a summary of redshift. It ends with the program’s extension, the lab notebook’s dual gravity term, which adds an a⁻⁶ correction to the effective Planck mass.

Redshift is the second page of established physics: 1 + z = a₀/a_e, the stretching of every wavelength by the ratio of scale factors, with the momentum of every free particle falling as 1/a. In wave terms, stretching a waveform compresses its spectrum by the same factor. The page derives the relation and its conditions, explains redshift as a clock and a distance, and lists the program’s uses: z as the coordinate of the dark-energy data, in which the second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a crossing of w = −1 above z ≃ 0.4, with flag 43 in review/wiki-build-flags.md on the crossing redshift; temperatures and gravitational-wave frequencies falling as 1/a in the early universe; and the popular article #12’s reading of redshift as a contracting momentum spectrum, which the wiki takes as a gloss on a·ã = 1.

The pages after these two are the program’s own. Their results are derived in the lab notebook (#1 §6–§7) and restated in the checkpoint (#11 §10); none is machine-checked.

The cosmological constraint a·ã = 1 is the program’s central cosmological result [D]. Rigidity of the symplectic form fixes the volume of a comoving phase-space cell, so the two scale factors are reciprocals; the derivation needs only ω, so it survived the change from the original P2 to P2′. The documents identify the relation with the |Z| = 1 locus of the modular field, a structural reading that awaits Paul’s ruling (review/wiki-build-flags.md item 23), and the page sets the program’s usage beside the scale-factor duality of string cosmology.

The duality dictionary w_eff = −w̃ follows from the constraint [D — corrected en route]. Conjugate-side energy with equation of state w̃ acts on our expansion like energy with −w̃, because its energy is counted in the dual frame’s shrinking volume before it is spread over ours. The page derives the rule step by step, explains each entry of the table, and records in a folded section the superseded heuristic −2 − w̃, which omitted that volume factor.

Obituary: conjugate collapse powers our expansion sits beside the dictionary that killed it. On the heuristic rule, collapsing conjugate matter acted on us like dark energy with w = −2, which is falsified; on the derived rule it acts as ordinary dust, which decelerates the expansion, so the mechanism disappeared. The claim died twice.

The turnaround as the conjugate bounce applies the constraint to the far future: if τ’s potential turns negative, our expansion halts and reverses, and since ã = 1/a our maximum is the conjugate description’s minimum, one event with two descriptions [D]. Its premise conflicts with the potential as the documents write it, V₀[1 + cos(τ/f)], which is never negative for V₀ > 0; no document resolves the conflict (flag 27 in review/wiki-build-flags.md). Its test is row 5, a long-horizon row whose near-term proxy is row 4.

Open: dual kinetic energy versus τ’s potential records a gap in the notebook’s motivation for dark energy: the dictionary suggests conjugate-side kinetic energy as its source, the program implements it as the potential energy of the field τ in our own description, and no document derives one from the other.

The phase-space diameter relation (chat-only) closes the section. It is a conjecture known only from the handoff’s summary of the original chat, D_x·D_p = h·N^{1/3} per dimension, with no definitions, derivation or script behind it; a note on it is in the current work queue.

Boundaries#

Dark energy itself — the field τ, its equation of state and the survey data that test it — is in Dark energy, including the open conflict over the sign of V(τ). The conjugate universe as a concept, one physical state with two descriptions, is the conjugate universe, and the founding suppositions behind these pages are on founding suppositions and their fates. The modular geometry whose |Z| = 1 locus the documents identify with a·ã = 1 (an identification queried in flag 23 of review/wiki-build-flags.md) is reciprocity as modular inversion. The early-universe consequence of the conjugate gravity term, a step in the effective strength of gravity, is row 10, with its background physics in early-universe cosmology. The cosmological solutions of the full two-metric theory are bimetric cosmology.