Scale-factor duality on The Broken-Born program

Friedmann cosmology and minisuperspace

Mon, 01 Jan 0001 00:00:00 +0000

Friedmann cosmology is the description of a universe that looks the same at every place and in every direction, so that its entire geometry reduces to one function of time, the scale factor a(t), which multiplies the distance between any two distant galaxies. The Friedmann equation ties the expansion rate to the energy density, and each kind of content, whether matter, radiation or a cosmological constant, thins out at its own rate as a grows. Minisuperspace is the name for gravity reduced to this one function, with the freedom to choose the time coordinate kept as a single function called the lapse. The program writes its cosmology in this reduced form and adds a second scale factor for the conjugate universe, the momentum-space description of the same state.

Redshift

Mon, 01 Jan 0001 00:00:00 +0000

Redshift is the stretching of the wavelength of light between its emission and its observation. In cosmology it is written z and measures how much the universe expanded while the light was travelling: 1 + z = a₀/a_e, the scale factor when the light arrives divided by the scale factor when it left, the scale factor being the number that multiplies the distances between distant galaxies. Every wavelength in a spectrum stretches by the same factor, the momentum of every freely moving particle falls in inverse proportion to the scale factor, and the frequency of gravitational waves and the temperature of freely travelling radiation fall with it; in wave terms, stretching a waveform compresses its spectrum by the same factor. Because the scale factor has grown throughout the history of the universe, a measured redshift dates the emission, and surveys report the expansion history as a function of z, including the equation of state w of dark energy, its pressure divided by its energy density. The program meets redshift there. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a w that lies below −1 at redshifts above about 0.4 and above −1 since; the program requires that w never fall below −1, so its dark-energy sector is under real pressure. The popular article #12 reads the redshift as the momentum spectrum of the universe contracting, which the wiki takes as a plain-language gloss on the program’s constraint a·ã = 1; the technical documents do not discuss the connection.

The cosmological constraint a·ã = 1

Mon, 01 Jan 0001 00:00:00 +0000

The cosmological constraint is the relation a·ã = 1 between the scale factor a of our universe and the scale factor ã of its momentum-space description, the conjugate universe: as our universe expands, its momentum-space description contracts by exactly the same factor. It follows from the program’s postulate that the symplectic form ω, the structure that fixes the size of a cell of phase space and with it the value of Planck’s constant, is rigid. A comoving cell therefore keeps its phase-space volume while position-space distances grow, which forces momentum-space distances to shrink; in signal terms, stretching a waveform compresses its spectrum. The relation is derived [D], and it turned one of the program’s founding suppositions, that the conjugate universe shrinks as ours expands, into a theorem.

The duality dictionary w_eff = −w̃

Mon, 01 Jan 0001 00:00:00 +0000

The duality dictionary is the rule that translates energy in the conjugate, momentum-space description of the universe (the conjugate universe) into the form in which it acts on our own expansion. Each kind of energy is labelled by its equation of state w, the ratio of its pressure to its energy density, and the rule is that conjugate-side energy with equation of state w̃ acts on us like energy with w = −w̃. Pure kinetic energy on the conjugate side therefore acts here like a cosmological constant, a conjugate cosmological constant acts like the stiffest possible fluid, and dust acts like dust. The rule is derived [D] from the cosmological constraint a·ã = 1, and it replaced an earlier heuristic that gave a different answer; #11 lists the change as one of the two forced corrections of the program’s hard core.

Obituary: conjugate collapse powers our expansion

Mon, 01 Jan 0001 00:00:00 +0000

This page records a dead claim: that the gravitational collapse of the conjugate universe, the momentum-space description of our own, is what drives the accelerating expansion of our universe. It was one of the program’s founding suppositions. Written down with the heuristic translation rule of the early discussion, it required dark energy with equation of state w = −2, a value the data exclude. With the translation rule derived properly, the duality dictionary, collapsing conjugate matter acts on our expansion as ordinary matter does, slowing it rather than accelerating it, so the mechanism disappeared. The claim died twice and stays dead.

The turnaround as the conjugate bounce

Mon, 01 Jan 0001 00:00:00 +0000

The turnaround is a far-future event the program predicts: our universe stops expanding and begins to contract, which happens if the potential energy of the dark-energy field τ becomes negative. Because our scale factor and that of the conjugate universe are reciprocals (the cosmological constraint a·ã = 1), the moment our universe is largest is the moment its momentum-space description is smallest, so the same event, read in momentum variables, is a contracting universe reaching its minimum size and rebounding. The program states this as one event with two descriptions, not as two linked events. It is a long-horizon prediction whose near-term proxy is the dark-energy drift that survey data are testing now, and whether τ’s potential, as the documents write it, can become negative at all is an unresolved conflict in the sources.

Open: dual kinetic energy versus τ's potential

Mon, 01 Jan 0001 00:00:00 +0000

This open problem is a gap between two explanations of dark energy that stand side by side in the program’s lab notebook. The first comes from the duality dictionary: kinetic energy in the conjugate, momentum-space description of the universe (the conjugate universe) acts on our expansion like a cosmological constant, so the notebook names it the natural source of a near-constant dark energy. The second is the explanation the program uses: dark energy is the potential energy of the field τ (hilltop quintessence), held near the top of its potential. No document derives the second from the first, so the step from the motivation to the identification of dark energy with τ is asserted rather than derived.

The phase-space diameter relation (chat-only)

Mon, 01 Jan 0001 00:00:00 +0000

The phase-space diameter relation is a conjecture from the program’s original chat sessions: the diameter of the universe in position space, multiplied by its diameter in momentum space, equals Planck’s constant times a large number fixed by how many photons the universe contains. It is a cosmological counterpart of the time–bandwidth product of signal analysis, which says that a pulse cannot be both very short and very narrow in frequency. The relation exists only in the chat, as summarized by the handoff document: no paper or note derives it and no script checks its numbers. A short note on it is in the current work queue.