Dark energy on The Broken-Born program

The cosmological constant Λ

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

The cosmological constant Λ is a constant term in Einstein’s equations of gravity that acts as an energy density of empty space: the same at every point and at every time, with a pressure equal to minus that density, so that its equation of state w, the ratio of pressure to energy density, is exactly −1. Matter and radiation thin out as the universe expands while this density does not, so a positive Λ comes to dominate, makes the expansion accelerate, and leaves an empty universe expanding exponentially forever, called de Sitter space; with a negative Λ, the expansion of a universe containing matter halts and reverses. Λ is the dark energy of the standard cosmological model, ΛCDM, a cosmological constant plus cold dark matter, and it is the standard expectation in two rows of the program’s prediction table: a constant w = −1 in row 4, and eternal expansion toward de Sitter space in row 5. The program assigns the observed dark energy to a field, τ, in its place, and data that settle on a constant w = −1 would kill that sector. The fits the program cites report evidence that dark energy is weakening, which a constant cannot do; the second data release of the Dark Energy Spectroscopic Instrument also favours an earlier phase with w below −1, which τ cannot have (DESI DR2’s phantom crossing). The vacuum energy of quantum fields acts exactly as a Λ does, and why it is not far larger than observed is a separate, unsolved problem.

De Sitter space

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

De Sitter space is the spacetime of a universe that contains nothing but a positive cosmological constant Λ, an energy density of empty space that stays the same as space expands. Its expansion rate, the Hubble rate H, never changes, so the distances between distant galaxies grow exponentially, and every observer is surrounded by a horizon beyond which events can never be seen. Its curvature is the same at every point and in every direction; anti-de Sitter space is the counterpart for a negative cosmological constant. With a positive cosmological constant as its dark energy, standard cosmology expects our universe to approach de Sitter space as matter thins: “Eternal de Sitter” is the standard expectation of the program’s row 5, where the program requires instead a far-future turnaround whose premise is unresolved. De Sitter space is also the background on which a massive spin-2 field, a graviton with mass, must satisfy the Higuchi bound, the setting of partially massless gravity, and, in two dimensions, the space of values of the field K in the program’s first toy model, where its metric, positive in one direction and negative in the other, would give a ghost: a mode with negative energy.

Quintessence and the equation of state

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

Quintessence is dark energy made of a scalar field, a quantity with a single value at each point of space, whose energy density changes slowly as the universe expands instead of staying fixed as a cosmological constant’s does. Its behaviour is summarized by the equation of state w, the ratio of pressure to energy density: a cosmological constant has w = −1 exactly, and a field whose energy is almost all potential energy comes close to that value. In a thawing model the expansion holds the field nearly still at early times and it starts to change late, so w begins near −1 and rises. The program’s dark-energy field τ is a model of this kind, with the periodic potential of an axion-like field, sitting near the potential’s maximum.

Thawing and freezing quintessence

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

Thawing and freezing are the two classes into which quintessence, dark energy made of a slowly changing scalar field, divides according to how its equation of state w, the ratio of its pressure to its energy density, changes with time. A thawing field is held almost still by the expansion of the universe at early times, so w stays close to −1, the value of a cosmological constant; the field starts to change only late, and w rises above −1 toward the present. A freezing field is already changing at early times and slows down, so w falls toward −1. A field that starts near the maximum of its potential, a hilltop model, thaws. The program’s dark-energy field τ is taken to be such a field, and row 4 of its predictions requires a thawing drift that never takes w below −1. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a w that was below −1 earlier in cosmic history, a phase that a field of this kind, with the standard kinetic energy, cannot produce, so the requirement is under real pressure.

Axion-like fields and pseudo-Goldstone bosons

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

An axion-like field is a scalar field whose value is an angle. Its interactions are unchanged when the field is shifted by a constant, φ → φ + c, except for a small explicit breaking of that symmetry, which leaves a potential that repeats as the angle goes around its circle, in its standard form V₀[1 + cos(φ/f)]. Such a field is a pseudo-Goldstone boson: the massless mode that a spontaneously broken continuous symmetry leaves behind, given a small mass by a slight explicit breaking. The shift symmetry keeps the field light and lets it couple to other matter only through how it changes in space and time. The axion of the strong interaction is the original example; it and lighter fields of the same kind are studied as dark matter, and the lightest as dark energy. The Broken-Born program’s dark-energy field τ is one: an angle with a cosine potential, starting near the potential’s maximum. The program also calls its momentum graviton X “axion-like” for its exact shift symmetry, and the worklog of its toy models describes the vacuum’s choice of position and momentum as “a pseudo-Goldstone polarization angle”.

The CPL parametrization w(a) = w₀ + wₐ(1 − a)

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

The Chevallier–Polarski–Linder (CPL) parametrization is the standard way to describe, with two numbers, how the equation of state w of dark energy, its pressure divided by its energy density, changes as the universe expands: w(a) = w₀ + wₐ(1 − a), where a is the scale factor, the size of the universe relative to its size today. w₀ is the value of w today and wₐ measures how it has drifted; a cosmological constant is the point w₀ = −1, wₐ = 0. Cosmological surveys commonly report their dark-energy results in this form, and the program states its dark-energy requirement in it: a thawing field, with w₀ > −1 and wₐ < 0, that is never phantom, meaning that w never falls below −1. The form is a straight line in the scale factor, an approximation to the true history rather than a physical model. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a w that crosses −1, lying below it earlier in cosmic history, at redshifts above about 0.4, and above it since, and calls that preference “rather independent of parametrization choices”. The crossing stands against the program’s requirement and puts its dark-energy sector under real pressure. In these terms the favoured shape has w₀ > −1 with w₀ + wₐ < −1, although a CPL fit in that region does not by itself show a crossing, since a line fitted to a field that never falls below −1 can lie there too (flag 43 in review/wiki-build-flags.md).

The phantom divide and the null energy condition

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

The phantom divide is the value w = −1 of the dark-energy equation of state, the value a cosmological constant has, and it separates two kinds of behaviour: above it, dark energy thins out as space expands; below it, in the regime called phantom, dark energy becomes denser as space expands. An ordinary scalar field cannot reach the phantom side, because the quantity that would have to become negative is the square of the field’s rate of change. That fact is a case of the null energy condition, a standard requirement on physically reasonable matter. A measured w below −1 would therefore point to something unusual, such as a field with negative kinetic energy, several interacting fields or a change to gravity. The program’s dark-energy field is an ordinary scalar, so a robust intrinsic w < −1 would kill the program’s framework outright.

DESI baryon acoustic oscillations

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

The Dark Energy Spectroscopic Instrument (DESI) is a survey that measures the redshifts of galaxies and quasars in order to map how the universe has expanded over a long span of its history. Its main tool is baryon acoustic oscillations (BAO): sound waves in the hot plasma of the early universe left a preferred separation between concentrations of matter, a fixed length that serves as a ruler at every later epoch. In Fourier terms the imprint is a single bump in the correlation between galaxy positions and a series of wiggles in its transform, the power spectrum. Measuring the ruler’s apparent size at different redshifts gives distances, and the dark-energy equation of state is inferred from those distances by fitting a model; it is not observed directly. DESI’s data releases are the external clock for the program’s dark-energy sector, and its second release already leans against the program’s requirement.

τ: hilltop quintessence on the Maslov circle

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

The program identifies dark energy with τ, the angle that sets how the vacuum divides phase space into position and momentum. Because τ is an angle, its potential energy must repeat when τ goes once around its circle, and the program writes it as a cosine, V = V₀[1 + cos(τ/f)]. Today τ is taken to sit near the top of that potential, where its energy is almost all potential energy and it acts almost like a cosmological constant, while drifting slowly away from the top, so that dark energy weakens over time and its equation of state w never falls below −1. Thawing hilltop quintessence of this kind is an established class of models; the program’s contribution is the identification of the field, which the popular article calls “the angle of the mirror”. The claim stands under real pressure: the second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a phase with w below −1, which τ cannot produce, and the next release is the decisive test.

Fifth-force safety of τ

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

Fifth-force safety is the requirement that a new light field not produce a detectable extra force between ordinary bodies. The program’s dark-energy field τ is extremely light, so if it coupled to ordinary matter through its value, as the gravitational potential does, it would add a long-range force of the kind that torsion-balance experiments search for. The lab notebook argues that τ is an approximate Goldstone field, one whose couplings are fixed by a symmetry under shifting its value, so it couples to matter only through its rate of change, which produces no such static force between ordinary bodies. The argument appears only in the lab notebook, in its first version; the current checkpoint does not restate it, and no script checks it.

The no-phantom bound w ≥ −1

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

The no-phantom bound is the program’s requirement that the dark-energy equation of state w never fall below −1, the value of a cosmological constant. It holds because the program’s dark-energy field τ is an ordinary (canonical) scalar field, whose energy density can fall or hold steady as space expands but never rise, while w < −1, the phantom regime, would make it rise. The bound makes the dark-energy sector a sharp test: a robust measurement of an intrinsic w < −1 would kill the program’s framework outright. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a phase with w < −1 at redshifts above about 0.4, so the bound stands against the direct reading of current data and is under real pressure; it survives only because fits without a phantom phase remain compatible and because w is inferred rather than observed.

DESI DR2's phantom crossing

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

DESI DR2’s phantom crossing is the finding, reported in 2025 from the second data release (Data Release 2, DR2) of the Dark Energy Spectroscopic Instrument (DESI), that the dark-energy equation of state favoured by the data lies above −1 at low redshift and crosses below −1, into the phantom regime, at redshifts above about 0.4, in every combination of datasets analysed. The program requires dark energy never to be phantom, so its requirement stands against the direct reading of these data, and its dark-energy sector is under real pressure. The sector survives on two stated points: fits without a phantom phase remain compatible at about two standard deviations, and the equation of state is inferred from distance measurements rather than observed. The third data release is the decisive test.

The dark-energy over-constraint

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

The dark-energy over-constraint is a counting argument: the program’s dark-energy model has fewer free numbers than the quantities it predicts, so the data can test the model against itself. The model has three free numbers, the height V₀ and scale f of the potential of the dark-energy field τ and τ’s initial value τᵢ, and the height is fixed by the measured amount of dark energy, which leaves two. Those two must account for three observables: today’s equation of state w₀, its rate of change wₐ, and the time of the future turnaround, t_turn. One relation among the three is therefore a prediction, which the checkpoint counts among the program’s progressive signs. The third observable lies in the far future, and the turnaround it times needs the potential to go negative, which the potential as written never does.

Open: the sign of V(τ) and the turnaround

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

The sign problem is a conflict in the program’s documents about the potential energy V(τ) of the dark-energy field τ. They write it as V = V₀[1 + cos(τ/f)], and since a cosine never goes below −1, this potential is never negative when V₀ is positive. Yet the program’s far-future turnaround, in which our expansion halts and reverses, requires the potential to become negative, and the published fits the program cites for that event use potentials with a negative minimum. None of the three documents that write the potential discusses the conflict. It was found while this wiki was being built and is recorded, for Paul’s ruling, as flag 27 in the build’s list of suspected errors in the sources. Until he rules, the turnaround, its reading as the conjugate bounce, and the turnaround time in the over-constraint all rest on an unresolved point. This page records the conflict and does not choose a side.