Quintessence and the equation of state

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.

Status: Established — [T] — textbook cosmology, written here from general knowledge, with the literature it names marked †; the fits that checkpoint #11 cites are live-searched [S] (#11 refs [19]–[21]).

Dark energy and the equation of state#

Expansion and the Friedmann equation [T]#

A spatially flat, homogeneous universe is described by a scale factor a(t), the factor by which distances between distant galaxies have grown. Its expansion rate H = ȧ/a is set by the total energy density ρ through the Friedmann equation, H² = ρ/(3M_P²), where M_P is the reduced Planck mass. Each component of the cosmic fluid obeys the continuity equation

ρ̇ = −3H(ρ + p)

where p is its pressure. Writing the pressure as p = wρ defines the equation of state w. For constant w the continuity equation integrates to ρ ∝ a^{−3(1+w)}: matter (w = 0) dilutes as a⁻³, radiation (w = 1/3) as a⁻⁴, and a fluid with w = −1 does not dilute at all [T]. The background is on Friedmann cosmology and minisuperspace.

Acceleration [T]#

The second Friedmann equation gives the acceleration, ä/a = −(ρ + 3p)/(6M_P²). The expansion accelerates when ρ + 3p < 0, which for a single dominant component means w < −1/3. Dark energy is the name for whatever component makes the present expansion accelerate.

The cosmological constant [T]#

The simplest dark energy is a cosmological constant Λ, an energy density of empty space that is the same everywhere and at all times: its pressure is minus its density, so w = −1 exactly and ρ stays constant as space expands. It is the standard expectation of the program’s prediction table, “w ≡ −1” (#11 §13 row 4). Its own page, the cosmological constant Λ, covers the rest, from the de Sitter future to the cosmological-constant problem.

A scalar field as dark energy#

Energy density and pressure [T]#

Quintessence replaces the constant by a scalar field φ (Ratra and Peebles †; Wetterich †). For a field that is uniform in space, the energy density and the pressure are

ρ = ½φ̇² + V(φ)
p = ½φ̇² − V(φ)

with V the field’s potential energy density, so that

w = (½φ̇² − V) / (½φ̇² + V)

When V ≥ 0, w lies between −1 (all potential energy, φ̇ = 0) and +1 (all kinetic energy, the stiff fluid of Friedmann cosmology). A field whose kinetic energy is small compared with its potential energy has w close to −1 and behaves almost like a cosmological constant. The combination ρ + p = φ̇² is never negative, which is why such a field never reaches w < −1 while its energy density is positive (the phantom divide and the null energy condition).

The program writes the same structure for its field τ, S_τ = ∫√−g[½(∂τ)² − V(τ)] (#1 §3, transcribed from its LaTeX): a canonical kinetic term plus a potential.

The field equation [T]#

A uniform field obeys

φ̈ + 3Hφ̇ + V′(φ) = 0

This is the equation of an oscillator whose restoring force is −V′ and whose damping coefficient is 3H, set by the expansion. While H is large compared with the field’s natural rate of change, √|V″|, the damping dominates and the field barely moves: it is “frozen” by the expansion, with φ̇ ≈ 0 and w ≈ −1. As H falls, the damping weakens and the field begins to respond to the slope of its potential.

Thawing and freezing [T]#

The two regimes of the field equation divide quintessence models into two classes (Caldwell and Linder †). A thawing field is frozen at early times, with w ≈ −1, and departs from −1 late, once H has fallen far enough, so w rises toward the present. A freezing field is already evolving at early times, with w > −1, and slows down, so w falls toward −1. The lab notebook places the program’s field in the first class: “Hilltop initial conditions give w ≈ −1 in the past, rising now” (#1 §7 item 2). The classes are set out on thawing and freezing quintessence: why a thawing field’s initial velocity drops out, hilltop models, where each class lies in the plane of the two numbers w₀ and wₐ summarized below, the program’s thawing corridor, and how the second data release of the Dark Energy Spectroscopic Instrument bears on the class.

Hilltop and axion-like potentials [T]#

Hilltop quintessence#

In a hilltop model the field begins near a maximum of its potential, where V′ ≈ 0 and V″ < 0 (Dutta and Scherrer †), and a small displacement from the top grows once the expansion damping weakens. Hilltop models are thawing models; how fast w rises depends on the curvature of the maximum compared with the expansion rate (hilltop models).

Axion-like fields#

An axion-like field, or pseudo-Nambu–Goldstone field, is an angle: its couplings are invariant under the shift φ → φ + constant, up to a small explicit breaking that leaves the periodic potential V(φ) = V₀[1 + cos(φ/f)], with period 2πf in φ (Frieman, Hill, Stebbins and Waga †). The shift symmetry keeps the field light, and the maximum at cos(φ/f) = 1 makes it a natural hilltop model. The mechanism, the axion itself, axion-like dark matter and dark energy, and the program’s uses of the idea are on axion-like fields and pseudo-Goldstone bosons.

The program’s choice#

#11 §10: “τ is hilltop quintessence on the Maslov circle, V = V₀[1 + cos(τ/f)], periodicity by topology”. In the program the period is not chosen: τ is an angle on a circle in the space of polarizations (τ as the Maslov phase of the boundary). The program’s version is on τ: hilltop quintessence on the Maslov circle, and among the fits of this class of potential to data is an axion-quintessence analysis of data from the Dark Energy Spectroscopic Instrument (DESI), Ureña-López et al. [20S] (DESI baryon acoustic oscillations).

The w₀–wₐ parametrization [T]#

Surveys summarize a changing w by the two-parameter Chevallier–Polarski–Linder (CPL) form w(a) = w₀ + wₐ(1 − a), with a = 1 today: w₀ is the present value, and a negative wₐ means that w has been rising as the universe expands (Chevallier and Polarski †; Linder †). A thawing field has w₀ > −1 and wₐ < 0, and thawing models lie near a line that the program calls the “thawing corridor wₐ ≈ −1.5(1 + w₀)” (#11 §10); no document gives a reference for it. Where the form comes from, its pivot version, what two numbers cannot represent, and how a CPL fit expresses the phantom crossing reported in DESI’s second data release (DESI DR2’s phantom crossing) are on the CPL parametrization. That page also reports flag 43 in review/wiki-build-flags.md: the corridor itself lies in the region of the w₀–wₐ plane in which a CPL line passes below −1 in the past, the region of the release’s fits, and what separates it from the release’s crossing is the slope, not the region (the corridor and DR2’s crossing).

Recollapse [T]#

If the total energy density of a flat universe reaches zero, the Friedmann equation gives H = 0: the expansion halts. A negative potential energy can bring this about, because the potential term does not dilute while the matter density falls as a⁻³. Since the acceleration is negative at that moment, the universe then contracts. A field that settles into a negative minimum of its potential acts as a negative cosmological constant, an anti-de Sitter phase, and the contraction ends in collapse [T].

#11 §10 cites “high posterior preference for a future anti-de Sitter transition and collapse in negative-minimum quintessence fits [19S, 20S, 21S]”. The lab notebook gives a figure for one of those fits, a “93.8% preference for a future transition to anti-de Sitter space with eventual cosmological collapse” (Gialamas et al., arXiv:2506.21542; #1 §7 item 3). That figure appears only in #1, and no verify/ script or later document checks it.

The program uses recollapse for its far-future turnaround (the turnaround as the conjugate bounce). Whether its own potential can go negative is unresolved: V₀[1 + cos(τ/f)] is never negative when V₀ > 0 (Open: the sign of V(τ) and the turnaround; flag 27 in review/wiki-build-flags.md).

Fits to current data [S]#

#11 §10 summarizes the fits of quintessence models to recent data: “compelling evidence for weakening dark energy with phantom behavior less significant (non-phantom compatible at ~2σ)”. The works it cites, all retrieved by live search:

  • [19][S] I. D. Gialamas, G. Hütsi, M. Raidal, J. Urrutia, M. Vasar, H. Veermäe, “Quintessence and phantoms in light of DESI 2025,” arXiv:2506.21542.
  • [20][S] L. A. Ureña-López et al. (DESI), axion-quintessence analysis of DESI data, arXiv:2503.20178. #11 gives a description, not the paper’s title.
  • [21][S] DESI Collaboration (M. Abdul-Karim et al.), “DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints,” arXiv:2503.14738 (2025).

The earlier documents cite the first two as [T]: #1 §7 tags them [T], and #9 §8 adds “[T — refresh citations before publication]”. The survey itself is described on DESI baryon acoustic oscillations.

Used by#

Sources#

  • papers/broken-born-checkpoint-v2.md §10 — the program’s potential, the thawing corridor, the state of the fits and the negative-minimum fits; §13 row 4 for the standard expectation w ≡ −1
  • papers/broken-born-checkpoint-v2.md refs [19]–[21] — the cited fits and the DESI release (live-searched)
  • notes/broken-born-theory.md §3 — the action S_τ with a canonical kinetic term
  • notes/broken-born-theory.md §7 items 2–3 — hilltop initial conditions and τ’s thawing shape, the fits with a negative minimum and the 93.8% figure (#1 only)
  • notes/broken-born-program-v1.md §8 — the same fits, marked “[T — refresh citations before publication]”
  • review/wiki-build-flags.md item 27 — the potential as written cannot turn negative; item 43 — the corridor in the region of the release’s fits, separated from its crossing by the slope
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