V2.141: Effective Dark Energy Equation of State from Horizon Thermodynamics

Status: COMPLETE

Question

V2.138 found 3.3-4.2 sigma DESI tension against w = -1. The framework predicts Lambda is constant. But the Clausius relation is applied at the apparent horizon, which CHANGES in LCDM (the universe is not exactly de Sitter). Does the time-evolving horizon produce an effective w(z) that deviates from -1? And is Lambda stable through the electroweak phase transition?

Method

Two analyses:

A) Non-equilibrium correction: Apply the Clausius relation at the evolving LCDM apparent horizon r_A = c/H(z). As the horizon area A(z) = 4pi/E^2(z) changes with redshift, the log correction S_log = delta * ln(A) also changes. Does this source an effective w != -1?

B) Phase transition stability: At the electroweak transition (~100 GeV), the field content changes: 4 massless gauge bosons + 4 scalars -> 1 massless + 3 massive vectors + 1 scalar (Goldstone bosons eaten). Does this change R = |delta|/(6*alpha)?

Results

Phase 1: LCDM Background

The apparent horizon shrinks dramatically at higher redshift:

z	H(z) km/s/Mpc	q(z)	r_A (Hubble units)
0	67.36	-0.527	1.000
1	120.66	+0.180	0.558
2	204.37	+0.389	0.330
3	307.85	+0.451	0.219

The transition from deceleration (q > 0) to acceleration (q < 0) occurs at z ~ 0.6.

Phase 2: Naive w(z) — the WRONG Answer

If one naively applies the self-consistency condition at each redshift, treating Lambda(z) ∝ E^2(z), the effective equation of state is:

z	w_naive
0	-0.685
0.5	-0.391
1.0	-0.213
2.0	-0.074

This gives huge deviations from w = -1. But this is WRONG. The self-consistency condition determines Lambda at the de Sitter ENDPOINT (z -> -1), not at each intermediate redshift. At finite z, the Clausius relation generates the full Friedmann equation with Lambda as a CONSTANT.

Phase 3: Correct w(z) — Non-equilibrium Correction

The correct treatment fixes Lambda at the de Sitter equilibrium. Non-equilibrium corrections from entropy production are second-order in deviations from de Sitter and suppressed by l_P^2/r_A^2 ~ 10^{-122}:

| z | w(z) | |w + 1| | |---|------|---------| | 0 | -1.000… | 7.8 x 10^{-124} | | 0.5 | -1.000… | 5.1 x 10^{-123} | | 1.0 | -1.000… | 1.6 x 10^{-122} | | 2.0 | -1.000… | 6.2 x 10^{-122} |

Maximum |w + 1| = 1.5 x 10^{-121}, utterly undetectable.

The suppression has a clear physical origin:

• Non-equilibrium entropy production is quadratic in deviations: ~(H_dot/H^2)^2
• The ratio of log to area corrections provides the factor: |delta|/(alpha * A/l_P^2) ~ 10^{-122}
• Combined: |w + 1| ~ Omega_m^2 * delta/(alpha * 10^{122}) ~ 10^{-122}

Phase 4: DESI Comparison

The framework predicts w = -1 exactly (to 10^{-120}). Against DESI:

Dataset	w0	wa	Tension
DR2 + PantheonPlus	-0.752	-1.01	4.2 sigma
DR2 + DESY5	-0.775	-0.84	3.3 sigma
DR1 + PantheonPlus	-0.827	-0.75	2.0 sigma

Phase 5: Electroweak Phase Transition Stability

Symmetric phase (T >> 100 GeV): 12 massless vectors + 4 scalars + 45 Weyls

• alpha_eff = 118, delta = -1991/180

Broken phase (T << 100 GeV): 9 massless + 3 massive vectors + 1 scalar + 45 Weyls

• alpha_eff = 118, delta = -1991/180

Both alpha and delta are IDENTICAL. The Goldstone equivalence theorem guarantees this: the 3 eaten Goldstone bosons become the longitudinal polarizations of W+, W-, Z. A massive Proca field = massless vector + scalar in terms of both alpha (heat kernel tr(1)) and delta (trace anomaly). So R = |delta|/(6*alpha) is exactly preserved.

Lambda is EXACTLY constant through the electroweak transition.

Key Findings

1. w = -1 to ~10^{-120} precision. Non-equilibrium corrections from the evolving LCDM horizon are suppressed by l_P^2/r_A^2 ~ 10^{-122}. The framework has NO mechanism to produce w != -1 at any detectable level.

2. Lambda is exactly stable through EWSB. The Goldstone equivalence theorem ensures alpha and delta are identical in symmetric and broken phases. R is exactly preserved.

3. Three independent upper bounds on |w + 1| from the framework:

   • Mass corrections: |w+1| < 10^{-32} (V2.130)
   • Non-equilibrium: |w+1| < 10^{-121} (this experiment)
   • Phase transitions: |w+1| = 0 exactly (this experiment)

4. DESI tension: up to 4.2 sigma. This is the framework’s most serious current challenge. If DESI confirms w != -1 at > 5 sigma, the framework is FALSIFIED. This is a genuine, sharp, falsifiable prediction.

Implications

The framework makes an extraordinarily rigid prediction: w = -1 with no room for deviation. There are no free parameters that can be adjusted, no higher-order corrections that can help, and no phase transition effects that could produce time-dependent dark energy.

This rigidity is both a strength (sharp falsifiability) and a vulnerability (no escape route if DESI w != -1 is confirmed). The framework essentially predicts that DESI’s evidence for dynamical dark energy will not survive to 5 sigma as more data accumulates.

Files

• run_experiment.py: 6-phase experiment driver
• src/horizon_thermodynamics.py: LCDM cosmology, horizon entropy, w(z) computations
• src/phase_transition.py: EW phase transition stability analysis
• tests/test_horizon_thermo.py: 17 tests (all pass)
• results/results.json: Full numerical data

References

• Cai & Kim (2005): First law at apparent horizon
• Akbar & Cai (2006): Friedmann equation from first law
• Jacobson (1995, 2016): Thermodynamic gravity
• DESI DR2 (2025): Evidence for dynamical dark energy