V2.671 - Lambda Phase Transition Invariance — The Goldstone Shield
V2.671: Lambda Phase Transition Invariance — The Goldstone Shield
Experiment
Question: Does the predicted cosmological constant change through SM phase transitions?
Method: Compute R = |δ|/(6α) = Ω_Λ for the SM field content in six cosmic epochs spanning the electroweak transition (T ~ 160 GeV), QCD confinement (T ~ 200 MeV), neutrino decoupling (T ~ 1 MeV), and the present epoch. Compare with standard QFT, which requires separate fine-tuning cancellations at each transition.
Core physics: Three interlocking theorems guarantee Lambda is constant:
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Adler-Bardeen non-renormalization: The trace anomaly coefficients (a, c) receive no radiative corrections. Therefore δ = -4a is topological and transition-invariant.
-
Goldstone equivalence theorem: The Higgs mechanism converts 3 Goldstone scalars into longitudinal W/Z polarizations. The trace anomaly decomposes as a(massive vector) = a(massless vector) + a(scalar), so the total is exactly preserved.
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‘t Hooft anomaly matching: UV and IR anomalies are identical. QCD confinement binds quarks into hadrons but cannot change the trace anomaly — the same dof exist, just rearranged.
Results
1. Exact invariance across all epochs
| Epoch | δ | N_eff | R |
|---|---|---|---|
| Above EW (T > 160 GeV) | -149/12 | 128 | 0.687749 |
| Below EW (1 GeV < T < 160 GeV) | -149/12 | 128 | 0.687749 |
| Above QCD (200 MeV < T < 160 GeV) | -149/12 | 128 | 0.687749 |
| Below QCD (T < 200 MeV) | -149/12 | 128 | 0.687749 |
| Below ν decoupling (T < 1 MeV) | -149/12 | 128 | 0.687749 |
| Present (T = 2.7 K) | -149/12 | 128 | 0.687749 |
R is EXACTLY constant (0σ variation) across all six epochs. Observed: Ω_Λ = 0.6847 ± 0.0073. Prediction: R = 0.6877 (+0.4σ).
2. Goldstone equivalence proof (algebraic)
Above EW: 12 massless vectors + 4 scalars → a_bosonic = 12 × (31/180) + 4 × (1/360) = 187/90
Below EW: 9 massless vectors + 3 massive vectors + 1 scalar = 9 × (31/180) + 3 × (31/180 + 1/360) + 1 × (1/360) = 9 × (31/180) + 3 × (31/180) + 3 × (1/360) + 1 × (1/360) = 12 × (31/180) + 4 × (1/360) = 187/90 ✓
The identity is EXACT in rational arithmetic. Not a numerical coincidence — it’s the Goldstone equivalence theorem expressed through trace anomaly coefficients.
3. Fine-tuning budget eliminated
| Transition | ΔV (standard QFT) | Fine-tuning required |
|---|---|---|
| Electroweak | 1.18 × 10⁸ GeV⁴ | 55 decimal places |
| QCD confinement | 1.19 × 10⁻² GeV⁴ | 45 decimal places |
| Chiral condensate | 1.58 × 10⁻⁴ GeV⁴ | 43 decimal places |
| Zero-point energy | 1.66 × 10⁷⁶ GeV⁴ | 123 decimal places |
Standard QFT requires up to 123 digits of cancellation. This framework requires ZERO.
The Higgs vev contributes (88 GeV)⁴ ≈ 10⁸ GeV⁴ to vacuum energy — a factor 4.2 × 10⁵⁴ above the observed cosmological constant. In this framework, this energy is already encoded in α (the area-law coefficient) and does not separately source Lambda. The trace anomaly δ, which determines Lambda, is immune to the Higgs condensate by the Adler-Bardeen theorem.
4. BSM sensitivity
| Model | R | ΔR | σ from obs |
|---|---|---|---|
| SM (baseline) | 0.6877 | — | +0.4σ |
| +1 real scalar | 0.6830 | -0.0047 | -0.2σ |
| +1 sterile ν (Weyl) | 0.6805 | -0.0072 | -0.6σ |
| +1 dark photon (vector) | 0.7147 | +0.0270 | +4.1σ |
| MSSM | 0.4030 | -0.2847 | -38.6σ |
Key: the transition-invariance prediction holds for BSM as well — any BSM field content gives a constant R, just at a different value. The MEASUREMENT of Ω_Λ then constrains what particles can exist. This is a joint constraint connecting particle physics to cosmology.
5. LISA connection
The Higgs vacuum energy ΔV ≈ 1.18 × 10⁸ GeV⁴ is 0.51% of the radiation energy density at T_EW ≈ 160 GeV. In standard QFT, this must be cancelled to match the tiny observed Λ.
In this framework, Λ is unchanged through the transition. If LISA detects a first-order EWPT (possible in BSM extensions like 2HDM or NMSSM), the gravitational wave spectrum encodes H(T_EW). Any deviation of H from the radiation-dominated prediction could signal Lambda variation — which our framework predicts is exactly zero.
What is unique about this prediction
| Framework | ΔΛ at EW transition | Mechanism |
|---|---|---|
| This framework | 0 exactly | Trace anomaly non-renormalization |
| ΛCDM | 0 (assumed) | Bare Λ is a constant by fiat |
| Quintessence | Generically ≠ 0 | Scalar field potential varies |
| Anthropic/landscape | ~0 (fine-tuned) | Environmental selection |
| Weinberg bound | ~0 (bounded) | Galaxy formation constraint |
Only this framework derives ΔΛ = 0 from first principles. ΛCDM assumes it. The distinction is scientifically meaningful: our prediction comes with a mechanism (three theorems), not just a parameter choice. And it is falsifiable: if evidence of Lambda variation across cosmic epochs were found, the framework is killed.
Falsification criteria
- LISA (2030s): If the EWPT gravitational wave spectrum requires H(T_EW) inconsistent with constant Λ, the framework is falsified.
- BBN consistency: If primordial element abundances require a different Λ at T ~ 1 MeV than observed today, falsified.
- CMB-S4: If large-scale CMB observations require time-varying dark energy (w ≠ -1) at >5σ, falsified.
- New particles: Every new light species shifts R by a calculable amount. A single new vector boson excludes the SM prediction at 4.1σ.
Interpretation
The “cosmological constant problem” is often stated as: why is Λ so small despite receiving contributions from every phase transition in the history of the universe? Standard QFT requires cancellations at 55+ decimal places at the EW scale alone.
This experiment demonstrates that the problem dissolves in the entanglement entropy framework. Lambda is determined by the trace anomaly (δ) and entanglement area-law coefficient (α), both of which are UV properties immune to phase transitions. The Goldstone equivalence theorem ensures that the Higgs mechanism preserves the total anomaly exactly. There is no fine-tuning because there is no cancellation — the vacuum energy simply does not source Lambda.
This is not a loophole or a redefinition. It is a concrete, testable prediction: Lambda was the same at T = 10¹⁶ GeV as it is today, and any experiment that detects Lambda variation across cosmic epochs falsifies the framework.
Files
src/phase_transition_lambda.py— Core computationtests/test_phase_transition.py— 27 tests (all passing)results.json— Full numerical results