V2.636 - Growth Rate f·σ₈(z) — Zero-Parameter Prediction vs DESI DR1
V2.636: Growth Rate f·σ₈(z) — Zero-Parameter Prediction vs DESI DR1
Question
The framework predicts Ω_Λ = 0.6877 with w = -1 exactly, fully determining the expansion history and linear growth of structure. Does this zero-parameter prediction survive confrontation with DESI DR1 f·σ₈ measurements? Does the framework resolve the S₈ tension between CMB and weak lensing surveys?
Method
- Solve the linear growth ODE for D(z) with Ω_m = 0.3123 (framework) and 0.3153 (Planck)
- Compute f(z) = d ln D / d ln a and f·σ₈(z) = f(z)·σ₈(0)·D(z)
- Compare against 7 DESI DR1 redshift bins (z = 0.3 to 2.3)
- Diagnose S₈ tension with KiDS-1000, DES-Y3, HSC-Y3
- Assess DESI w₀-wₐ evolving dark energy preference
Using σ₈(0) = 0.8120 from Planck 2018 (framework predicts same CMB physics). RK45 ODE integration from z = 10 with matter-domination initial conditions.
Results
1. f·σ₈(z) vs DESI DR1: excellent zero-parameter fit
| z | DESI f·σ₈ | ± | Framework | Pull (σ) | Planck | Pull (σ) |
|---|---|---|---|---|---|---|
| 0.295 | 0.408 | 0.040 | 0.4725 | +1.6 | 0.4735 | +1.6 |
| 0.510 | 0.436 | 0.027 | 0.4740 | +1.4 | 0.4743 | +1.4 |
| 0.706 | 0.424 | 0.026 | 0.4617 | +1.4 | 0.4617 | +1.4 |
| 0.930 | 0.444 | 0.038 | 0.4397 | −0.1 | 0.4393 | −0.1 |
| 1.317 | 0.404 | 0.032 | 0.3956 | −0.3 | 0.3949 | −0.3 |
| 1.491 | 0.395 | 0.044 | 0.3761 | −0.4 | 0.3754 | −0.4 |
| 2.330 | 0.346 | 0.088 | 0.2973 | −0.6 | 0.2965 | −0.6 |
χ²/dof = 7.25/7 = 1.04 (p = 0.40) — zero free parameters, textbook fit.
2. Framework ≈ Planck ΛCDM for growth rate
The maximum difference between framework and Planck f·σ₈ predictions is 0.43% across all redshifts. The Δ(Ω_Λ) = 0.003 difference is undetectable in current growth rate measurements. DESI DR2 (expected ~2× smaller errors) will not distinguish them either.
3. S₈ tension: honest null result
| Survey | S₈ observed | Framework tension | Planck tension |
|---|---|---|---|
| KiDS-1000 | 0.766±0.020 | +3.1σ | +3.3σ |
| DES-Y3 | 0.776±0.017 | +3.1σ | +3.3σ |
| HSC-Y3 | 0.769±0.034 | +1.7σ | +1.8σ |
Framework S₈ = 0.829, Planck S₈ = 0.832.
The framework does NOT resolve the S₈ tension. Its S₈ is marginally lower than Planck (by 0.4%) due to lower Ω_m, but this is negligible compared to the ~8% discrepancy with weak lensing. The S₈ tension, if real, requires non-linear or baryonic physics — not a modified Ω_Λ.
4. Growth index γ confirms GR
| z | f(z) | Ω_m(z) | γ_eff |
|---|---|---|---|
| 0.0 | 0.525 | 0.312 | 0.554 |
| 0.5 | 0.759 | 0.605 | 0.549 |
| 1.0 | 0.875 | 0.784 | 0.547 |
| 2.0 | 0.958 | 0.925 | 0.546 |
GR prediction: γ = 6/11 ≈ 0.5455. Framework gives γ(z=0) = 0.554, within 1.6% of the GR value. This is expected — the framework produces a cosmological constant, not modified gravity.
5. DESI w₀-wₐ: the sharpest threat
DESI DR1 (2024): w₀ = −0.55 ± 0.21, wₐ = −1.32 ± 0.68 (2.5–3.9σ preference for evolving dark energy depending on data combination)
The framework requires w = −1 exactly. The trace anomaly δ is topological (protected by the Gauss-Bonnet theorem), so dark energy from entanglement entropy cannot evolve. Current tension: 2.1σ on w₀ alone.
Falsification threshold: if DESI DR2 + Euclid confirm w ≠ −1 at 5σ, the framework is falsified.
Key Findings
-
χ² = 7.25/7 (p = 0.40): The framework’s zero-parameter f·σ₈(z) prediction is an excellent fit to DESI DR1, with no individual bin beyond 1.6σ.
-
S₈ tension is NOT resolved — framework S₈ = 0.829, still 3σ above weak lensing. Honest null result.
-
w = −1 is the framework’s most falsifiable prediction — DESI’s evolving DE preference at 2.5–3.9σ is the sharpest near-term threat. DESI DR2 (2025–2026) will be decisive.
-
Framework and Planck ΛCDM are indistinguishable for growth rate (max 0.43% difference). The framework’s unique content is the derivation of Ω_Λ = 0.6877, not a different phenomenology.
Significance
This experiment establishes the framework’s compatibility with large-scale structure growth data. The key result is negative in the most interesting sense:
- The framework passes the f·σ₈ test (χ² = 1.04 per dof)
- It fails to resolve S₈ tension (same as standard ΛCDM)
- Its sharpest threat is the DESI w₀-wₐ preference, not growth rate
The framework makes exactly one unique large-scale-structure prediction: Ω_Λ = 0.6877, w = −1. This is consistent with all current data but faces a clear falsification path via DESI DR2 + Euclid.
Technical Notes
- Growth ODE solved via RK45, rtol = 10⁻¹⁰, atol = 10⁻¹²
- Initial conditions at z = 10 (matter domination: D = a, f = 1)
- D(z=0) = 1 normalisation
- σ₈(0) = 0.8120 (Planck 2018 TT,TE,EE+lowE+lensing)
- DESI DR1 data from arXiv:2404.03002 (Table 3)
- S₈ data: KiDS-1000 (arXiv:2007.15633), DES-Y3 (arXiv:2105.13549), HSC-Y3 (arXiv:2304.00701)