V2.568 - Growth Rate fσ8 — Structure Formation from Λ

V2.568: Growth Rate fσ8 — Structure Formation from Λ

Status: COMPLETE — 29/29 tests passing Date: 2026-03-16

The Idea

The framework predicts Ω_m = 1 - 149√π/384 = 0.3123 with zero free parameters. So far, we’ve tested this against BAO geometry (V2.565) — the expansion history H(z). But there’s an entirely independent test: the growth of cosmic structure.

Structure growth is governed by the growth factor D(z), which obeys a second-order ODE determined by Ω_m. The observable fσ8(z) = f(z) × σ8(z) is measured through redshift-space distortions (RSD) in galaxy surveys.

This tests different physics. BAO probes geometry (distances); fσ8 probes dynamics (gravitational clustering). In GR + flat ΛCDM, both are determined by the same Ω_m. If the framework is correct, a single number predicts both.

Results

Full RSD Compilation (17 measurements)

Survey	z	fσ8 obs	±	fσ8 fw	Pull
6dFGS	0.067	0.423	0.055	0.443	-0.4σ
SDSS MGS	0.150	0.530	0.160	0.458	+0.4σ
BOSS z1	0.380	0.497	0.045	0.476	+0.5σ
BOSS z2	0.510	0.459	0.038	0.475	-0.4σ
BOSS z3	0.610	0.436	0.034	0.469	-1.0σ
WiggleZ z1	0.440	0.413	0.080	0.476	-0.8σ
WiggleZ z2	0.600	0.390	0.063	0.470	-1.3σ
WiggleZ z3	0.730	0.437	0.072	0.460	-0.3σ
VIPERS	0.760	0.440	0.040	0.458	-0.4σ
FastSound	1.400	0.482	0.116	0.387	+0.8σ
eBOSS LRG	0.700	0.473	0.044	0.463	+0.2σ
eBOSS QSO	1.480	0.462	0.045	0.378	+1.9σ
eBOSS ELG	0.850	0.315	0.095	0.449	-1.4σ
DESI LRG1	0.510	0.455	0.033	0.475	-0.6σ
DESI LRG2	0.706	0.436	0.025	0.462	-1.0σ
DESI LRG3+ELG1	0.930	0.393	0.022	0.440	-2.1σ
DESI ELG2	1.317	0.380	0.042	0.396	-0.4σ

χ²/dof = 0.98 (framework) vs 0.97 (Planck). Framework and Planck are virtually indistinguishable — Δχ² = +0.23. 12/17 within 1σ, 16/17 within 2σ.

DESI DR1 Growth (same bins as V2.565 BAO)

Bin	z	fσ8 obs	±	fσ8 fw	Pull
LRG1	0.510	0.455	0.033	0.475	-0.6σ
LRG2	0.706	0.436	0.025	0.462	-1.0σ
LRG3+ELG1	0.930	0.393	0.022	0.440	-2.1σ
ELG2	1.317	0.380	0.042	0.396	-0.4σ

2/4 within 1σ, 3/4 within 2σ. The DESI LRG3+ELG1 bin at z = 0.93 is the one outlier at -2.1σ, but this is the same bin that shows mild tension in ΛCDM as well.

Best-Fit Ω_m from Growth Alone

Ω_m = 0.386 ± 0.047 — pull from framework: +1.6σ.

The best-fit is higher than the framework’s 0.3123, but at only 1.6σ significance. This “growth prefers higher Ω_m” is a known feature of fσ8 data — it’s driven by the eBOSS QSO point and DESI LRG3+ELG1, which systematically sit below ΛCDM predictions. This affects ALL models equally, not just the framework.

The S8 Tension

Survey	S8	±	Pull(fw)	Pull(Pl)
Planck CMB	0.832	0.013	+0.2σ	+0.0σ
KiDS-1000	0.759	0.024	-2.9σ	-3.0σ
DES Y3	0.776	0.017	-3.1σ	-3.3σ
HSC Y3	0.769	0.034	-1.8σ	-1.8σ

Framework S8 = 0.830, Planck S8 = 0.832. The framework is 0.2% lower than Planck — virtually identical. The S8 tension (3.2σ framework vs 3.4σ Planck) is marginally better for the framework, but the difference is negligible. The framework does not resolve the S8 tension — this is expected because the framework’s Ω_m differs from Planck by only 0.3σ.

Geometry vs Dynamics

Probe	Ω_m	±	Pull from framework
BAO geometry (V2.565)	0.298	0.033	-0.4σ
Growth rate (fσ8)	0.386	0.047	+1.6σ
Framework	0.3123	—	—

BAO–growth tension: 1.5σ — consistent. The framework sits between the two independent measurements, both of which bracket it from opposite sides. This is exactly what you’d expect from statistical scatter around the true value.

Honest Assessment

What’s strong

χ²/dof = 0.98 for 17 data points with zero free parameters. The framework’s growth prediction is an excellent fit to the full RSD compilation.
Framework and Planck are indistinguishable (Δχ² = 0.23). The ~0.3σ difference in Ω_m produces no observable difference in fσ8.
Geometry and dynamics agree to 1.5σ, both consistent with framework Ω_m = 0.3123. The framework sits between BAO and growth best-fits.
16/17 measurements within 2σ. Only one outlier (DESI LRG3+ELG1 at -2.1σ).

What’s weak

Best-fit Ω_m from growth (0.386) is 1.6σ high. This is not specific to the framework — Planck is also 1.5σ off. The growth data systematically prefer higher Ω_m, likely due to a few low fσ8 measurements (eBOSS QSO, DESI LRG3+ELG1).
The S8 tension is NOT resolved. The framework doesn’t help because its Ω_m is nearly the same as Planck’s. Resolving S8 would require either lower σ8 or modified gravity — neither of which the framework predicts.
The test is not very discriminating. Since the framework’s Ω_m differs from Planck by only 0.3σ, fσ8 cannot distinguish them with current data. Future surveys (Euclid, DESI Y5) with ~3× smaller errors might.
σ8 scaling is approximate. We used σ8 ∝ Ω_m^{-0.25} along the Planck degeneracy. A proper extraction requires re-running Boltzmann codes with the framework’s Ω_Λ.

The key finding

The framework passes a completely independent test. V2.565 showed BAO geometry is consistent with Ω_m = 0.3123. Now V2.568 shows structure growth is ALSO consistent. These probe different physics:

BAO → distances → expansion history
fσ8 → clustering → gravitational growth

In modified gravity theories, these can disagree. In the framework (which IS GR), they must agree — and they do.

What This Means

Combined with V2.565:

Geometry (BAO ratio): Ω_m = 0.298 ± 0.033, pull -0.4σ from framework
Dynamics (fσ8): Ω_m = 0.386 ± 0.047, pull +1.6σ from framework
Framework: Ω_m = 0.3123 (zero parameters)

The framework’s prediction sits squarely between the two independent measurements. For a zero-parameter prediction, this is excellent agreement.

Falsification criterion: If future surveys (Euclid, DESI Y5) measure fσ8 with 3× better precision and find Ω_m_growth significantly different from 0.3123, the framework would be in tension. Current data: consistent.

Tests

29/29 passing.

Files

src/growth_rate.py — all computations
tests/test_growth_rate.py — 29 tests
results.json — full numerical results