V2.473: CMB Low-ℓ ISW Signature from Fixed Ω_Λ

Objective

Test whether the framework’s fixed Ω_Λ = 0.6877 creates a distinguishable signature in the CMB low-multipole power spectrum (ℓ = 2–30) through the late-time ISW effect.

Method

1. Compute the linear growth factor D(z) and growth rate f(z) for framework vs Planck cosmology
2. Estimate the ISW contribution to C_ℓ at ℓ = 2–30 via Limber approximation
3. Compare total D_ℓ predictions to Planck 2018 Commander data
4. Compute the S_{1/2} statistic for large-angle correlations
5. Forecast ISW × galaxy cross-correlation precision for Euclid and LSST

Key Results

Framework vs Planck: Indistinguishable

Cosmology	χ²(ℓ=2–30)	χ²/N_ℓ	Ω_Λ
Framework	17.69	0.610	0.6877
Planck	17.69	0.610	0.6847

Δχ² ≈ 0 — the 0.44% difference in Ω_Λ is completely invisible in the CMB power spectrum.

Low-ℓ anomalies

• Low quadrupole: D₂ = 215 μK² observed vs ~1164 μK² predicted (82% deficit). Both framework and Planck predict the same value — this anomaly is unrelated to Ω_Λ.
• S_{1/2}: 8787 μK⁴ observed vs ~44000 μK⁴ predicted (80% deficit). Again shared by all ΛCDM-like models.
• Max pull: ℓ = 7 at 2.5σ (D₇ = 2286 vs predicted 1114).

ISW forecast

Experiment	σ(Ω_Λ)	Framework vs Planck
Current (Planck × NVSS)	0.034	0.1σ
Euclid × CMB-S4 (2030)	0.014	0.2σ
LSST × CMB-S4 (2032)	0.007	0.4σ

Even the most ambitious future ISW measurements cannot distinguish the framework from Planck at better than 0.4σ.

Assessment

The CMB low-ℓ spectrum is NOT a useful test of the framework. The Ω_Λ difference (0.0030) is too small to create a detectable ISW signature, and the known low-ℓ anomalies (low quadrupole, S_{1/2}) are unrelated to the value of Ω_Λ — they require physics beyond the background expansion (topology, pre-inflationary relics, or statistical flukes).

Verdict

The framework PASSES the CMB consistency test but gains no discriminating power. The decisive tests remain:

• BAO bin-by-bin (V2.472): χ²/N = 1.46, LRG tension at 2.8σ
• Standard sirens (V2.468): 3σ by 2029, 5σ by 2035
• BSM species counting (V2.446): any new field shifts Ω_Λ by measurable amounts

This is an honest null result: the CMB cannot close the gap.