V2.682: CMB Acoustic Scale Test — The Most Precise Confrontation

Status: COMPLETE — Framework passes at -0.05σ (shift parameter R)

The Question

The CMB acoustic scale θ* = r_s(z*)/D_M(z*) is measured to 0.03% by Planck — thirty times more precise than Ω_Λ itself. The framework predicts Ω_Λ = 0.6840 (V2.676) with zero free parameters. Combined with CMB-measured ω_m h² = 0.1430 and ω_b h² = 0.02237, this FULLY DETERMINES θ*, the shift parameter R, and the acoustic scale l_A. Does the framework survive this precision gate?

Method

Implemented a Boltzmann-lite cosmology solver:

• Friedmann equation with radiation, matter, and Λ components
• Sound horizon via numerical integration: r_s(z*) = (c/H₀) ∫_{z*}^{∞} c_s(z)/E(z) dz
• Comoving distance: D_M(z*) = (c/H₀) ∫_0^{z*} dz/E(z)
• Recombination redshift from Hu & Sugiyama (1996) fitting formula
• Shift parameter R = √(Ω_m) · H₀ · D_M(z*) / c (independent of sound horizon)
• Compressed likelihood: θ*, R, l_A against Planck 2018 data with correlations

Key inputs (from CMB, NOT predicted):

• ω_m h² = 0.1430 ± 0.0011 (Planck 2018)
• ω_b h² = 0.02237 ± 0.00015 (Planck 2018)
• T_CMB = 2.7255 K, N_eff = 3.044

Framework input (predicted, zero free parameters):

• Ω_Λ = 0.6840 (V2.676, with interaction corrections)

Key Results

1. Shift Parameter R — The Clean Test

The shift parameter R = √(Ω_m) · H₀ · D_M(z*) / c depends only on the geometry of expansion (comoving distance), NOT on sound horizon physics. It has no systematic from our simplified Boltzmann code.

Model	R	σ from Planck
Framework (Ω_Λ = 0.6840)	1.7500	-0.05σ
Planck LCDM (Ω_Λ = 0.6847)	1.7496	-0.13σ
Planck measurement	1.7502 ± 0.0046	—

The framework is closer to the Planck measurement than Planck’s own best-fit LCDM. At -0.05σ, this is essentially perfect agreement.

2. Differential θ* — Framework vs Planck LCDM

Our simplified Boltzmann (without helium recombination corrections) gives a ~0.36% systematic offset in r_s, causing a -12σ absolute offset in θ* for BOTH the framework AND Planck LCDM. The proper test is the DIFFERENTIAL:

Quantity	Framework − Planck LCDM	In sigma units
Δ(θ*)	-2.22 × 10⁻⁶	-0.72σ
Δ(R)	+0.0004	+0.08σ
Δ(l_A)	+0.065	+0.72σ

The 0.10% difference in Ω_Λ (0.6840 vs 0.6847) produces only a 0.02% shift in θ*. The framework is fully consistent with Planck LCDM at the sub-sigma level.

3. Derived H₀

Model	H₀ (km/s/Mpc)
Framework	67.27
Planck LCDM	67.35
Planck measured	67.36 ± 0.54

The framework predicts H₀ = 67.27 km/s/Mpc, only 0.09 km/s/Mpc below Planck’s value (0.17σ). This is on the “CMB side” of the Hubble tension — the framework does NOT resolve the tension with SH0ES (H₀ ≈ 73).

4. Sensitivity Analysis

A 1σ change in Ω_Λ (±0.0073) produces a 7.5σ change in θ*. This amplification factor means θ* is a FAR more powerful discriminator than Ω_Λ itself. The framework’s prediction Ω_Λ = 0.6840 is within the θ*-allowed band.

5. Ω_Λ Scan

From the θ* vs Ω_Λ scan (using shift parameter R, which has no systematic):

Ω_Λ	σ(R) from Planck
0.6700	+1.55σ
0.6840	-0.05σ
0.6847	-0.13σ
0.7000	-1.96σ

The framework sits at the sweet spot of the shift parameter constraint. The 2σ-allowed range from R alone is approximately Ω_Λ ∈ [0.67, 0.70].

Honest Assessment

What This Shows

1. The framework’s zero-parameter Ω_Λ = 0.6840 is fully consistent with Planck’s CMB acoustic scale measurements.
2. The shift parameter R (no sound horizon systematic) is at -0.05σ — essentially perfect.
3. The differential θ* test gives -0.72σ — well within measurement uncertainty.
4. The derived H₀ = 67.27 km/s/Mpc agrees with Planck at 0.17σ.

What This Does NOT Show

1. The θ test is not very constraining for Ω_Λ.* A 1σ change in θ* maps to a ~0.13% change in Ω_Λ. The Ω_Λ constraint from θ* alone (2σ band ≈ 0.67–0.70) is wider than the direct Planck measurement. θ* is precise but not very sensitive to Ω_Λ — it constrains the whole expansion history, not just dark energy.
2. Our simplified Boltzmann has a 0.36% systematic in r_s. This is due to missing helium recombination physics. The differential comparison eliminates this, but it means we cannot claim sub-percent absolute precision on θ*.
3. The shift parameter R is essentially the same test as Ω_Λ itself. R = √(Ω_m) · D_M/D_H, so it’s a geometric repackaging of the distance-redshift relation. The -0.05σ result on R is CONSISTENT with the Ω_Λ = 0.6840 prediction being within Planck’s error bars, but doesn’t add much independent information.
4. This does not address the Hubble tension. The framework predicts H₀ = 67.27, firmly on the Planck side. It does not resolve the discrepancy with local H₀ ≈ 73.

What Would Strengthen the Argument

• Full Boltzmann code (CLASS/CAMB): Eliminate the 0.36% r_s systematic and confront the full CMB power spectrum, not just compressed likelihood.
• BAO data (DESI): r_d/D_V(z) at multiple redshifts provides independent distance-ladder tests.
• CMB power spectrum shape: The framework predicts w = -1 exactly (V2.676), which affects the ISW effect and CMB lensing. These provide independent tests beyond θ*.

The Bottom Line

The framework passes the CMB acoustic scale test. The shift parameter R deviates by only -0.05σ from Planck — the framework’s zero-parameter prediction is closer to the measurement than Planck’s own 6-parameter fit. However, this test is not as constraining as the 0.03% precision of θ* might suggest: the sensitivity of θ* to Ω_Λ is moderate (~7.5× amplification), and the 2σ allowed band (Ω_Λ ∈ 0.67–0.70) accommodates both the framework and LCDM comfortably. The real discriminating power will come from BAO data at multiple redshifts and the full CMB power spectrum shape.