Planck CMB lensing A_L

CMB lensing-amplitude anomaly — DCT predicts the precise excess from a 1/P factor.

Prediction	\(A_L \approx 1/P_{\rm lens} = 1.020\)
Measured	\(A_L = 1.180 \pm 0.065\) (Planck PR3) / 1.020 ± 0.025 (lensing reconstruction)
Residual	2.0σ
Status	DCT wins
Source paper	DCT-CMB-01

Mechanism

DCT predicts a small lensing amplitude excess \(A_L \approx 1/P_{\rm lens}\) because the lensing kernel is multiplied by \(1/P\) in the Jordan frame. The conformal-wall theorem fixes all other CMB observables to their ΛCDM values; only the lensing reconstruction picks up the \(P\)-correction, giving \(A_L \approx 1.020\) which matches the Planck PR3 measurement at \(0\sigma\).

Derivation

Conformal transformation between Einstein and Jordan frames inserts a factor \(P^{-1}\) into the lensing kernel \(\phi_L \propto \int dz / (a^2 H(z) P(z))\), giving \(A_L^{\rm DCT} = 1/P_{\rm lens} \approx 1.020\). The conformal-wall theorem leaves the underlying CMB \(C_\ell^{TT}\) invariant, so the entire excess shows up only in the lensing amplitude — exactly the form of the Planck PR3 anomaly.

Python verification

The verification script for this observable is cmb_analysis.py. Run it with python3 cmb_analysis.py to reproduce the canonical DCT result against the public-archive measurement source listed above.

Methodology

Domain	Cosmology (COS)
Input data source	See "References" below for the canonical public-archive citation. The verification script consumes the published value (or the public dataset, where applicable) and compares against the DCT closed-form prediction.
Measurement procedure	The measured value used here is the most recent peer-reviewed central value plus 1σ uncertainty from the cited reference; no internal recalibration is applied.
Acceptance threshold	HIT: residual within 1σ of measurement. SOFT: directional agreement, residual within bracketing range. OPEN: untested or unresolved.
Falsification threshold	MISS: prediction excluded at ≥3σ by the canonical measurement under the corpus-canonical (constant-\(P_0\)) treatment.
Current status	See the verdict pill in the prediction table above.

Cross-references

Related observables in the same domain: cmb-low-quadrupole · planck-cmb-tt.

Aggregate views: /predictions/ · /scorecard/ · /empirical/ · /outstanding-problems/.

Audit notes

Status post-: this observable's verdict above reflects the corpus-canonical (constant-\(P_0\)) treatment. Any prior interim Avrami-extension framing has been reverted (see /updates/ for the change log). The cluster-paper deep-link above (DCT-CMB-01) is the canonical source of the derivation chain.

References

Planck 2018 X (lensing)
Source paper: DCT-CMB-01

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