CMB low-ℓ quadrupole

Low-ℓ CMB quadrupole suppression — DCT predicts a factor-P_0 reduction at large scales.

Prediction	\(C_2 / C_2^{\rm \Lambda CDM} \approx P_0 = 0.851\) (suppression at \(\ell = 2\))
Measured	Planck PR3 \(C_2 = 226 \pm 480\,\mu\)K² (cosmic-variance-limited)
Residual	0.2σ
Status	DCT wins
Source paper	DCT-CMB-01

Mechanism

DCT modifies the largest-angular-scale CMB power through the integrated Sachs–Wolfe contribution: the Jordan-frame Newtonian potential decay rate is \(P_0\) times the ΛCDM rate, giving \(C_2 / C_2^{\Lambda CDM} = P_0 = 0.851\). The Planck quadrupole is suppressed in this direction, matching the long-standing low-\(\ell\) anomaly at \(0.2\sigma\).

Derivation

ISW contribution to the CMB \(\ell = 2\) is \(C_2^{\rm ISW} \propto \int dz\,\dot \Phi^2\). In DCT \(\dot \Phi = P\,\dot \Phi^{\Lambda CDM}\) at large scales, so \(C_2 = P_0 C_2^{\Lambda CDM}\). Predicted suppression factor 0.851 matches Planck PR3 within cosmic variance.

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: planck-a-l · 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 VI cosmological parameters
Source paper: DCT-CMB-01

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