Cosmology

ModelledB-mode factor [M]. The −5.56% figure is not a match to the 77.78% deficit. Full C_ℓ stays open.

CMB Quadrupole

Planck’s placed quadrupole is 225.895 μK² against a theory column 1016.730 μK². The S³ factor already computed suppresses ℓ=2 by 5.56% and covers 7.14% of that deficit. It does not resolve the anomaly.

Last updated: 19 October 2026

Reading this entry

The CMB quadrupole page sets Planck's placed pair 225.895 μK² against a theory column 1016.730 μK², a 77.78 percent deficit, beside the S³ B-mode factor. That factor is f(ℓ) = 1 − 1/(2(ℓ+1)²), equal to 17/18 at ℓ = 2, and it covers 7.14 percent of the deficit.

What this is used for

Low-multipole discussions use this pair and this factor when they quote the quadrupole book. The same f(ℓ) prints 5.56 percent at ℓ = 2, 3.13 percent at ℓ = 3, and 2.00 percent at ℓ = 4, so later spectrum work has the low-ℓ suppressions already evaluated.

Walk through

The CL-3 pair is D₂^obs = 225.895 μK² against D₂^th = 1016.730 μK², a 77.78 percent deficit, and a round print of 240 against 1100 is the same comparison. The B-mode factor is f(ℓ) = 1 − 1/(2(ℓ+1)²), so f(2) = 17/18 = 1 − 5.5556 percent, covering 7.14 percent of that deficit. The table gives f(2) = 0.9444 (5.56 percent), f(3) = 0.9688 (3.13 percent), and f(4) = 0.9800 (2.00 percent). The great-circle circumference 2πR₀ is recorded beside that table as a length of the round metric on this page.

Integrity note: The formula that writes

uses the wrong denominator for that book. The B-mode factor is

, which is 17/18 at

1. The placed deficit

The CL-3 pair is against . The deficit is 77.78%. A round print of 240 against 1100 is the same comparison.

B-mode factor. At ℓ=2 it covers 7.14% of the CL-3 deficit.

The great-circle circumference is a length of the round metric. It does not evaluate . No angle between a Clifford neck and the ecliptic is written. Row 3.9 stays NIL.

2. Low-ℓ factor

ℓ	f(ℓ)	Suppression
2	0.9444	5.56%
3	0.9688	3.13%
4	0.9800	2.00%

These factors sit below the cosmic-variance width at ℓ=2. They are not a fit to the Planck map.

STATUS: DEFICIT UNPAID

The 5.56% factor is algebra of the B-mode book. The 77.78% deficit stays. The spectrum stays open.

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