The Low Quadrupole in Numbers: Cosmic-Variance Tail Probability of the Observed Quadrupole Power and What Would Change It

The observed quadrupole power of the microwave sky, D2 = 225.9 microK squared in Planck PR3 bookkeeping, sits well below the value the best-fit cosmological model leads one to expect, and the size of that shortfall is often described in words rather than numbers. This paper computes the exact cosmic-variance tail probability. For a central stated theory value of 1150 microK squared, the probability that a Gaussian isotropic sky returns a quadrupole this low or lower is 0.036, about one chance in 27.8; across the published range of theory values the probability runs from 0.0605 to 0.0234. To make the observed value a routine five per cent event, the theory expectation would have to fall to 986.1 microK squared, far outside any plausible revision of the spectrum's large-angle end. The low quadrupole is thus a genuine, quantified tension of the mild kind: unlikely enough to record, likely enough to have happened somewhere among the low multipoles. The study is computed, not measured.