The low level of correlation observed in the CMB sky at large angular scales and the low quadrupole variance

Abstract

The angular two-point correlation function of the temperature of the cosmic microwave background (CMB), as inferred from nearly all-sky maps, is very close to zero on large angular scales. A statistic invented to quantify this feature, $S_{1/2}$, has a value sufficiently low that only about 7 in 1000 simulations generated assuming the standard cosmological model have lower values; i.e., it has a $p$-value of 0.007. As such, it is one of several unusual features of the CMB sky on large scales, including the low value of the observed CMB quadrupole, whose importance is unclear: are they multiple and independent clues about physics beyond the cosmological standard model, or an expected consequence of our ability to find signals in Gaussian noise? We find they are not independent: using only simulations with quadrupole values near the observed one, the $S_{1/2}$ $p$-value increases from 0.007 to 0.08. We also find strong evidence that corrections for a "look-elsewhere effect" are large. To do so, we use a one-dimensional generalization of the $S_{1/2}$ statistic, and select along the one dimension for the statistic that is most extreme. Subjecting our simulations to this process increases the $p$-value from 0.007 to 0.03; a result similar to that found in Planck XVI (2016). We argue that this optimization process along the one dimension provides an $underestimate$ of the look-elsewhere effect correction for the historical human process of selecting the $S_{1/2}$ statistic from a very high-dimensional space of alternative statistics $after$ having examined the data.