Angular Power Spectra with Finite Counts

TL;DR

This paper investigates the variance in angular power spectra estimates from cosmic radiation maps, revealing significant higher-order effects that can impact the interpretation of anisotropy measurements.

Contribution

It derives an exact variance formula including higher-order effects and demonstrates their importance through simulations, improving the accuracy of anisotropy analyses.

Findings

Higher-order effects can increase variance estimates by orders of magnitude.

Neglecting these effects may lead to false detections of anisotropy.

The formalism applies to gamma-ray, neutrino, and cosmic ray data.

Abstract

Angular anisotropy techniques for cosmic diffuse radiation maps are powerful probes, even for quite small data sets. A popular observable is the angular power spectrum; we present a detailed study applicable to any unbinned source skymap S(n) from which N random, independent events are observed. Its exact variance, which is due to the finite statistics, depends only on S(n) and N; we also derive an unbiased estimator of the variance from the data. First-order effects agree with previous analytic estimates. Importantly, heretofore unidentified higher-order effects are found to contribute to the variance and may cause the uncertainty to be significantly larger than previous analytic estimates---potentially orders of magnitude larger. Neglect of these higher-order terms, when significant, may result in a spurious detection of the power spectrum. On the other hand, this would indicate the presence of higher-order spatial correlations, such as a large bispectrum, providing new clues about the sources. Numerical simulations are shown to support these conclusions. Applying the formalism to an ensemble of Gaussian-distributed skymaps, the noise-dominated part of the power spectrum uncertainty is significantly increased at high multipoles by the new, higher-order effects. This work is important for harmonic analyses of the distributions of diffuse high-energy gamma-rays, neutrinos, and charged cosmic rays, as well as for populations of sparse point sources such as active galactic nuclei.