Angular Power Spectra with Finite Counts

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.

27 pages, 8 figures

References in corpus (16)

• Cosmological Constraints from the SDSS Luminous Red Galaxies
• Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis
• Use and Abuse of the Fisher Information Matrix in the Assessment of Gravitational-Wave Parameter-Estimation Prospects
• Camparison of the Hanbury Brown-Twiss effect for bosons and fermions
• Indications of Intermediate-Scale Anisotropy of Cosmic Rays with Energy Greater Than 57 EeV in the Northern Sky Measured with the Surface Detector of the Telescope Array Experiment
• Dark matter annihilation or unresolved astrophysical sources? Anisotropy probe of the origin of cosmic gamma-ray background
• Joint anisotropy and source count constraints on the contribution of blazars to the diffuse gamma-ray background
• Observation of Small-scale Anisotropy in the Arrival Direction Distribution of TeV Cosmic Rays with HAWC
• Diffuse gamma-ray emission from galactic pulsars
• Searches for small-scale anisotropies from neutrino point sources with three years of IceCube data
• Multipole analysis of IceCube data to search for dark matter accumulated in the Galactic halo
• Magnetic Deflections of Ultra-High Energy Cosmic Rays from Centaurus A
• Fermi-LAT gamma-ray anisotropy and intensity explained by unresolved Radio-Loud Active Galactic Nuclei
• Effects of Velocity-Dependent Dark Matter Annihilation on the Energy Spectrum of the Extragalactic Gamma-ray Background
• A comparison of CMB Angular Power Spectrum Estimators at Large Scales: the TT case
• Large-scale distribution of cosmic rays in right ascension as observed by the Yakutsk array at energies above $10^{18}$ eV

Cited by in corpus (1)

• Sensitivity of full-sky experiments to large scale cosmic ray anisotropies