Peaks in the CMBR power spectrum. I. Mathematical analysis of the associated real space features

TL;DR

This paper explores the mathematical origins of peaks in the CMBR power spectrum, linking real space features to oscillations in the spectrum using Fourier analysis and simulations.

Contribution

It provides a mathematical framework connecting real space correlation features to power spectrum oscillations, including the effects of disk distributions on the sky.

Findings

Oscillations in the power spectrum are linked to derivative discontinuities in the correlation function.

Localized real space features can generate the observed spectral peaks.

Smooth distributions of temperature excesses can produce similar spectral features.

Abstract

The purpose of our study is to understand the mathematical origin in real space of modulated and damped sinusoidal peaks observed in cosmic microwave background radiation anisotropies. We use the theory of the Fourier transform to connect localized features of the two-point correlation function in real space to oscillations in the power spectrum. We also illustrate analytically and by means of Monte Carlo simulations the angular correlation function for distributions of filled disks with fixed or variable radii capable of generating oscillations in the power spectrum. While the power spectrum shows repeated information in the form of multiple peaks and oscillations, the angular correlation function offers a more compact presentation that condenses all the information of the multiple peaks into a localized real space feature. We have seen that oscillations in the power spectrum arise when there is a discontinuity in a given derivative of the angular correlation function at a given angular distance. These kinds of discontinuities do not need to be abrupt in an infinitesimal range of angular distances but may also be smooth, and can be generated by simply distributing excesses of antenna temperature in filled disks of fixed or variable radii on the sky, provided that there is a non-null minimum radius and/or the maximum radius is constrained.