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