Cosmic Microwave Background Acoustic Peak Locations
  The Planck collaboration has measured the temperature and polarization of the
cosmic microwave background well enough to determine the locations of eight
peaks in the temperature (TT) power spectrum, five peaks in the polarization
(EE) power spectrum and twelve extrema in the cross (TE) power spectrum. The
relative locations of these extrema give a striking, and beautiful,
demonstration of what we expect from acoustic oscillations in the plasma; e.g.,
that EE peaks fall half way between TT peaks. We expect this because the
temperature map is predominantly sourced by temperature variations in the last
scattering surface, while the polarization map is predominantly sourced by
gradients in the velocity field, and the harmonic oscillations have temperature
and velocity 90 degrees out of phase. However, there are large differences in
expectations for extrema locations from simple analytic models vs. numerical
calculations. Here we quantitatively explore the origin of these differences in
gravitational potential transients, neutrino free-streaming, the breakdown of
tight coupling, the shape of the primordial power spectrum, details of the
geometric projection from three to two dimensions, and the thickness of the
last scattering surface. We also compare the peak locations determined from
Planck measurements to expectations under the $\Lambda$CDM model. Taking into
account how the peak locations were determined, we find them to be in
agreement.
