  Fluctuations in the cosmic neutrino background are known to produce a phase
shift in the acoustic peaks of the cosmic microwave background. It is through
the sensitivity to this effect that the recent CMB data has provided a robust
detection of free-streaming neutrinos. In this paper, we revisit the phase
shift of the CMB anisotropy spectrum as a probe of new physics. The phase shift
is particularly interesting because its physical origin is strongly constrained
by the analytic properties of the Green's function of the gravitational
potential. For adiabatic fluctuations, a phase shift requires modes that
propagate faster than the speed of fluctuations in the photon-baryon plasma.
This possibility is realized by free-streaming relativistic particles, such as
neutrinos or other forms of dark radiation. Alternatively, a phase shift can
arise from isocurvature fluctuations. We present simple models to illustrate
each of these effects. We then provide observational constraints from the
Planck temperature and polarization data on additional forms of radiation. We
also forecast the capabilities of future CMB Stage IV experiments. Whenever
possible, we give analytic interpretations of our results.
