Probing Decoupling in Dark Sectors with the Cosmic Microwave Background
  The acoustic peaks in the angular power spectrum of cosmic microwave
background (CMB) temperature and polarization anisotropies play an important
role as a probe of the nature of new relativistic particles contributing to the
radiation density in the early universe, parametrized by $\Delta N_{eff}$. The
amplitude and phase of the acoustic oscillations provide information about
whether the extra species are free-streaming particles, like neutrinos, or
tightly-coupled, like the photons, during eras probed by the CMB. On the other
hand, some extensions of the Standard Model produce new relativistic particles
that decouple from their own non-gravitational interactions after neutrinos,
but prior to photons. We study the signature of new relativistic species that
decouple during this intermediate epoch. We argue that the decoupling species
will cause a scale-dependent change in the amplitude and phase shift of the
acoustic oscillations, different from the usual constant shifts on small
scales. For intermediate decoupling times, the phase and amplitude shifts
depend not only on $\Delta N_{eff}$ but the redshift $z_{dec,X}$ at which the
new species decoupled. For $\Delta N_{eff} >0.334$, a Stage IV CMB experiment
could determine $N_{eff}$ at the percent level and $z_{dec,X}$ at the $\sim
10\%$ level. For smaller values, $\Delta N_{eff}\sim 0.1$, constraints on
$z_{dec,X}$ weaken but remain $\sim 20-50\%$ for $z_{dec,X} \sim
\mathcal{O}(10^3-10^4)$. As an application, we study the contributions to
$\Delta N_{eff}$ and determine the $z_{dec,X}$ values for simple
implementations of the so-called $N$naturalness model.
