Cosmic Neutrino Flavor Ratios with Broken $\nu_\mu-\nu_\tau$ Symmetry
  Reactor neutrino experiments have now observed a nonzero value for
$\theta_{13}$ at $5\sigma$, and global fits to data imply a nonzero value above
$10\sigma$. Nonzero values for $\theta_{13}$ and/or $\theta_{32}-\pi/4$ break a
$\nu_\mu-\nu_\tau$ symmetry, which has qualitative as well as quantitative
implications for the time-evolution of neutrino flavors. In particular, the
large-distance flavor evolution matrix, non-invertible with $\nu_\mu-\nu_\tau$
symmetry, is now invertible. This means that measurements of neutrino flavor
ratios at Earth can now be inverted to directly reveal the flavor ratios
injected at cosmically distant sources. With the updated values of the three
neutrino mixing angles, we obtain the inverted large-distance evolution matrix
and use it to derive several phenomenological relations between the injection
flavor ratios and the observable ratios at Earth. Taking the three popular
injection models as examples, we also exhibit the shift of Earthly observed
flavor ratios from the corresponding values in the case with $\nu_\mu-\nu_\tau$
symmetry.
