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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 quantitativeimplications for the time-evolution of neutrino flavors. In particular, thelarge-distance flavor evolution matrix, non-invertible with $\nu_\mu-\nu_\tau$symmetry, is now invertible. This means that measurements of neutrino flavorratios at Earth can now be inverted to directly reveal the flavor ratiosinjected at cosmically distant sources. With the updated values of the threeneutrino mixing angles, we obtain the inverted large-distance evolution matrixand use it to derive several phenomenological relations between the injectionflavor ratios and the observable ratios at Earth. Taking the three popularinjection models as examples, we also exhibit the shift of Earthly observedflavor ratios from the corresponding values in the case with $\nu_\mu-\nu_\tau$symmetry.

Cosmic Neutrino Flavor Ratios with Broken $\nu_\mu-\nu_\tau$ Symmetry