Neutrino Flavor Ratios Modified by Cosmic Ray Secondary Acceleration

Acceleration of $\pi$'s and $\mu$'s modifies the flavor ratio at Earth (at astrophysical sources) of neutrinos produced by $\pi$ decay, $\nu_e:\nu_{\mu}:\nu_{\tau}$, from $1:1:1$ ($1:2:0$) to $1:1.8:1.8$ ($0:1:0$) at high energy, because $\pi$'s decay more than $\mu$'s during secondary-acceleration. The neutrino spectrum accompanies a flat excess, differently from the case of energy losses. With the flavor spectra, we can probe timescales of cosmic-ray acceleration and shock dynamics. We obtain general solutions of convection-diffusion equations and apply to gamma-ray bursts, which may have the flavor modification at around PeV -- EeV detectable by IceCube and next-generation experiments.