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Neutrino Flavor Ratios Modified by Cosmic Ray Secondary Acceleration

Acceleration of $π$'s and $μ$'s modifies the flavor ratio at Earth (at astrophysical sources) of neutrinos produced by $π$ decay, $ν_e:ν_μ:ν_τ$, from $1:1:1$ ($1:2:0$) to $1:1.8:1.8$ ($0:1:0$) at high energy, because $π$'s decay more than $μ$'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.