shock wave propagation, which is in reasonable agreement with 56.7 % decrease as shown by their calculations. We thus conclude that their theoretical calculations and ours of the neutrino event rates agree with each other despite several essential differences: We solved in Eq. (1) both flavor conversion of neutrinos and supernova density profile numerically in our method described in Sec. 1, while Fogli et al. [27] applied an artificial model of density profile of shock wave propagation without solving dynamical supernova explosion. 

For further comparison, let us analytically estimate the decrease in the neutrino events in the non-adiabatic case (with shock) compared with the adiabatic case (without shock). In the non-adiabatic case, we define the survival probability of ¯ ν$_{e}$ as ¯ P = 0 . 7 in our model, and we use the result of ¯ ν$_{e}$ in the case of normal hierarchy in Fogli’s model. We set ¯ P = 0 in the adiabatic case. Thus estimated analytical results show 62.3 % and 63.0 % decrease in the neutrino events from adiabatic to non-adiabatic cases in our model and their model, respectively. These values are close enough to each other, and ∼ 60% are not very different from ∼ 55 . 0% which was inferred from numerical calculations as discussed in the previous paragraph. This fact justifies that our treatment of analytical estimates are reasonable. Therefore, regardless of all possible differences between the two models, the supernova neu- trinos provide a powerful tool to indicate the shock wave propagation inside supernova if sin $^{2}$2 θ$_{13}$ is large in the inverted hierarchy. 

We assumed that CP violating phase is zero and sin $^{2}$2 θ$_{23}$ = 1 throughout this study. However, CP-phase is unknown and there is an uncertainty in the mixing angle θ$_{23}$ . Here we discuss the dependence of the shock effect on neutrino flavor transitions on CP-phase and θ$_{23}$ . Transition probabilities of neutrinos in arbitrary density profile have been studied theoretically in [65], where it was found that the transition probabilities of ν$_{e}$ → ν$_{e}$ and ¯ ν$_{e}$ → ¯ ν$_{e}$ do not depend on CP-phase and mixing angle θ$_{23}$ at all. Furthermore, one can show from their formula in [65] that the sum of the transition probabilities of ν$_{µ}$ → ν$_{e}$ and ν$_{τ}$ → ν$_{e}$ (¯ ν$_{µ}$ → ¯ ν$_{e}$ and ¯ ν$_{τ}$ → ¯ ν$_{e}$ ) are totally free from CP-phase and θ$_{23}$ . Since the spectra of ν$_{µ}$ and ν$_{τ}$ (¯ ν$_{µ}$ and ¯ ν$_{τ}$ ) are the same at the neutrino sphere, the transition probability from ν$_{µ}$ or ν$_{τ}$ (¯ ν$_{µ}$ or ¯ ν$_{τ}$ ) to ν$_{e}$ (¯ ν$_{e}$ ) can be written as a half of the sum of the transition probabilities of ν$_{µ}$ → ν$_{e}$ and ν$_{τ}$ → ν$_{e}$ (¯ ν$_{µ}$ → ¯ ν$_{e}$ and ¯ ν$_{τ}$ → ¯ ν$_{e}$ ). The transition probability from ν$_{µ}$ or ν$_{τ}$ (¯ ν$_{µ}$ or ¯ ν$_{τ}$ ) to ν$_{e}$ (¯ ν$_{e}$ ) does not depend on CP-phase and θ$_{23}$ . We thus conclude that there is no effect of CP-phase and mixing angle θ$_{23}$ on the supernova neutrino spectra. 

If the influence of the shock wave is seen very early ( t ≤ 1 s ) in the observation of the 

