as ¯ ν$_{e}$ even after the neutrinos get out of the star. As the result, the fraction of ¯ ν$_{e}$ converted from ¯ ν$_{x}$ is small. The average energies of the three flavor neutrinos just emitted from the proto-neutron star satisfy the following relation; ¯ E$_{ν}$$_{e}$ < ¯ E$_{¯}$ ν$_{e}$ < ¯ E$_{ν}$$_{x}$ . Therefore, the average energy of ¯ ν$_{e}$ in adiabatic case becomes higher than that in non-adiabatic case. As the main reaction cross section for ¯ ν$_{e}$ is proportional to E 2 ν as discussed in section II C, the detection probability of high-energy neutrinos is larger than that of low-energy neutrinos. Therefore, the event rate of ¯ ν$_{e}$ for large θ$_{13}$ (adiabatic case) becomes much larger than that for small θ$_{13}$ (non-adiabatic case). This is the reason for large split between the expected event rates for large θ$_{13}$ (i.e. sin $^{2}$2 θ$_{13}$ = 10 − 2 and 10 − $^{3}$) and small θ$_{13}$ (i.e. sin $^{2}$2 θ$_{13}$ = 10 − 4 and 10 − $^{5}$). We find difference of event rates by ∼ 1 , 000 at t = 0 between the two cases. 

Furthermore, the ratio of the event rates for ¯ ν$_{e}$ which exhibits the shock wave propagation effect changes dramatically when sin $^{2}$2 θ$_{13}$ is large (right lower panel of Figure 7). The ratio of ¯ ν$_{e}$ increases at around 2 s, and decreases after 3 s, which is similar to ν$_{e}$ event rates in normal hierarchy (left lower panel of Figure 6). The mechanism is quite similar to each other so that the shock wave propagation through the H-resonance starts from low energy neutrinos to high energy neutrinos, which eventually decreases the event rates. However, the decrease in the ratio of the ¯ ν$_{e}$ event rates is remarkable by at most 30 % for sin $^{2}$2 θ$_{13}$ = 10 − $^{3}$. On the other hand, the ratio changes weakly for the other θ$_{13}$ values because H-resonance is non-adiabatic regardless of the shock wave, although its degree is still larger than the ν$_{e}$ event rate (see left lower panel of Figure 6). 

The event rate of ¯ ν$_{e}$ is extremely larger than that of ν$_{e}$ (for a given neutrino oscillation parameter set). This is mainly because the total cross section of ¯ ν$_{e}$ -induced reactions is about 10 2 times larger than that of ν$_{e}$ . Moreover, the total cross section of ¯ ν$_{e}$ -induced reactions is almost proportional to the square of neutrino energy, but the total cross section of ν$_{e}$ - induced reactions is linearly proportional to the neutrino energy. Owing to this different energy dependence of the cross sections, one can expect that the effect of the shock wave should be identified more clearly in the ¯ ν$_{e}$ events. 

In practice, since the shock effect is folded in the observed event rates of ν$_{e}$ and ¯ ν$_{e}$ , we should carry out careful theoretical analysis. The observed event rates are statistically uncertain of the order of their square root. As for ν$_{e}$ in normal hierarchy, the statistical error is the same order as the change in the ratio ∼ 15% of the event rates with and without the shock effect because absolute number of expected event rates are small dN dt ≲ 70 (Figure 

