Neutrino oscillation and expected event rate of supernova neutrinos in adiabatic explosion model

We study how the influence of the shock wave appears in neutrino oscillations and the neutrino spectrum using density profile of adiabatic explosion model of a core-collapse supernova which is calculated in an implicit Lagrangian code for general relativistic spherical hydrodynamics. We calculate expected event rates of neutrino detection at SK and SNO for various theta_{13} values and both normal and inverted hierarchies. The predicted event rates of bar{nu}_e and nu_e depend on the mixing angle theta_{13} for the inverted and normal hierarchies, respectively, and the influence of the shock appears for about 2 - 8 s when sin^2 2 theta_{13} is larger than 10^{-3}. These neutrino signals for the shock propagation is decreased by < 30 % for bar{nu}_e in inverted (SK) or by < 15 % for nu_e in normal hierarchy (SNO) compared with the case without shock. The obtained ratio of the total event for high-energy neutrinos (20 MeV < E_{nu} < 60 MeV) to low-energy neutrinos (5 MeV < E_{nu} < 20 MeV) is consistent with the previous studies in schematic semi-analytic or other hydrodynamic models of the shock propagation. The time dependence of the calculated ratio of the event rates of high-energy to low-energy neutrinos is a very useful observable which is sensitive to theta_{13} and hierarchies. Namely, time-dependent ratio shows clearer signal of the shock propagation that exhibits remarkable decrease by at most factor \sim 2 for bar{nu}_e in inverted (SK), whereas it exhibits smaller change by \sim 10 % for nu_e in normal hierarchy (SNO). Observing time-dependent high-energy to low-energy ratio of the neutrino events thus would provide a piece of very useful information to constrain theta_{13} and mass hierarchy, and eventually help understanding the propagation how the shock wave propagates inside the star.