Testing MSW effect in Supernova Explosion with Neutrino event rates

Flavor transitions in supernova neutrinos are yet to be determined. We present a method to probe whether or not the Mikheyev-Smirnov-Wolfenstein effects occur as SN neutrinos propagate outward from the SN core by investigating time evolutions of neutrino event rates for different flavors in different kinds of detectors. As the MSW effect occurs, the $ u_e$ flux swaps with the $ u_x$ flux, which represents any one of $ u_{mu}$, $ u_{tau}$, $bar{ u}_{mu}$, and $bar{ u}_{tau}$ flux, either fully or partially depending on the neutrino mass hierarchy. During the neutronization burst, the $ u_e$ emission evolves in a much different shape from the emissions of $bar{ u}_e$ and $ u_x$ while the latter two evolve in a similar pattern. Meanwhile, the luminosity of the the $ u_e$ emission is much larger than those of the $bar{ u}_e$ and $ u_x$ emissions while the latter two are roughly equal. As a consequence, the time-evolution pattern of the $ u_e{rm Ar}$ event rates in the absence of the MSW effect will be much different from that in the occurrence of the MSW effect, in either mass hierarchy. With the simulated SN neutrino emissions, the $ u_e{rm Ar}$ and inverse beta decay event rates are evaluated. The ratios of the two cumulative event rates are calculated for different progenitor masses up to $100~{rm ms}$. We show that the time evolutions of this cumulative ratios can effectively determine whether MSW effects really occur for SN neutrinos or not.