Effects of Shock Waves on Neutrino Oscillations in Three Supernova   Models
  It has been realized that the shock wave effects play an important role in neutrino oscillations during the supernova explosion. In recent years, with the development of simulations about supernova explosion, we have a better understanding about the density profiles and the shock waves in supernovae than before. It has been shown that the appearance of shock waves not only varies with time, but is also affected by the mass of the supernova. When the mass of the supernova happens to be in a certain range (e.g. it equals 10.8 times the mass of the sun), there might be a reverse shock wave, another sudden change of density except the forward shock wave, emerging in the supernova. In addition, there are some other time-dependent changes of density profiles in different supernova models. Because of these complex density profiles, the expression of the crossing probability at the high resonance, $P_H$, which we used previously would be no longer applicable. In order to get more accurate and reasonable results, we use the data of density profiles in three different supernova models obtained from simulations to study the variations of $P_s$ (the survival probability of $\nu_e\rightarrow\nu_e$), as well as $P_c$ (the conversion probability of $\nu_x\rightarrow\nu_e$). It is found that the mass of the supernova does make a difference on the behavior of $P_s$, and affects $P_c$ at the same time. With the results of $P_s$ and $P_c$, we can estimate the number of $\nu_e$ remained after they go through the matter in the supernova. 
Jing Xu, Li-Jun Hu, Rui-Cheng Li, Xin-Heng Guo and Bing-Lin Young
hep-ph astro-ph.HE