Identifying Earth matter effects on supernova neutrinos at a single detector

The neutrino oscillations in Earth matter introduce modulations in the supernova neutrino spectra. These modulations can be exploited to identify the presence of Earth effects on the spectra, which would enable us to put a limit on the value of the neutrino mixing angle $\theta_{13}$ and to identify whether the mass hierarchy is normal or inverted. We demonstrate how the Earth effects can be identified at a single detector without prior assumptions about the flavor-dependent source spectra, using the Fourier transform of the ``inverse-energy'' spectrum of the signal. We explore the factors affecting the efficiency of this method, and find that the energy resolution of the detector is the most crucial one. In particular, whereas water Cherenkov detectors may need a few ten thousand events to identify the Earth effects, a few thousand may be enough at scintillation detectors, which generically have a much better energy resolution. A successful identification of the Earth effects through this method can also provide $\Delta m^2_\odot$ to a good accuracy. The relative strength of the detected Earth effects as a function of time provides a test for supernova models.