The matter fluctuation effect to T violation at a neutrino factory

We derived an analytic formula for T violation by using the perturbation theory for small quantities, $\Delta m_{21}^2L/2E$ and $\delta a(x)L/2E$, where $\delta a(x)$ represents symmetric and asymmetric matter fluctuations, i.e., deviations from the average density. We analyzed the effect of matter fluctuations to T violation, by assuming PREM profile of earth matter density. We found that matter fluctuations do not give any viable contribution for $L<$6000km, while the fluctuation effect becomes large due to resonances for $L>$7000km. For 7000km$<L<$8000km, matter fluctuations contribute destructively to the average density term and the net result is small, while for $L>$8000km, the contribution from matter fluctuations becomes large but contributes constructively.