Constraints on neutrino masses from the lensing dispersion of type ia supernovae

We investigate how accurately the total mass of neutrinos is constrained from the magnitude dispersion of SNe Ia due to the effects of gravitational lensing. For this purpose, we use the propagation equation of light bundles in a realistic inhomogeneous universe and propose a sample selection for supernovae to avoid difficulties associated with small-scale effects such as strong lensing or shear effects. With a fitting formula for the nonlinear matter power spectrum taking account of the effects of massive neutrinos, we find that in our model it is possible to obtain the upper limit $\Sigma m_{\nu }\simeq 1.0\left[eV\right]$ for future optical imaging surveys with the Wide-Field InfraRed Survey Telescope and Large Synoptic Survey Telescope. Furthermore, we discuss how far we need to observe SNe Ia and to what extent we have to reduce the magnitude error except for lensing in order to realize the current tightest limit $\Sigma m_{\nu }<0.2\left[eV\right]$.