Cosmology plays a fundamental role to determine the neutrino mass , therefore also to determine its mass hierarchy , since the massive neutrino contributes to the total matter density in the Universe at the background and perturbation levels , once it becomes non-relativistic .
After the non-relativistic transition the fluctuations are smashed out at the scales k \gg k _ { fs } .
Therefore , the missing fluctuation in the total matter is imprinted on the large scale structure , say the suppression of the matter power spectrum \Delta P / P \approx - 8 f _ { \nu } at the scales k \gg k _ { fs } .
In this paper , instead of considering the linear perturbation theory , which is well understood in the presence of neutrino , we propose to use the cross correlation between the Rees-Sciama effect and weak lensing to probe the neutrino mass .
At the small scales , the density contrast grows faster than the background scale factor \delta \sim a , that makes a sign flipping on \Phi ^ { \prime } \propto \mathcal { H } \delta d \ln ( \delta / a ) / d \ln a , which happens only in the non-linear regime .
We show that the flipping scale in the cross power spectrum between the Rees-Sciama effect and weak lensing depends on the neutrino mass by assuming the shallow and deep weak lensing surveys .
Our analysis shows that the Deep survey has larger signal-to-noise ratio S / N \sim 160 .
Finally , we use the Fisher information matrix to forecast constraint on the neutrino mass .