Probing the largest scale structure in the universe with polarization map of galaxy clusters

We introduce a new formalism to describe the polarization signal of galaxy clusters on the whole sky. We show that a sparsely sampled, half-sky map of the cluster polarization signal at $z\sim 1$ would allow to better characterize the very large scale density fluctuations. While the horizon length is smaller in the past, two other competing effects significantly remove the contribution of the small scale fluctuations from the quadrupole polarization pattern at $z\sim 1$. For the standard Lambda-CDM universe with vanishing tensor mode, the quadrupole moment of the temperature anisotropy probed by WMAP is expected to have a ~32% contribution from fluctuations on scales below 6.3h^{-1}Gpc. This percentage would be reduced to ~2% level for the quadrupole moment of polarization pattern at $z\sim 1$. A cluster polarization map at $z \sim 1$ would shed light on the potentially anomalous features of the largest scale structure in the observable universe.