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 at z approximately 1 would allow us 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 approximately 1. For the standard LambdaCDM universe with vanishing tensor mode, the quadrupole moment of the temperature anisotropy at z = 0 is expected to have an approximately 32% contribution from fluctuations on scales below 6.3 h(-1) Gpc. This percentage would be reduced to approximately 2% level for the quadrupole moment of polarization pattern at z approximately 1. A cluster polarization would shed light on the potentially anomalous features of the largest scale fluctuations.