Soft gravitational effects in Kadanoff-Baym approach

In de Sitter space, the gravitational fluctuation at the super-horizon scale may make physical quantities time dependent by breaking the de Sitter symmetry. We adopt the Kadanoff-Baym approach to evaluate soft gravitational effects in a matter system at the sub-horizon scale. This investigation proves that only the local terms contribute to the de Sitter symmetry breaking at the one-loop level. The IR singularities in the non-local terms cancel after summing over degenerate states between real and virtual processes. The corresponding IR cut-off is given by the energy resolution like QED. Since the energy resolution is physical and independent of cosmic evolution, the non-local contributions do not induce the de Sitter symmetry breaking. We can confirm that soft gravitational effects preserve the effective Lorentz invariance.