Separate universe and the back reaction of long wavelength fluctuations

We investigate the back reaction of cosmological long wavelength perturbations on the evolution of the Universe. By applying the renormalization group method to a Friedmann-Robertson-Walker universe with long wavelength fluctuations, we demonstrate that the renormalized solution with the back reaction effect is equivalent to that of the separate universe. Then, using the effective Friedmann equation, we show that only the nonadiabatic mode of long wavelength fluctuations affects the expansion law of the spatially averaged universe.