A Note on the Consistency Condition of Primordial Fluctuations

Abstract: We show that the squeezed limit of (N+1)-point functions of primordial correlation functions in which one of the modes has a very small wavenumber can be inferred from the spatial variation of locally measured N-point function. We then show how in single clock inflation a long wavelength perturbation can be re-absorbed in the background cosmology and how in computing correlation functions the integrals of the interaction Hamiltonian are dominated by conformal times of order of the short wavelength modes, when the long mode is already outside of the horizon. This allows us to generalize the consistency condition for N-point functions to the case in which the short wavelength fluctuations are inside the horizon and derivatives acts on them. We further discuss the consistency condition in the soft internal squeezed limit in which in an (N+M)-point function with (N+M) short modes the sum of the first N modes is a very soft momentum. These results are very useful to study infrared effects in Inflation.