Consistency relations of cosmological correlators in the squeezed limit, the analog of soft theorems in scattering amplitudes, play an important role in revealing the underlying theory and symmetry. The speaker explores the consequence of symmetries with non-linear transformation and derives an identity which relates squeezed (N +1)-point function to a symmetry transformation of the N -point function. It is different form ordinary equal time soft theorem since the soft mode is inserted in the far past. Nevertheless, experiments can only measure equal time correlators in the late universe, therefore whether the identity can constrain late time correlators has to be examined. It turns out that when an adiabatic mode exists, the late-initial identity can be promoted to an equal late time identity. He also perturbatively proves that only symmetries associated with adiabatic modes possess equal time soft theorem in single field model. It resolves the puzzle of why the ultra slow roll model has dilation as a symmetry but violates the corresponding equal time consistency relation.
