Single-field consistency relations of large scale structure

We derive consistency relations for the late universe (CDM and Lambda CDM): relations between an n-point function of the density contrast delta and an (n + 1)-point function in the limit in which one of the (n + 1) momenta becomes much smaller than the others. These are based on the observation that a long mode, in single-field models of inflation, reduces to a diffeomorphism since its freezing during inflation all the way until the late universe, even when the long mode is inside the horizon (but out of the sound horizon). These results are derived in Newtonian gauge, at first and second order in the small momentum q of the long mode and they are valid non-perturbatively in the short-scale d. In the non-relativistic limit our results match with [1, 2]. These relations are a consequence of diffeomorphism invariance; they are not satisfied in the presence of extra degrees of freedom during inflation or violation of the Equivalence Principle (extra forces) in the late universe.