On Soft Limits of Large-Scale Structure Correlation Functions
  We study soft limits of correlation functions for the density and velocity
fields in the theory of structure formation. First, we re-derive the (resummed)
consistency conditions at unequal times using the eikonal approximation. These
are solely based on symmetry arguments and are therefore universal. Then, we
explore the existence of equal-time relations in the soft limit which, on the
other hand, depend on the interplay between soft and hard modes. We scrutinize
two approaches in the literature: The time-flow formalism, and a background
method where the soft mode is absorbed into a locally curved cosmology. The
latter has been recently used to set up (angular averaged) `equal-time
consistency relations'. We explicitly demonstrate that the time-flow relations
and `equal-time consistency conditions' are only fulfilled at the linear level,
and fail at next-to-leading order for an Einstein de-Sitter universe. While
applied to the velocities both proposals break down beyond leading order, we
find that the `equal-time consistency conditions' quantitatively approximates
the perturbative results for the density contrast. Thus, we generalize the
background method to properly incorporate the effect of curvature in the
density and velocity fluctuations on short scales, and discuss the reasons
behind this discrepancy. We conclude with a few comments on practical
implementations and future directions.
