Power spectrum response of large-scale structure in 1D and in 3D: tests
  of prescriptions for post-collapse dynamics
  The power spectrum response function of the large-scale structure of the
Universe describes how the evolved power spectrum is modified by a small change
in initial power through non-linear mode coupling of gravitational evolution.
It was previously found that the response function for the coupling from small
to large scales is strongly suppressed in amplitude, especially at late times,
compared to predictions from perturbation theory (PT) based on the
single-stream approximation. One obvious explanation for this is that PT fails
to describe the dynamics beyond shell-crossing. We test this idea by comparing
measurements in $N$-body simulations to prescriptions based on PT but augmented
with adaptive smoothing to account for the formation of non-linear structures
of various sizes in the multi-stream regime. We first start with
one-dimensional (1D) cosmology, where the Zel'dovich approximation provides the
exact solution in the single stream regime. Similarly to the 3D case, the
response function of the large-scale modes exhibits a strong suppression in
amplitude at small scales which cannot be explained by the Zel'dovich solution
alone. However, by performing adaptive smoothing of initial conditions to
identify haloes of different sizes and solving approximately post-collapse
dynamics in the 3-streams regime, agreement between theory and simulations
drastically improves. We extend our analyses to the 3D case using PINOCCHIO
algorithm, in which similar adaptive smoothing is implemented on the Lagrangian
PT fields to identify haloes and is combined with a spherical halo prescription
to account for post-collapse dynamics. Again, a suppression is found in the
coupling between small- and large-scale modes and the agreement with
simulations is improved.
