Spherical collapse, formation hysteresis and the deeply non-linear
  cosmological power spectrum
  I examine differences in non-linear structure formation between cosmological
models that share a $z=0$ linear power spectrum in both shape and amplitude,
but that differ via their growth history. $N$-body simulations of these models
display an approximately identical large-scale-structure skeleton, but reveal
deeply non-linear differences in the demographics and properties of haloes. I
investigate to what extent the spherical-collapse model can help in
understanding these differences, in both real and redshift space. I discuss how
this is difficult to do if one attempts to identify haloes directly, because in
that case one is subject to the vagaries of halo finding algorithms. However, I
demonstrate that the halo model of structure formation provides an accurate
non-linear response in the power spectrum, but only if results from spherical
collapse that include formation hysteresis are properly incorporated. I comment
on how this fact can be used to provide per cent level accurate matter power
spectrum predictions for dark energy models for $k\leq5\,h\mathrm{Mpc}^{-1}$ by
using the halo model as a correction to accurate $\Lambda$CDM simulations. In
the appendix I provide some fitting functions for the linear-collapse threshold
($\delta_\mathrm{c}$) and virialized overdensity ($\Delta_\mathrm{v}$) that are
valid for a wide range of dark energy models. I also make my spherical-collapse
code available at https://github.com/alexander-mead/collapse
