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I examine differences in non-linear structure formation between cosmologicalmodels 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 modelsdisplay an approximately identical large-scale-structure skeleton, but revealdeeply non-linear differences in the demographics and properties of haloes. Iinvestigate to what extent the spherical-collapse model can help inunderstanding these differences, in both real and redshift space. I discuss howthis is difficult to do if one attempts to identify haloes directly, because inthat case one is subject to the vagaries of halo finding algorithms. However, Idemonstrate that the halo model of structure formation provides an accuratenon-linear response in the power spectrum, but only if results from sphericalcollapse that include formation hysteresis are properly incorporated. I commenton how this fact can be used to provide per cent level accurate matter powerspectrum predictions for dark energy models for $k\leq5\,h\mathrm{Mpc}^{-1}$ byusing the halo model as a correction to accurate $\Lambda$CDM simulations. Inthe appendix I provide some fitting functions for the linear-collapse threshold($\delta_\mathrm{c}$) and virialized overdensity ($\Delta_\mathrm{v}$) that arevalid for a wide range of dark energy models. I also make my spherical-collapsecode available at https://github.com/alexander-mead/collapse

Spherical collapse, formation hysteresis and the deeply non-linear cosmological power spectrum