Excursion Sets with a "Perfect" Collapse Model

Abstract

The $\Lambda$CDM model predicts structure formation across a vast mass range, from massive clusters ($\sim 10^{15}{\text{M}}_{\odot }$) to Earth-mass micro-haloes ($\sim 10^{-6}{\text{M}}_{\odot }$), resolving which far exceeds the capabilities of current simulations. Excursion set models are the most efficient theoretical tool to disentangle this hierarchy in mass. We test the excursion set paradigm by combining smoothed initial density fields with a "perfect" collapse model -- $N$-body simulations. We find that a core excursion set assumption -- small-scale perturbations do not impact larger-scale collapse -- is approximately fulfilled but exhibits small quantitative violations dependent on the smoothing filter. For a sharp $k-$space cut-off $\sim 20\%$ of mass elements revert collapse as the smoothing scale decreases, while only $3.5\%$ do for a Gaussian and $5\%$ for a top-hat. Further, we test the simple deterministic mass-mapping $M\propto R^3$ (first-crossing scale to halo mass) relation. We find that particles that are first accreted into haloes at the same smoothing scale may end up in haloes of significantly different masses, with a scatter of 0.4-0.8 dex. We also demonstrate that the proportionally constant of this relation should be considered as a degree of freedom. Finally, we measure the mass fraction in different structure morphologies (voids, pancakes, filaments and haloes) as a function of filter scale. Typical particles appear to be part of a large-scale pancake, a smaller-scale filament and a notably smaller halo. We conclude that validating predictions of excursion set models on a particle-by-particle basis against simulations may enhance their realism.

18 pages, 15 figures

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