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The first generation of haloes forms from the collapse of the smallest peaksin the initial density field. $N$-body simulations of this process suggest aprompt formation of a steep power-law cusp, but these calculations are plaguedby numerical artifacts which casts some doubt on this result. Here, we developnew simulation methods based on the dark matter phase-space sheet approach andpresent results which are entirely free of artificial clumps. We find that acusp with density $\rho\propto r^{-1.5}$ is indeed formed promptly,subsequently accreting a more extended halo and participating in thehierarchical growth of later halo generations. However, our simulations alsosuggest that the presence of artificial clumps just before peak collapse cansignificantly shallow the inner profiles of the cusps. We use $N$-bodysimulations with controlled amounts of small-scale power to place aconservative upper limit on the scales affected by artificial clumps. Finally,we used these results to simulate the collapse of the first generation of peaksof various types and in different cosmologies, finding prompt cusps to form inall cases. We conclude that prompt cusps are a generic feature of the collapseof peaks on the free-streaming scale of the initial density field, and theirstructure can safely be studied using $N$-body simulations provided care istaken to excise the region potentially affected by artificial clumps.

Phase-space simulations of prompt cusps: simulating the formation of the first haloes without artificial fragmentation