Individual dark matter halos in cosmological simulations vary widely in their detailed structural properties, properties such as concentration, shape, spin, and degree of internal relaxation. Recent non-parametric (principal component) analyses suggest that a few principal components explain a large fraction of the scatter in these structural properties. The main principal component is closely aligned with concentration, which in turn is known to be related to the mass accretion history (MAH) of the halo, as described by its merger tree. Here, we examine more generally the connection between the MAH and structural parameters. The space of mass accretion histories has principal components of its own. The strongest, accounting for almost 60% of the scatter between individual histories, can be interpreted as the age of the system. We give an analytic fit for this first component, which provides a rigorous way of defining the dynamical age of a halo. The second strongest component, representing acceleration or deceleration of growth at late times, accounts for 25% of the scatter. Relating structural parameters to formation history, we find that concentration correlates strongly with the early history of the halo, while shape and degree of relaxation or dynamical equilibrium correlate with the later history. We examine the inferences about formation history that can be drawn by splitting halos into sub-samples based on observable properties such as concentration and shape. Applications include the definition young and old samples of galaxy clusters in a quantitative way, or empirical tests of environmental processing rates in clusters.
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