The coarse geometry of merger trees in Λ cold dark matter

We introduce the contour process to describe the geometrical properties of merger trees. The contour process translates a tree structure into a one-dimensional object: the contour walk. We characterize the contour walk measuring its length and action. The length is proportional to the number of progenitors in the tree, and the action is a proxy for the mean length of a branch in the tree.
We construct the contour walk for merger trees extracted from the public data base of the Millennium Run and also for merger trees constructed with a public Monte Carlo code which implements a Markovian algorithm. The trees correspond to haloes of final masses between 10¹¹ and  10¹⁴  h ⁻¹ M_⊙  . We study how the length and action scale with the mass of the final halo.
In all cases, except for the action measured from Markovian trees, we find a transitional scale around  1–6 × 10¹²  h ⁻¹ M_⊙  , which can be interpreted as the point where the halo growth starts to be dominated by mergers and not accretion. The transitional scale depends weakly on the mass resolution and time discretization of the merging trees.