Constructing merger trees that mimic N-body simulations

We present a simple and efficient empirical algorithm for constructing dark matter halo merger trees that reproduce the distribution of trees in the Millennium cosmological N-body simulation. The generated trees are significantly better than EPS trees. The algorithm is Markovian, and it therefore fails to reproduce the non-Markov features of trees across short time-steps, except for an accurate fit to the evolution of the average main progenitor. However, it properly recovers the full main-progenitor distribution and the joint distributions of all the progenitors over long-enough time-steps, Δω ≃ Δ z > 0.5, where ω ≃ 1.69/D(t) is the self-similar time variable and D(t) refers to the linear growth of density fluctuations. We find that the main-progenitor distribution is lognormal in the variable σ²(M), the variance of linear density fluctuations in a sphere encompassing mass M. The secondary progenitors are successfully drawn one by one from the remaining mass using a similar distribution function. These empirical findings may be clues to the underlying physics of merger-tree statistics. As a byproduct, we provide useful, accurate analytic time-invariant approximations for the main-progenitor accretion history and for halo merger rates.