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, $\Delta \omega \simeq \Delta z>0.5$, where $\omega \simeq 1.69/D(t)$ 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 log-normal in the variable $\sigma^2(M)$, 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.