We have developed a new method to extract halo merger rates from the Millennium Simulation. First, by removing superfluous mergers that are artifacts of the standard friends-of-friends (FOF) halo identification algorithm, we find a lower merger rate compared to previous work. The reductions are more significant at lower redshifts and lower halo masses, and especially for minor mergers. Our new approach results in a better agreement with predictions from the extended Press-Schechter model. Second, we find that the FOF halo finder overestimates the halo mass by up to 50% for halos that are about to merge, which leads to an additional ≈20% overestimate of the merger rate. Therefore, we define halo masses by including only particles that are gravitationally bound to their FOF groups. We provide new best-fitting parameters for a global formula to account for these improvements. In addition, we extract the merger rate per progenitor halo, as well as per descendant halo. The merger rate per progenitor halo is the quantity that should be related to observed galaxy merger fractions when they are measured via pair counting. At low-mass/redshift, the merger rate increases moderately with mass and steeply with redshift. At high enough mass/redshift (for the rarest halos with masses a few times the "knee" of the mass function), these trends break down, and the merger rate per progenitor halo decreases with mass and increases only moderately with redshift. Defining the merger rate per progenitor halo also allows us to quantify the rate at which halos are being accreted onto larger halos, in addition to the minor and major merger rates. We provide an analytic formula that converts any given merger rate per descendant halo into a merger rate per progenitor halo. Finally, we perform a direct comparison between observed merger fractions and the fraction of halos in the Millennium Simulation that have undergone a major merger during the recent dynamical friction time, and find a fair agreement, within the large uncertainties of the observations. Our new halo merger trees are available at http://www.mpe.mpg.de/ir/MillenniumMergerTrees/.
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