This page contains a collection of results used to assess the convergence properties of merger trees built using as implemented in Galacticus by the mergerTreeBuilderCole2000 and mergerTreeBranchingProbabilityParkinsonColeHelly classes.

All calculations were performed using merger trees each of mass .

Convergence was tested for the following parameters:

The mass resolution sets the minimum halo mass that is tracked in a merger tree.

Convergence results are shown relative to the highest resolution model run (). Perfect convergence would have all points at zero on the y-axis. The completeness of the progenitor mass function due to resolution effects is modeled as (shown as lines in the following figures):

where $$w(t)=\delta_\mathrm{c}(t)/D(t)$$ is the usual time variable used in Press-Schechter theory.

The mergeProbability numerical parameter controls the size of timesteps taken when constructing merger trees using the Cole et al. (2000) algorithm, implemented in Galacticus by the mergerTreeBuilderCole2000 class. Specifically, mergeProbability sets the maximum probability for a binary branching event to occur in a timestep. A smaller value of mergeProbability will therefore result in smaller timesteps, and reduce the likelihood that multiple mergers that should have occurred in a timestep are missed.

Convergence results are shown relative to a default reference model. Perfect convergence would have all points at zero on the y-axis.

The accretionLimit numerical parameter controls the size of timesteps taken when constructing merger trees using the Cole et al. (2000) algorithm, implemented in Galacticus by the mergerTreeBuilderCole2000 class. Specifically, accretionLimit sets the maximum fraction of mass in subresolution accretion that is allowed occur in a timestep. A smaller value of accretionLimit will therefore result in smaller timesteps, and increase the accuracy of the mass evolution along each branch.

Convergence results are shown relative to a default reference model. Perfect convergence would have all points at zero on the y-axis.

The accuracyFirstOrder numerical parameter controls the size of timesteps taken when constructing merger trees using branching rate of Parkinson, Cole & Helly. (2008), implemented in Galacticus by the mergerTreeBranchingProbabilityParkinsonColeHelly class. Specifically, accuracyFirstOrder limits the timestep to accuracyFirstOrder\(\sqrt{2[\sigma^2(M_2/2)-\sigma^2(M_2)]}\), which ensures the the first order expansion of the merging rate that is assumed is accurate. A smaller value of accuracyFirstOrder will therefore result in smaller timesteps, and increase the accuracy of the mass evolution along each branch.

Convergence results are shown relative to a default reference model. Perfect convergence would have all points at zero on the y-axis.

The branchIntervalStep numerical parameter controls the algorithm used for taking steps when constructing merger trees using the Cole et al. (2000) algorithm, implemented in Galacticus by the mergerTreeBuilderCole2000 class. Specifically, if branchIntervalStep=true timesteps are drawn from a negative exponential distribution following the algorithm of Benson, Ludlow & Cole (2019), otherwise the original algorithm of Cole et al. (2000) is used instead.

Convergence results are shown relative to a default reference model. Perfect convergence would have all points at zero on the y-axis.

The massThreshold numerical parameter controls the halo mass below which subsampling of merger tree branches occurs, following the algorithm of Menker & Benson (2024). Below this mass, a branch of the merger tree is kept with probability \( p = M/\mathtt{[massThreshold]}\,\mathrm{M}_\odot \). The progenitor mass function is weighted by the subsampling weight (as defined in Menker & Benson; 2024)) to correct for the effects of subsampling.

Convergence results are shown relative to a default reference model. Perfect convergence would have all points at zero on the y-axis.