This page contains a collection of results used to assess the convergence properties of halo scale radii (and, therefore, concentrations) as implemented in Galacticus by the darkMatterProfileScaleRadiusJohnson2021 class which implements the model of Johnson, Benson & Grin (2021).

All calculations were performed using merger trees at the masses and redshifts shown for each convergence plot. Parameters for the darkMatterProfileScaleRadiusJohnson2021 model were set to:

energyBoost:
massExponent:
peakHeightExponent:
scatterExcess:
unresolvedEnergy:
factorMassResolution:

Note that this model does not currently produce scale radii which match those measured in N-body simulations - it has not been calibrated to do so. The goal here is simply to explore the convergence properties.

Information on the initial conditions model (used to set concentrations in insufficiently well-resolved branches of merger trees) can be found below.

Convergence was tested for the following parameters:

The mass resolution sets the minimum halo mass that is tracked in a merger tree. Here, the minimum halo mass is defined as \( f_\mathrm{res} M \) where \( M \) is the mass of the merger tree, and \( f_\mathrm{res} \) is therefore the fractional resolution.

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, and were run with . Convergence requires that all points agree (within the statistical uncertainties).

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, and were run with . Convergence requires that all points agree (within the statistical uncertainties).

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, and were run with . Convergence requires that all points agree (within the statistical uncertainties).

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, and were run with . Convergence requires that all points agree (within the statistical uncertainties).

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, and were run with . Convergence requires that all points agree (within the statistical uncertainties).

Initial Conditions

For insufficiently well-resolved branches of a merger tree (any branch for which the halo masses are below factorMassResolution times the mass resolution of the tree), the scale radius is set by an initial conditions model. This must produce results that are consistent with what the darkMatterProfileScaleRadiusJohnson2021 class would produce in these halos if they were sufficiently well resolved. We use a simple power-law model of the form

$$ r_\mathrm{s} = r(\nu) \left(\frac{M}{M_0}\right)^{\alpha(\nu)} (1+z)^{-\beta(\nu)} $$

where \(r(\nu)\), \(\alpha(\nu)\), and \(\beta(\nu)\) are sigmoid functions of the peak height, \(\nu\), of the form:

$$ y(x) = y_0+(y_1-y_0)/(1+\exp[-(x-x_\nu)/\Delta x]), $$

where \(r_0\)=radiusLow, \(r_1\)=radiusHigh, \(r_\nu\)=radiusTransition, \(\Delta r\)=radiusWidth, \(\alpha_0\)=massLow, \(\alpha_1\)=massHigh, \(\alpha_\nu\)=massTransition, \(\Delta \alpha\)=massWidth, \(\beta_0\)=expansionFactorLow, \(\beta_1\)=expansionFactorHigh, \(\beta_\nu\)=expansionFactorTransition, and \(\Delta \beta\)=expansionFactorWidth with values determined by fitting this model to the results from the darkMatterProfileScaleRadiusJohnson2021 class in well-resolved branches, as given in the following table:

radiusLow:
radiusHigh:
radiusTransition:
radiusWidth:
massLow:
massHigh:
massTransition:
massWidth:
expansionFactorLow:
expansionFactorHigh:
expansionFactorTransition:
expansionFactorWidth:

Correlation Model

A correlated set of random, log-normal deviates are applied to the scale radii of these halos. That is, the scale radius of the ith halo in such a sub-branch will be \( r_\mathrm{s} = \bar{r}_{\mathrm{s}, i} 10^{x_i} \) where \( x_i \) is a normally-distributed random variate with mean zero and dispersion \( \sigma^\prime \) = modelScatter. The deviates \( x_i \) are assumed to be correlated with correlation matrix: $$ C_{i,j} = \exp\left( -\gamma \left| \log_{10} \frac{M_i}{M_j} \right|^\mu \right), $$

where \( \gamma \) = correlationRateDecay, and \( \mu \) = correlationExponent with values found by fitting to the behavior measured from the darkMatterProfileScaleRadiusJohnson2021 class in well-resolved branches, as given in the following table, and shown in the following plot:

correlationRateDecay:
correlationExponent:
modelScatter: