Constraints: Dark matter halo mass function - galacticusorg/galacticus GitHub Wiki
To constrain the dark matter halo mass function model in Galacticus we calibrate to the suite of MultiDark Planck cold dark matter N-body simulations.
We make use of the Rockstar halo catalogs/merger trees for the VSMDPL, SMDPL, MDPL2, BigMDPL, and HugeMDPL simulations. We remove backsplash halos following the methodology of Benson (2017), and then construct halo mass functions by binning halos by mass. This workflow is implemented in the constraints/pipelines/darkMatter/haloMassFunctionPreProcess.pl script.
We use the Sheth-Tormen functional form for the mass function with an environmental dependence based on the peak--background split model (e.g. Bardeen et al. 1986). When fitting to the N-body data this mass function is first averaged over all environments then convolved with the N-body halo mass distribution function (Trenti et al. 2010), and finally averaged over each bin of the N-body halo mass function. A particle swarm optimization algorithm is then used to optimize the parameters of the Sheth-Tormen mass function to maximize the log-likelihood of the model given the N-body data. In constructing this log-likelihood we assume that the number of N-body halos in each bin of our mass function obeys Poisson statistics, and exclude any bin which corresponds to halos of fewer than 3,000 particles (below which we find that systematic deviations from higher resolution simulations become apparent), or for which the simulation has fewer than 30 halos (for which the uncertainties become large). The likelihoods of the five MultiDark Planck simulations are assumed to be independent, and so we sum the log-likelihoods to obtain the final log-likelihood which we maximize. This workflow is part of the dark matter constraints pipeline implemented in the constraints/pipelines/darkMatter/pipeline.pl script.
The resulting optimized parameters for the Sheth-Tormen mass function are:
a=0.75836
p=0.33112
A=0.28990
The following figure shows the resulting halo mass function. Each color corresponds to an individual MultiDark Planck N-body simulation. Open circles with error bars show the N-body halo mass function. The lines indicate the Sheth-Tormen model (these differ slightly for each simulation due to the different particle mass used in the model for the N-body mass uncertainty), with the small crosses on the lines indicating the bin-averaged value. The vertical dotted lines show the lowest mass bin used when fitting to each mass function.
The next two figures show the fractional residuals (i.e. the difference between N-body and model mass functions, divided by the model mass function), and the normalized residuals (i.e. the difference between N-body and model mass functions, divided by the uncertainty in the N-body mass function). Colors are as in the above figure.
