Measuring Galaxy Clustering with Fewer Simulated Merger Trees

Why clustering is expensive

Galaxies populate dark matter haloes, whose growth and mergers influence the galaxies they contain. Semi-analytic models follow that history through merger trees, using prescriptions for processes such as gas cooling, star formation and feedback.

Clustering provides a test of those prescriptions. The two-point correlation function measures how often galaxy pairs occur at a given separation compared with a random distribution. But using it to tune a model can require producing thousands of galaxy catalogues, one for each combination of model parameters.

In their preprint, Hickman, Sownak Bose, Kai Wang, Carlton M. Baugh and Baojiu Li describe the Sparse and Unbiased Clustering Estimator, or SaUCE. It reduces the number of merger trees that must be populated with galaxies. It does not remove the cost of the underlying dark matter simulation.

Why ordinary subsampling goes wrong

The approach exploits the fact that standard semi-analytic models process merger trees independently. The authors divide those trees into groups called partitions, then run the galaxy model on only a randomly chosen subset.

In their tests, each partition contains trees scattered throughout the simulation volume, rather than occupying one small region. This lets a sparse selection sample separations across the full box.

However, selecting whole trees is not equivalent to selecting individual galaxies randomly. Galaxies sharing a halo are retained or discarded together. A pair inside one halo survives whenever that halo is selected; a pair spanning different partitions survives only when both partitions are selected.

Without a correction, this imbalance exaggerates small-scale clustering. When eight of 1,024 partitions are used, the authors show that the naive estimate overstates clustering by about a factor of 128 where pairs within individual haloes dominate.

A correction based on selection probabilities

SaUCE separates pairs according to whether their galaxies belong to the same partition or different partitions. It weights these two classes using their known probabilities of selection. The weights depend only on the number of partitions available and the number selected, with no fitted parameters.

The authors derive exactly unbiased corrected pair counts, after accounting for the overall sampling fraction. Here, “unbiased” means correct on average across random selections, not exact for every sparse catalogue.

There is a further qualification: the correlation function divides those counts by a normalisation calculated from the selected galaxy population. Because that population fluctuates, the correlation function itself is unbiased only to leading order. The authors’ numerical checks find its mean offset consistent with zero.

What the tests recover

The team tested two versions of the GALFORM galaxy formation model on three dark matter simulations, comparing sparse estimates with catalogues generated from every partition.

Their headline result comes from P-Millennium at redshift zero, for a sample with number density 0.01 galaxies per cubic h-scaled megaparsec. Processing 75 of 1,024 partitions—7.3%—met their accuracy criterion across separations of 0.03–30 h-scaled megaparsecs. That represents roughly fourteen times fewer trees to process.

The criterion allowed a deviation of 0.05 dex, approximately 12%. At least 68% of random selections fell within that tolerance in each separation bin. This is not a guarantee that 68% matched every bin simultaneously; the authors explicitly distinguish those statements.

More demanding samples required more computation. At redshift about 1.5 and a tenfold lower number density, the required fraction rose to 29%. For the sparsest sample, matching the full separation range required about 96%, leaving almost no saving.

Where the savings have limits

The two model variants produced similar sampling scatter when matched in number density, but they also populated haloes similarly. The authors caution that models with substantially different halo populations could behave differently.

The method also requires independent tree processing and intact merger histories. Processes that couple separate trees can break that assumption. Spatially grouped partitions remain usable, but produce greater scatter, so the reported sampling fractions cannot simply be transferred to them.

Errors at different separations are strongly correlated: selecting a particular group of haloes changes many pair counts together. Model fitting must therefore use the full covariance, not treat each bin as independent. Observational uncertainty and variation between simulation realisations must be included separately.

The preprint establishes a route to cheaper clustering calculations, not a universal precision guarantee. Its practical value depends on the galaxy sample, scales and tolerance required—especially when distinguishing models whose predictions differ by less than the study’s 12% benchmark.