Measuring the HMF: halo finders (FOF vs SO)

To measure a mass function from a simulation you must first decide which particles form a halo — and different rules give different answers, especially for fuzzy dark matter's filament-linked halos. This article covers the two standard finders and the systematic they introduce.

Two definitions of a halo

Halos are not sharply bounded, so ‘halo’ is a matter of definition. Friends-of-friends (FOF) links any two particles closer than a chosen linking length and calls each connected group a halo. Spherical overdensity (SO) grows a sphere around a density peak until the enclosed mean density drops to a threshold (e.g. $200\bar\rho$). They agree for isolated halos but diverge at the low-mass end and for bridged systems (figure).

The FDM complication

May & Springel (2023) point out that FDM halos are strung together by dense, smooth filaments — unlike the clumpier $\Lambda$CDM web — which makes halo identification genuinely harder and more finder-dependent. FOF can join two clumps linked by such a filament into one object; SO enforces a spherical boundary.

Two halo definitions: FOF (left) links neighbouring particles and can bridge clumps; SO (right) grows spheres to a density threshold. They disagree most at the low-mass end — the origin of the finder systematic.

The systematic in our comparison

Worked example — the finder caveat on our head-to-head

Our GADGET-4 control uses FOF (91,058 halos at $z=6$); our GAMER FDM census uses SO (292 halos). Comparing the two codes at the low-mass end therefore carries a finder systematic that can reach tens of percent. The honest statement — the one we make — is to compare each code against its own theory, not directly against each other, and to flag the cross-code low-mass ratio as finder-dependent.

Why this matters for blind science

Naming this systematic is part of the campaign's discipline: rather than quote a clean-looking but finder-contaminated cross-code ratio, we state the caveat and lean on the each-vs-own-theory comparison. It is the difference between an honest measurement and a misleading one.

FOF and SO halo finders diverge at the low-mass end — the systematic behind our each-vs-own-theory comparison.

In our research

This is exactly the caveat on our sim-vs-theory figure (Topic 7.2): GADGET-4 (FOF) and GAMER (SO) are each shown against their own theory, and the ‘finder systematic’ note on that figure is this issue. It is also why the filament-linked FDM web (May & Springel) makes FDM halo-finding a live research problem.

Key references

Davis et al. (1985), The evolution of large-scale structure (FOF), ApJ 292, 371.
Knebe et al. (2013), Halo-finder comparison, MNRAS 435, 1618.
May & Springel (2023), MNRAS 524, 4256 (arXiv:2209.14886).

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