I have previously blogged about Structure finding in cosmological simulations and the Haloes Going Mad conference in Madrid this past spring–there you will find the basic background if you are unfamiliar with the subject and I’ll skip that in this post. The result of this conference was a Halo-finder comparison project and its findings were recent posted on the physics arxiv in the paper Haloes gone MAD: The Halo-Finder Comparison Project1. I’ll summarize them here.

The paper concentrates on comparing codes on given test data, not comparing the results of various codes to observations, provides a standard test suite and proposes a standard methodology of comparison, both on test, isolated, halos and on a simulation of cosmological volume. Shown is a figure from the paper presenting various halo finders used in the community; those in bold are included in the comparison paper. The paper divides, broadly, the algorithms into two distinct categories, those which involve some density peak locations, and those which collect particles based on “closeness” in 3D space or 6D phase space and includes a short overview of each code included as well as links to the original code papers, where relevant.

The mock data includes, as mentioned well delineated isolated mock halos, some with substructure, following standard density profiles (NFW and Plummer), as well as a cosmological simulation focusing on large-scale structure. Not all finders could participate in both isolated and cosmological comparisons due to the nature of their design, these are indicated in the respective plots. The workshop and the paper focused first on addressing the hairy question of what exactly is a halo-while the proper definition is a “gravitationally bound object” as the halo finders employ various approximations to this definition for the purposes of this paper the peak of the rotation curve $v_{\text{max}}$ and the radial location of this peak $R_{\text{max}}$ were used.

# Property comparison

The codes were then compared on their determinations of the following properties of a host halo and subhalo:

1. Center determination-Figure 2. Differences seen for the comparison types. NFW density profile deviations from actual result small, there are outliers for a few finders which are primarily for the FOF finders which use center of mass instead of density peaks to determine the center. Plummer profile deviations from actual for non-FOF finders increase and some finders are unable to locate host at all.
2. Halo bulk velocity-Figure 3. errors in this indicate contamination from particles not belonging to the halo. Error is smaller than 3% with few outliers. SKID displays very significant contamination.
3. Number of particles-Figure 4. variations in mass, hence number of particles are greater for NFW than for Plummer. There is a wide scatter in relative mass of the largest subhalo.
4. Mass-Figure 5. The density profile was calculated, then it is determined where it drops below $200 \rho_{\text{crit}}$ which is used to define $M_{200}$. Not the correct definition for subhalo mass but used for easy comparison. Differences are better as the difference in definition for the edge of a halo have been removed. There’s still a suprisingly wide variation.
5. Maximum of rotation curve-Figure 6. Subsubhalo still has not been recovered correctly in most cases.

## Static

And additionally traces the radial dependence of the determination of the following subhalo’s

1. Number of particles-Figure 7. Only phase space finders are capable of finding subhalo when it’s directly in the center (no surprise). For the NFW haloes the number of particles drops as subhalo approaches center. For Plummer sphere subhalo appears to be well recovered.
2. Maximum of rotation curve-Figure 9. The maximum of the rotation curve $v_{\text{max}}$ is a good proxy for the mass and this quantity is much less affected by the radial position of the subhalo only in the case of the centers of the two overlapping causing an issue.

## Dynamic

A subhalo dynamically infalling

1. Evolution of number of particles-Figure 10. All non-phase space finders loose subhalo and find it again as it passes through the center. All observe a drop of particles due to stripping of the subhalo but this drop can also be a result of the subhalo moving to the denser region of the host. The drop has a different shape depending on the halo finder.
2. Evolution of Maximum of rotation curve-Figure 11. Number of particles is less reliable than the maximum of the rotation curve, as the particle list must be postprocessed for many finders. For the more reliable measure of the maximum of the rotation curve, all halo finders perform equally well starting with the analytic value and dropping by the same amount once the subhalo leaves the center region. However many finders also found a jump in $v_{\text{max}}$ right after the center passage. This is indicative of problems in the unbinding procedure. This figure also shows that most halos recover the same distance for the subhalo.

# Resolution

A resolution study of subhalo

1. Number of particles-Figure 14. All halo finders perform equally well, only the HOT algorithms show deviation as they don’t have an unbinding procedure.
2. Contamination by host particles-Figure 14. Most finders don’t assign host particles to subhalo, but a few do (e.g. AdaptaHOP) esp those without an unbinding procedure.
3. Maximum of Rotation Curve-Figure 15. Except for those finders which showed contamination, all finders recover the maximum correctly if the subhalo is itself recovered.

# Cosmological simulation

In practice, halo finders are deployed on real cosmological simulations embedding in the large-scale structure of the universe. For simplicity for most the paper the finders were compared on cleaner situations but in the last section the finders were compared in their more natural application. Each finder was compared at the highest resolution analysis it was, due to performance reasons, able to perform from a range of $256^3 - 1024^3$ particles.

1. Cumulative mass function-Figure 17. No large differences. A few stray finders have problems.
2. Maximum of Rotation Curve-Figure 18. Cumulative distribution of $v_{\text{max}}$. Finders recover the correct $v_{\text{max}}$ values, apart from a flattening that is to be expected due to resolution
3. 2-point correlation function-Figure 19. Agreement for the most massive objects.
4. Bulk velocities of haloes-Figure 20. Agreement.

# Moral of the Story

18 halo finders were compared representing the state of the art in the field. Halo finders showed largely expected deviations the quantities relevant for analysis of cosmological simulations and comparison with observation were stable. The comparison illuminated and clarified limitations of particular techniques and room for improvement. Phase space finders found overlapping subhalo and host halo centers with little problem but have problems with assigning particles to the center found correctly, which is important in deriving the actual halo’s properties in the end.

Overall the comparison data set and methodology and comprehensive comparisons provide a standard to reference in future halo finder development. It’s encouraging that the most relevant properties seem stable across finders but there is still much work to be done, and the outliers in the various plots indicate which techniques might go astray  in which conditions. I have hopes this paper will spur a new wave of halo finder development.

# References

1. Alexander Knebe, Steffen R. Knollmann, Stuart I. Muldrew, Frazer R. Pearce, Miguel Angel Aragon-Calvo, Yago Ascasibar, Peter S. Behroozi, Daniel Ceverino, Stephane Colombi, Juerg Diemand, Klaus Dolag, Bridget L. Falck, Patricia Fasel, Jeff Gardner, Stefan Gottloeber, Chung-Hsing Hsu, Francesca Iannuzzi, Anatoly Klypin, Zarija Lukic, Michal Maciejewski, Cameron McBride, Mark C. Neyrinck, Susana Planelles, Doug Potter, Vicent Quilis, Yann Rasera, Justin I. Read, Paul M. Ricker, Fabrice Roy, Volker Springel, Joachim Stadel, Greg Stinson, P. M. Sutter, Victor Turchaninov, Dylan Tweed, Gustavo Yepes, & Marcel Zemp (2011). Haloes gone MAD: The Halo-Finder Comparison Project Mon. Not. R. Astron. arXiv: 1104.0949v1