Gravitationally Consistent HALO Catalogs and Merger Trees for Precision Cosmology Page: 4 of 16
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BEHROOZI ET AL
easy parallel analysis. For each group, particle positions and
velocities are normalized by the group position and veloc-
ity dispersions, giving a natural phase-space metric. Then,
the algorithm adaptively chooses a phase-space linking length
such that 70% of the group's particles are linked together
into subgroups. This process repeats for each subgroup-
renormalization, a new linking-length, and a new level of
substructure calculated-until a full hierarchy of particle sub-
groups is created. Seed halos are then placed in the dens-
est subgroups, and particles are assigned hierarchically to the
closest seed halo in phase space (see Behroozi et al. 2011 for
full details). Finally, once particles have been assigned to ha-
los, unbound particles are removed and halo properties (posi-
tions, velocities, spherical masses, radii, spins, etc.) are cal-
culated. If halos at the previous snapshot are available, they
are used to determine the host halo / subhalo relationships in
cases (such as major mergers) where they are ambiguous.
3.4. The BDM Halo Finder
The basic technique of the BDM halo finder is described
in Klypin & Holtzman (1997); a more detailed description
is given in Riebe et al. (2011), and tests and comparisons
with other codes are presented in Knebe et al. (2011). The
code uses a spherical 3D overdensity algorithm to identify
halos and subhalos. It starts by finding the density for each
individual particle; the density is defined using a top-hat fil-
ter with a given number of particles Nfilter, which typically is
Nfilter = 20. The code finds all density maxima, and for each
maximum it finds a sphere containing a given overdensity
mass MA = (47r/3)ApeR3 , where pcr is the critical density
of the Universe and A is the specified overdensity.
Among all overlapping spheres the code finds the one that
has the deepest gravitational potential. The density maximum
corresponding to this sphere is treated as the center of a dis-
tinct halo. Thus, by construction, a center of a distinct halo
cannot be inside the radius of another one. However, periph-
eral regions can still partially overlap, if the distance between
centers is less than the sum of halo radii. The radius and mass
of a distinct halo depend on whether the halo overlaps or not
with other distinct halos. The code takes the largest halo and
identifies all other distinct halos inside a spherical shell with
distances R = (1 - 2)Rcenter from the large host halo, where
Reenter is the radius of the largest halo. For each halo selected
within this shell, the code finds two radii. The first is the dis-
tance Rbig to the surface of the large halo: Rbig = R -Reenter.
The second is the distance Rmax to the nearest density max-
imum in the shell with the inner radius min(Rbig,RA) and
the outer radius max(Rbig, RA) from the center of the selected
halo. If there are no density maxima within that range, then
Rmax = RA. The radius of the selected halo is the maximum
of Rbig and Rmax. Once all halos around the large halo are
processed, the next largest halo is taken from the list of dis-
tinct halos and the procedure is applied again. This setup is
designed to make a smooth transition of properties of small
halos when they fall into a larger halo and become subhalos.
The bulk velocity of either a distinct halo or a subhalo is
defined as the average velocity of the 100 most bound parti-
cles of that halo or by all particles, if the number of particles
is less than 100. The number 100 is a compromise between
the desire to use only the central (sub)halo region for the bulk
velocity and the noise level.
The gravitational potential is found by first finding the mass
in spherical shells and then by integration of the mass profile.
The binning is done in log radius with a very small bin size ofAlog(R) = 0.01.
Centers of subhalos can only be found among density max-
ima, but not all density maxima are subhalos. An important
construct for finding subhalos are barrier points: a subhalo ra-
dius cannot be larger than the distance to the nearest barrier
point times a numerical tuning factor called an overshoot fac-
tor foeer ~ 1.1-1.5. The subhalo radius can be smaller than
this distance. Barrier points are centers of previously identi-
fied (sub)halos. For the first subhalo, the barrier point is the
center of the distinct halo. For the second subhalo, it is the
first barrier point and the center of the first subhalo, and so on.
The radius of a subhalo is the minimum of (a) the distance to
the nearest barrier point times foVer and (b) the distance to its
most remote bound particle.
3.5. Particle-Based Merger Trees
As part of the algorithm process for generating gravita-
tionally consistent trees, we computed simple particle-based
merger trees for both Bolshoi and Consuelo. The algorithm
assigns a descendant to a halo based on which halo at the
next timestep receives the largest fraction of the halo's par-
ticles (excluding substructure). In principle, this method is
sufficient to correctly predict the vast majority of halo descen-
dants (although of course it cannot identify cases where a halo
should never have existed, or where a halo was missed). An
algorithm based on a fixed fraction of the most-bound parti-
cles would be superior in cases where a subhalo is undergoing
rapid stripping (e.g., when it loses more than half of its par-
ticles to its host); however, the algorithm we present in this
paper can very effectively correct for such cases.
Indeed, for the BDM analysis of Bolshoi, only the 250 most-
bound particles were available; as such, we created initial
merger trees based on only these 250 particles for each halo.
For small halos (<2000 particles), this approach worked well;
however, there were more issues for large halos and halos un-
dergoing major mergers. Specifically, the use of incomplete
particle information resulted in a small fraction (~ 0.2-0.5%
between timesteps) of halos without descendants and a large
fraction (10%) of spurious links (see 5.3). By comparison,
the fraction of spurious links in the ROCKSTAR particle trees
was between 1-3%, depending on redshift. The initial BDM
tree thus represents a more challenging set of initial condi-
tions, but it serves as an ideal proving ground for the efficacy
of our algorithm.
4. GRAVITATIONAL HALO EVOLUTION EQUATIONS
4.1. Overview
To solve the problems identified in 2.2, it is necessary to
enforce consistency in the halo catalog across timesteps. No
matter how well-written the halo finder is, there will always
be halos which (for example) cross the threshold of detection
in one timestep and then disappear in the next. It is impossi-
ble to tell whether those cases are statistical fluctuations or not
based only on the information available at a single timestep-
otherwise, presumably, appropriate logic could be added to
the halo finder to account for them. Thus, the presence or ab-
sence of a halo in adjacent timesteps lends otherwise unavail-
able evidence which helps determine whether the halo should
be present in the current timestep.
Knowing the positions, velocities, and mass profiles of ha-
los at one timestep, we may use the laws of gravity and inertia
to predict their properties at adjacent timesteps. By compar-
ing the predicted halo catalogs with the actual ones, and by4
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Behroozi, Peter S.; Wechsler, Risa H.; Wu, Hao-Yi; Busha, Michael T.; Klypin, Anatoly A. & Primack, Joel R. Gravitationally Consistent HALO Catalogs and Merger Trees for Precision Cosmology, article, December 21, 2012; United States. (https://digital.library.unt.edu/ark:/67531/metadc835861/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.