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The Friends of Friends Method


In the friends of friends (FOF) method of halo finding [5], one specifies a linking length, tex2html_wrap_inline1972, and identifies all pairs of particles with a separation of tex2html_wrap_inline1972 or less. Such pairs are designated friends, and halos are defined as sets of particles that are connected by one or more friendship relations, i.e., friends of friends. The linking length is usefully parameterized as a density, and following [12], we define a density threshold:
where tex2html_wrap_inline1982 is the average particle mass in the simulation. That is, tex2html_wrap_inline1984 is the density of a sphere of radius tex2html_wrap_inline1972 containing two average mass particles. In regions where the density is greater than tex2html_wrap_inline1984, particles will tend to be closer together than tex2html_wrap_inline1972, and will be linked together in the FOF method. Notice that the mechanics of the FOF method do not involve density, but the effect of the method is to identify density peaks above the threshold tex2html_wrap_inline1984.

A second parameter in FOF is the minimum number of particles, tex2html_wrap_inline1958, in a halo. The purpose of tex2html_wrap_inline1958 is to reject spurious halos -- i.e., groups of friends that do not form persistent objects in the simulation. Chance associations involving larger numbers of particles are less likely than those involving fewer, so by setting tex2html_wrap_inline1958 sufficiently large one hopes to avoid most spurious halos.

Figures 1 and 2 show two sets of results, using tex2html_wrap_inline1958 = 10 and tex2html_wrap_inline1958 = 30, from FOF on Model 3.

Figure: Particles in Model 3, and the central halo found by FOF in this model. These panels illustrate the difficulty of graphically representing data that samples a six-dimensional space (three position and three velocity components). In each panel the horizontal and vertical axes each represent one degree of freedom, and each particle contributes a single dot. In all panels, the horizontal axis is an arbitrary spatial coordinate (in this case, the y-coordinate). The panels on the left show all of the particles in the model; those on the right show the particles in the single most massive halo found by FOF. The upper panels show particle positions, projected into the spatial y-z plane. The middle panels show the same spatial coordinate and one velocity coordinate. The lower panels show the density (as calculated by IsoDen according to equation 2; see Section 4) and the same spatial coordinate. The lower panels emphasize the spatial variation and complex hierarchy of densities. It is clear from the lower right panel that the FOF method has failed to identify some of the substructure in the model, combining several distinct halos into a single halo. The horizontal line in the lower panels indicates the value of tex2html_wrap_inline1984 corresponding to the tex2html_wrap_inline1972 used for FOF.

Figure: As Figure 1, showing halos other than the central halo, as found by FOF with tex2html_wrap_inline1958 = 10 (on the left) and tex2html_wrap_inline1958 = 30 (on the right). The number of halos in each case is indicated at the top left. The large spread in the velocity coordinate in the middle panels indicates that even though some groupings of particles have high spatial density, they may not be gravitationally bound, and hence do not constitute a persistent object in the simulation. Thus, the middle panels suggest that many of the selected halos on the left, and at least one of those on the right are spurious. The lower and upper panels, however, indicate likely structures on the left (identifiable with tex2html_wrap_inline1958 = 10), which are rejected by the choice of tex2html_wrap_inline1958 = 30 on the right. Thus, neither value of tex2html_wrap_inline1958 is entirely satisfactory.

These figures demonstrate two problems with FOF: joining halos together, and poor distinction of small halos from noise. The first problem is that at the center of the cluster, FOF finds one large halo which is clearly composed of several distinct halos. This is because everything in a region where the density is above tex2html_wrap_inline1984 is joined into a single halo, whether or not the region includes objects which are distinct at some higher density -- a problem first noted by [3]. From the density plots it is seen that there is no value of tex2html_wrap_inline1984 which will distinguish the halos in the high density region without missing some of the lower density halos.

The second problem is the arbitrariness and ineffectiveness of the tex2html_wrap_inline1958 parameter. In our test, the value of tex2html_wrap_inline1958 = 10 is too small, since many of the small halos found have high internal velocity scatter, and hence are not bound (c.f. figure 2 left middle panel) At a higher tex2html_wrap_inline1958 of 30, (figure 2 right panels) most of the spurious halos are rejected, but one remains, and in addition several real (but small) halos have been rejected.

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Next: Rejecting halos based on Up: Halo World: Tools for Previous: Clustered PCs for Space

John Salmon
Sat Sep 27 18:44:36 PDT 1997