In any local spatial region of the simulation there will be statistical fluctuations in the number of particles. These fluctuations lead to errors in the calculated density field (relative to the hypothetical underlying mass distribution being sampled by the N-body particles). Such errors will result in ``noise peaks'', and so as with FOF, IsoDen requires a method for rejecting spurious halos. The evaporative method discussed for FOF is effective for this purpose, but requires considerable computation. As an alternative, we describe a simple statistical method which is unique to IsoDen and quite effective.
The statistical method requires that one be able to calculate the statistical uncertainty of the density estimate at each particle. For the kernel density estimation described above, the uncertainty can be estimated by assuming that the underlying density distribution is roughly uniform on scales that contain particles, and that the particle positions are sampled at random from this density field. Then the uncertainty in the density is just due to Poisson noise, and the statistical uncertainty, , is simply . That is, the relative uncertainty, , is a constant, . This is an important feature of nearest neighbor density estimation: in high density regions the spatial resolution is improved (i.e. smaller ) while maintaining constant relative uncertainty in the density estimates.
When each new halo is created, the halo is designated tentative,
and the particle that creates it defines the halo's
central (peak) density, .
Tentative halos will become either genuine halos, or will be
eliminated based on a statistical criterion.
When a tentative halo overlaps with another halo we apply a somewhat
ad hoc criterion akin to a statistical significance test.
, with , the density at which
the overlap is detected, plus , the statistical
uncertainty at the overlap density: i.e.,
we accept the peak as genuine. Otherwise it is rejected. Since the probability distribution of is somewhat difficult to define, we cannot precisely define the significance of this test. Empirically, we find that ``three sigma'' peaks, i.e., n=3 are almost always genuine in the sense that they pass the physically motivated evaporative test.
If a tentative halo passes this test, it becomes genuine and is recorded as an independent object (a leaf of the halo-tree) which is contained within the larger composite object that is created by the overlap. If it fails the test, the tentative halo is rejected. In either case, all particles in the overlapping halos are renumbered with the new composite halo-number. A composite halo is genuine if and only if any of its component halos are genuine.