Numerical estimation of densities

We present a novel technique, dubbed FiEstAS , to estimate the underlying density field from a discrete set of sample points in an arbitrary multidimensional space. FiEstAS assigns a volume to each point by means of a binary tree. Density is then computed by integrating over an adaptive kernel. As a first test, we construct several Monte Carlo realizations of a Hernquist profile and recover the particle density in both real and phase space. At a given point, Poisson noise causes the unsmoothed estimates to fluctuate by a factor of ∼2 regardless of the number of particles. This spread can be reduced to about 1 dex (∼26 per cent) by our smoothing procedure. The density range over which the estimates are unbiased widens as the particle number increases. Our tests show that real‐space densities obtained with an SPH kernel are significantly more biased than those yielded by FiEstAS . In phase space, about 10 times more particles are required in order to achieve a similar accuracy. As a second application we have estimated phase‐space densities in a dark matter halo from a cosmological simulation. We confirm the results of Arad, Dekel & Klypin that the highest values of f are all associated with substructure rather than the main halo, and that the volume function v ( f ) ∼ f −2.5 over about four orders of magnitude in f . We show that a modified version of the toy model proposed by Arad et al. explains this result and suggests that the departures of v (  f ) from power‐law form are not mere numerical artefacts. We conclude that our algorithm accurately measures the phase‐space density up to the limit where discreteness effects render the simulation itself unreliable. Computationally, FiEstAS is orders of magnitude faster than the method based on Delaunay tessellation that Arad et al. employed, making it practicable to recover smoothed density estimates for sets of 10 9 points in six dimensions.