A COUNT PROBABILITY COOKBOOK - SPURIOUS EFFECTS AND THE SCALING MODEL

We study the errors brought by finite volume effects and dilution effects on the practical determination of the count probability distribution function P-N(n,l), which is the probability of having N objects in a cell of volume l(3) for a set of average number density n. Dilution effects are particularly relevant to the so-called sparse sampling strategy. This work is mainly done in the framework of the Balian and Schaeffer scaling model, which assumes that the e-body correlation functions obey the scaling relation xi(Q)(lambda r(1),...,lambda r(Q)) = lambda (-(Q-1)gamma)xi(Q)(r(1),...,r(Q)). We use three synthetic samples as references to perform our analysis: a fractal generated by a Rayleigh-Levy random walk with similar to 3 X 10(4) objects, a sample dominated by a spherical power-law cluster with similar to 3 X 10(4) objects and a cold dark matter (CDM) universe involving similar to 3 X 10(5) matter particles. The void probability, P-0, is seen to be quite weakly sensitive to finite sample effects, if P(0)Vl(-3)greater than or similar to 1, where Vis the volume of the sample (but P-0 is not immune to spurious grid effects in the case of numerical simulations from such quiet initial conditions). If this condition is met, the scaling model can be tested with a high degree of accuracy. Still, the most interesting regime, when the scaling predictions are quite unambiguous, is reached only when nl(0)(3) greater than or similar to 30-50, where l(0) is the (pseudo-)correlation length at which the averaged two-body correlation function over a cell is unity. For the galaxy distribution, this corresponds to n greater than or similar to 0.02-0.03 h(3)Mpc(-3). The count probability distribution for N not equal 0 is quite sensitive to discreteness effects. Furthermore, the measured large N tail appears increasingly irregular with N, until a sharp cutoff is reached. These wiggles and the cutoff are finite volume effects. It is still possible to use the measurements to test the scaling model properties with a good accuracy, but the sample has to be as dense and large as possible. Indeed the condition nl(0)(3) greater than or similar to 80-120 is required, or equivalently n greater than or similar to 0.04-0.06 h(3) Mpc(-3). The number densities of the current three-dimensional galaxy catalogs are thus not large enough to test fairly the predictions of the scaling model. Of course, these results strongly argue against sparse sampling strategies.