A count probability cookbok: Spurious effects and the scaling model
S. Colombi, François R. Bouchet, Richard W. Schaeffer · The Astrophysical Journal Supplement Series · 1995
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 & Schaeffer scaling model, which assumes that the Q-body correlation functions obey the scaling relation ξ_Q_(λr_1_,...,λr_Q_) = λ^-(Q- 1)γxi_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 ~3 x 10^4^ objects, a sample dominated by a spherical power-law cluster with ~3 x 10^4^ objects and a cold dark matter (CDM) universe involving ~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^ ~> 1, where V is 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^ ~> 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 ~> 0.02-0.03 h^3^ Mpc^-3^. The count probability distribution for N not equal to 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^ ~> 80-120 is required, or equivalently n ~> 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.