We measure the scaling properties of the probability distribution of the smoothed density eld in N -body simulations of expanding universes with scale-free initial power-spectra, hj k j 2 i / k n , with particular attention to the predictions of the stable clustering hypothesis.We concentrate our analysis on the ratios S Q (`) Q = Q 1 2 , where Q is the averaged Q-body correlation function over a cell of radius`. According to the stable clustering hypothesis, S Q should not depend on scale. We measure directly the functions S Q (`) for Q 5. The behavior of the higher order correlations is studied through that of the void probability distribution, P 0 , which is the probability of nding an empty cell of radius`. If the stable clustering hypothesis applies, the function P 0 should also exhibit remarkable scaling properties.In our analysis, we carefully account for various spurious e ects, such as initial grid contamination, loss of dynamics due to the short range softening of the forces, and nite volume size of our simulations. Only after correcting for the latter do we nd agreement of the measured S Q , 3 Q 5 with the expected self-similar solution S Q (`; t) = S Q ( 2 ) = S Q (`=`0(t)), 0 (t) / t 4=(9+3n) . The void probability is only weakly sensitive to such defects and closely follows the expected self-similar behavior.As functions of 2 , the quantities S Q , 3 Q 5, exhibit two plateaus separated by a smooth transition around 2 1. In the weakly nonlinear regime, 2 < 1, the results are in reasonable agreement with the predictions of perturbation theory. In the nonlinear regime, 2 > 1, the function S Q ( 2 ) is larger than in the weakly nonlinear regime, and increasingly so with n. It is well-tted by the expression S Q = ( 2 =100) 0:045(Q 2) e S Q for all n. This weak dependence on scale proves a small, but signi cant departure from the stable clustering predictions at least for n = 0 and n = +1. It is thus also at variance with the predictions of the hierarchical model.The analysis of P 0 con rms that the expected scale-invariance of the functions S Q is not exactly attained in the part of the nonlinear regime we probe, except possibly for n = 2 and marginally for n = 1. In these two cases, our measurements are not accurate enough to be discriminant. On the other hand, we could demonstrate that the observed power-law behavior of S Q cannot be generalized as such to arbitrary order in Q. Indeed this would induce scaling properties of P 0 incompatible with those measured.

We study the errors brought by nite volume e ects and dilution e ects on the practical determination of the count probability distribution function P N (n;`), which is the probability of having N objects in a cell of volume`3 for a set of average number density n. Dilution e ects are particularly relevant to the so-called sparse sampling strategy. This work is mainly done in the framework of the scaling model (Balian & Schae er 1989), which assumes that the Qbody correlation functions obey the scaling relation Q ( r 1 ; :::; r Q ) = (Q 1) N (r 1 ; :::; r Q ).We use three synthetic samples as references to perform our analysis: a fractal generated by a Rayleigh-L evy random walk with 3:10 4 objects, a sample dominated by a spherical power-law cluster with 3:10 4 objects and a cold dark matter (CDM) universe involving 3:10 5 matter particles.The void probability, P 0 , is seen to be quite weakly sensitive to nite sample e ects, if P 0 V` 3 > 1, where V is the volume of the sample (but P 0 is not immune to spurious grid e ects 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 n`3 0 > 30 50, where`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 0:03h 3 Mpc 3 .The count probability distribution for N 6 = 0 is quite sensitive to discreteness e ects. Furthermore, the measured large N tail appears increasingly irregular with N , till a sharp cuto is reached. These wiggles and the cuto are nite volume e ects. 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 n`3 0 > 80 120 is required, or equivalently n > 0:04 0:06h 3 Mpc 3 . The number densities of the current three dimensional galaxy catalogues are thus not large enough to test fairly the predictions of the scaling model. Of course, these results strongly argue against sparse sampling strategies.

I propose a method to t the probability distribution function (hereafter PDF) of the large scale density eld , motivated by a Lagrangian version of the continuity equation. It consists in applying the Edgeworth expansion to the quantity log hlog i. The method is tested on the matter particle distribution in two cold dark matter N -body simulations of di erent physical sizes to cover a large dynamic range. It is seen to be very e cient, even in the non-linear regime, and may thus be used as an analytical tool to study the e ect on the PDF of the transition between the weakly non-linear regime and the highly non-linear regime.Subject headings: cosmology: large-scale structure of universe { galaxies: clustering 1

We present an analytical calculation of the extreme value statistics for dark matter haloes – i.e., the probability distribution of the most massive halo within some region of the universe of specified shape and size. Our calculation makes use of the counts‐in‐cells formalism for the correlation functions, and the halo bias derived from the Sheth–Tormen mass function. We demonstrate the power of the method on spherical regions, comparing the results to measurements in a large cosmological dark matter simulation and achieving good agreement. Particularly good fits are obtained for the most likely value of the maximum mass and for the high‐mass tail of the distribution, relevant in constraining cosmologies by observations of most massive clusters.

We study cell count moments up to fifth order of the distributions of haloes, of halo substructures as a proxy for galaxies, and of mass in the context of the halo model and compare theoretical predictions to the results of numerical simulations. On scales larger than the size of the largest cluster, we present a simple point cluster model in which results depend only on cluster–cluster correlations and on the distribution of the number of objects within a cluster, or cluster occupancy. The point cluster model leads to expressions for moments of galaxy counts in which the volume‐averaged moments on large scales approach those of the halo distribution and on smaller scales exhibit hierarchical clustering with amplitudes Sk determined by moments of the occupancy distribution. In this limit, the halo model predictions are purely combinatoric, and have no dependence on halo profile, concentration parameter or potential asphericity. The full halo model introduces only two additional effects: on large scales, haloes of different mass have different clustering strengths, introducing relative bias parameters; and on the smallest scales, halo structure is resolved and details of the halo profile become important, introducing shape‐dependent form factors. Because of differences between discrete and continuous statistics, the hierarchical amplitudes for galaxies and for mass behave differently on small scales even if galaxy number is exactly proportional to mass, a difference that is not necessarily well described in terms of bias.