Vlasov-Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory

We study analytically the collapse of an initially smooth, cold, self-gravitating collisionless system in one dimension. The system is described as a central S shape in phase-space surrounded by a nearly stationary halo acting locally like a harmonic background on the S. To resolve the dynamics of the S under its self-gravity and under the influence of the halo, we introduce a novel approach using post-collapse Lagrangian perturbation theory. This approach allows us to follow the evolution of the system between successive crossing times and to describe in an iterative way the interplay between the central S and the halo. Our theoretical predictions are checked against measurements in entropy conserving numerical simulations based on the waterbag method. While our post-collapse Lagrangian approach does not allow us to compute rigorously the long term behavior of the system, i.e. after many crossing times, it explains the close to power-law behavior of the projected density observed in numerical simulations. Pushing the model at late time suggests that the system could build at some point a very small flat core, but this is very speculative. This analysis shows that understanding the dynamics of initially cold systems requires a fine grained approach for a correct description of their very central part. The analyses performed here can certainly be extended to spherical symmetry.

Vlasov-Poisson in 1D: waterbags

We revisit in one dimension the waterbag method to solve numerically Vlasov-Poisson equations. In this approach, the phase-space distribution function $f(x,v)$ is initially sampled by an ensemble of patches, the waterbags, where $f$ is assumed to be constant. As a consequence of Liouville theorem it is only needed to follow the evolution of the border of these waterbags, which can be done by employing an orientated, self-adaptive polygon tracing isocontours of $f$. This method, which is entropy conserving in essence, is very accurate and can trace very well non linear instabilities as illustrated by specific examples. As an application of the method, we generate an ensemble of single waterbag simulations with decreasing thickness, to perform a convergence study to the cold case. Our measurements show that the system relaxes to a steady state where the gravitational potential profile is a power-law of slowly varying index $beta$, with $beta$ close to $3/2$ as found in the literature. However, detailed analysis of the properties of the gravitational potential shows that at the center, $beta > 1.54$. Moreover, our measurements are consistent with the value $beta=8/5=1.6$ that can be analytically derived by assuming that the average of the phase-space density per energy level obtained at crossing times is conserved during the mixing phase. These results are incompatible with the logarithmic slope of the projected density profile $beta-2 simeq -0.47$ obtained recently by Schulz et al. (2013) using a $N$-body technique. This sheds again strong doubts on the capability of $N$-body techniques to converge to the correct steady state expected in the continuous limit.

Cell Count Moments in the Halo Model

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 $S_k$ 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.

Accurate estimators of power spectra in N-body simulations

abridged] A method to rapidly estimate the Fourier power spectrum of a point distribution is presented. This method relies on a Taylor expansion of the trigonometric functions. It yields the Fourier modes from a number of FFTs, which is controlled by the order N of the expansion and by the dimension D of the system. In three dimensions, for the practical value N=3, the number of FFTs required is 20. We apply the method to the measurement of the power spectrum of a periodic point distribution that is a local Poisson realization of an underlying stationary field. We derive explicit analytic expression for the spectrum, which allows us to quantify--and correct for--the biases induced by discreteness and by the truncation of the Taylor expansion, and to bound the unknown effects of aliasing of the power spectrum. We show that these aliasing effects decrease rapidly with the order N. The only remaining significant source of errors is reduced to the unavoidable cosmic/sample variance due to the finite size of the sample. The analytical calculations are successfully checked against a cosmological N-body experiment. We also consider the initial conditions of this simulation, which correspond to a perturbed grid. This allows us to test a case where the local Poisson assumption is incorrect. Even in that extreme situation, the third-order Fourier-Taylor estimator behaves well. We also show how to reach arbitrarily large dynamic range in Fourier space (i.e., high wavenumber), while keeping statistical errors in control, by appropriately folding the particle distribution.

Cosmic velocity--gravity relation in redshift space

We propose a simple way to estimate the parameter beta = Omega_m^(0.6)/b from three-dimensional galaxy surveys. Our method consists in measuring the relation between the cosmological velocity and gravity fields, and thus requires peculiar velocity measurements. The relation is measured *directly in redshift space*, so there is no need to reconstruct the density field in real space. In linear theory, the radial components of the gravity and velocity fields in redshift space are expected to be tightly correlated, with a slope given, in the distant observer approximation, by g / v = (1 + 6 beta / 5 + 3 beta^2 / 7)^(1/2) / beta. We test extensively this relation using controlled numerical experiments based on a cosmological N-body simulation. To perform the measurements, we propose a new and rather simple adaptive interpolation scheme to estimate the velocity and the gravity field on a grid. One of the most striking results is that nonlinear effects, including `fingers of God, affect mainly the tails of the joint probability distribution function (PDF) of the velocity and gravity field: the 1--1.5 sigma region around the maximum of the PDF is *dominated by the linear theory regime*, both in real and redshift space. This is understood explicitly by using the spherical collapse model as a proxy of nonlinear dynamics. Applications of the method to real galaxy catalogs are discussed, including a preliminary investigation on homogeneous (volume limited) `galaxy samples extracted from the simulation with simple prescriptions based on halo and sub-structure identification, to quantify the effects of the bias between the galaxy and the total matter distibution, and of shot noise (ABRIDGED).