ColDICE: a parallel Vlasov-Poisson solver using moving adaptive simplicial tessellation

Resolving numerically Vlasov-Poisson equations for initially cold systems can be reduced to following the evolution of a three-dimensional sheet evolving in six-dimensional phase-space. We describe a public parallel numerical algorithm consisting in representing the phase-space sheet with a conforming, self-adaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six- and four-dimensional phase-space. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase-space sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincar\'e invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli (1993) generalised to linear order. As preliminary tests of the code, we study in four dimensional phase-space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves. We also perform a "warm" dark matter simulation in six-dimensional phase-space that we use to check the parallel scaling of the code.Comment: Code and illustration movies available at: http://www.vlasix.org/index.php?n=Main.ColDICE - Article submitted to Journal of Computational Physic

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).Comment: 24 pages, 10 figures. Matches the version accepted for publication in MNRAS. The definitive version is available at http://www.blackwell-synergy.co

Cosmic Error and the Statistics of Large Scale Structure

We examine the errors on counts in cells extracted from galaxy surveys. The measurement error, related to the finite number of sampling cells, is disentangled from the ``cosmic error'', due to the finiteness of the survey. Using the hierarchical model and assuming locally Poisson behavior, we identified three contributions to the cosmic error: The finite volume effect is proportional to the average of the two-point correlation function over the whole survey. It accounts for possible fluctuations of the density field at scales larger than the sample size. The edge effect is related to the geometry of the survey. It accounts for the fact that objects near the boundary carry less statistical weight than those further away from it. The discreteness effect is due to the fact that the underlying smooth random field is sampled with finite number of objects. This is the ``shot noise'' error. Measurements of errors in artificial hierarchical samples showed excellent agreement with our predictions. The probability distribution of errors is increasingly skewed when the order $N$ and/or the cell size increases. The Gaussian approximation is valid only in the weakly non-linear regime, otherwise it severely underestimates the true errors. We study the concept of ``number of statistically independent cells'' This number is found to depend highly on the statistical object under study and is generally quite different from the number of cells needed to cover the survey volume. In light of these findings, we advocate high oversampling for measurements of counts in cells.Comment: 28 pages, ps with figures included, except figure 5, which is available at http://fnas08.fnal.gov/Publication

Most massive halos with Gumbel Statistics

We present an analytical calculation of the extreme value statistics for dark matter halos - that is, 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.Comment: Accepted to MNRA

The gravitational force field of proto-pancakes

It is well known that the first structures that form from small fluctuations in a self-gravitating, collisionless and initially smooth cold dark matter (CDM) fluid are pancakes. We study the gravitational force generated by such pancakes just after shell-crossing, and find a simple analytical formula for the force along the collapse direction, which can be applied to both the single- and multi-stream regimes. The formula is tested on the early growth of CDM protohaloes seeded by two or three crossed sine waves. Adopting the high-order Lagrangian perturbation theory (LPT) solution as a proxy for the dynamics, we confirm that our analytical prediction agrees well with the exact solution computed by direct resolution of the Poisson equation, as long as the caustic structure remains locally sufficiently one-dimensional. These results are further confirmed by comparisons of the LPT predictions performed this way to measurements in Vlasov simulations performed with the public code ColDICE. We also show that the component of the force orthogonal to the collapse direction preserves its single stream nature by not changing qualitatively before and after the collapse, allowing sufficiently high-order LPT acceleration to be used to approximate it accurately as long as the LPT series converges. As expected, solving Poisson equation on the density field generated with LPT displacement provides a more accurate force than the LPT acceleration itself, as a direct consequence of the faster convergence of the LPT series for the positions than for the accelerations. This may provide a clue on improving standard LPT predictions. Our investigations represent a very needed first step to study analytically gravitational dynamics in the multi-stream regime, by estimating, at leading order in time and space the proper backreaction on the gravitational field inside the pancakes.Comment: 16 pages, 9 figure

Fast and accurate collapse-time predictions for collisionless matter

We consider the gravitational collapse of collisionless matter seeded by three crossed sine waves with various amplitudes, also in the presence of a linear external tidal field. We explore two theoretical methods that are more efficient than standard Lagrangian perturbation theory (LPT) for resolving shell-crossings, the crossing of particle trajectories. One of the methods completes the truncated LPT series for the displacement field far into the UV regime, thereby exponentially accelerating its convergence while at the same time removing pathological behavior of LPT observed in void regions. The other method exploits normal-form techniques known from catastrophe theory, which amounts here to replacing the sine-wave initial data by its second-order Taylor expansion in space at shell-crossing location. This replacement leads to a speed-up in determining the displacement field by several orders of magnitudes, while still achieving permille-level accuracy in the prediction of the shell-crossing time. The two methods can be used independently, but the overall best performance is achieved when combining them. Lastly, we find accurate formulas for the nonlinear density and for the triaxial evolution of the fluid in the fundamental coordinate system, as well as report a newly established exact correspondence between perfectly symmetric sine-wave collapse and spherical collapse.Comment: 30 pages, 24 figures, v2: fixed minor notational inconsistency in equation 2.8, optimised colour scale for figure 1

Cosmic Statistics of Statistics

The errors on statistics measured in finite galaxy catalogs are exhaustively investigated. The theory of errors on factorial moments by Szapudi & Colombi (1996) is applied to cumulants via a series expansion method. All results are subsequently extended to the weakly non-linear regime. Together with previous investigations this yields an analytic theory of the errors for moments and connected moments of counts in cells from highly nonlinear to weakly nonlinear scales. The final analytic formulae representing the full theory are explicit but somewhat complicated. Therefore as a companion to this paper we supply a FORTRAN program capable of calculating the described quantities numerically (abridged).Comment: 18 pages, 9 figures, Latex (MN format), published in MNRAS 310, 428 with slight correction

LyMAS: Predicting Large-Scale Lyman-alpha Forest Statistics from the Dark Matter Density Field

[abridged] We describe LyMAS (Ly-alpha Mass Association Scheme), a method of predicting clustering statistics in the Ly-alpha forest on large scales from moderate resolution simulations of the dark matter distribution, with calibration from high-resolution hydrodynamic simulations of smaller volumes. We use the "Horizon MareNostrum" simulation, a 50 Mpc/h comoving volume evolved with the adaptive mesh hydrodynamic code RAMSES, to compute the conditional probability distribution P(F_s|delta_s) of the transmitted flux F_s, smoothed (1-dimensionally) over the spectral resolution scale, on the dark matter density contrast delta_s, smoothed (3-dimensionally) over a similar scale. In this study we adopt the spectral resolution of the SDSS-III BOSS at z=2.5, and we find optimal results for a dark matter smoothing length sigma=0.3 Mpc/h (comoving). In extended form, LyMAS exactly reproduces both the 1-dimensional power spectrum and 1-point flux distribution of the hydro simulation spectra. Applied to the MareNostrum dark matter field, LyMAS accurately predicts the 2-point conditional flux distribution and flux correlation function of the full hydro simulation for transverse sightline separations as small as 1 Mpc/h, including redshift-space distortion effects. It is substantially more accurate than a deterministic density-flux mapping ("Fluctuating Gunn-Peterson Approximation"), often used for large volume simulations of the forest. With the MareNostrum calibration, we apply LyMAS to 1024^3 N-body simulations of a 300 Mpc/h and 1.0 Gpc/h cube to produce large, publicly available catalogs of mock BOSS spectra that probe a large comoving volume. LyMAS will be a powerful tool for interpreting 3-d Ly-alpha forest data, thereby transforming measurements from BOSS and other massive quasar absorption surveys into constraints on dark energy, dark matter, space geometry, and IGM physics.Comment: Accepted for publication in ApJ (minor corrections from the previous version). Catalogs of mock BOSS spectra and relevant data can be found at: http://www2.iap.fr/users/peirani/lymas/lymas.ht

Extreme value statistics of smooth random Gaussian fields

We consider the Gumbel or extreme value statistics describing the distribution function p_G(x_max) of the maximum values of a random field x within patches of fixed size. We present, for smooth Gaussian random fields in two and three dimensions, an analytical estimate of p_G which is expected to hold in a regime where local maxima of the field are moderately high and weakly clustered. When the patch size becomes sufficiently large, the negative of the logarithm of the cumulative extreme value distribution is simply equal to the average of the Euler Characteristic of the field in the excursion x > x_max inside the patches. The Gumbel statistics therefore represents an interesting alternative probe of the genus as a test of non Gaussianity, e.g. in cosmic microwave background temperature maps or in three-dimensional galaxy catalogs. It can be approximated, except in the remote positive tail, by a negative Weibull type form, converging slowly to the expected Gumbel type form for infinitely large patch size. Convergence is facilitated when large scale correlations are weaker. We compare the analytic predictions to numerical experiments for the case of a scale-free Gaussian field in two dimensions, achieving impressive agreement between approximate theory and measurements. We also discuss the generalization of our formalism to non-Gaussian fields.Comment: 10 pages, 2 figures, accepted for publication in MNRA