49 research outputs found
Reply to the comment on ''On the problem of initial conditions in cosmological N-body simulations''
We reply to some comments in astro-ph/0309381 concerning the problem of
setting-up initial conditions in cosmological N-body simulationsComment: 2 pages, 1 postscript figure, style epl.cl
Force distribution in a randomly perturbed lattice of identical particles with pair interaction
We study the statistics of the force felt by a particle in the class of
spatially correlated distribution of identical point-like particles,
interacting via a pair force (i.e. gravitational or Coulomb), and
obtained by randomly perturbing an infinite perfect lattice. In the first part
we specify the conditions under which the force on a particle is a well defined
stochastic quantity. We then study the small displacements approximation,
giving both the limitations of its validity, and, when it is valid, an
expression for the force variance. In the second part of the paper we extend to
this class of particle distributions the method introduced by Chandrasekhar to
study the force probability density function in the homogeneous Poisson
particle distribution. In this way we can derive an approximate expression for
the probability distribution of the force over the full range of perturbations
of the lattice, i.e., from very small (compared to the lattice spacing) to very
large where the Poisson limit is recovered. We show in particular the
qualitative change in the large-force tail of the force distribution between
these two limits. Excellent accuracy of our analytic results is found on
detailed comparison with results from numerical simulations. These results
provide basic statistical information about the fluctuations of the
interactions (i) of the masses in self-gravitating systems like those
encountered in the context of cosmological N-body simulations, and (ii) of the
charges in the ordered phase of the One Component Plasma.Comment: 23 pages, 10 figure
Towards quantitative control on discreteness error in the non-linear regime of cosmological N body simulations
The effects of discreteness arising from the use of the N-body method on the
accuracy of simulations of cosmological structure formation are not currently
well understood. After a discussion of how the relevant discretisation
parameters introduced should be extrapolated to recover the Vlasov-Poisson
limit, we study numerically, and with analytical methods we have developed
recently, the central issue of how finite particle density affects the
precision of results. In particular we focus on the power spectrum at
wavenumbers around and above the Nyquist wavenumber, in simulations in which
the force resolution is taken smaller than the initial interparticle spacing.
Using simulations of identical theoretical initial conditions sampled on four
different "pre-initial" configurations (three different Bravais lattices, and a
glass) we obtain a {\it lower bound} on the real discreteness error. With the
guidance of our analytical results, we establish with confidence that the
measured dispersion is not contaminated either by finite box size effects or by
subtle numerical effects. Our results show notably that, at wavenumbers {\it
below} the Nyquist wavenumber, the dispersion increases monotonically in time
throughout the simulation, while the same is true above the Nyquist wavenumber
once non-linearity sets in. For normalizations typical of cosmological
simulations, we find lower bounds on errors at the Nyquist wavenumber of order
of a percent, and larger above this scale. The only way this error may be
reduced below these levels at these scales, and indeed convergence to the
physical limit firmly established, is by extrapolation, at fixed values of the
other relevant parameters, to the regime in which the mean comoving
interparticle distance becomes less than the force smoothing scale.Comment: 26 pages, 15 figures, minor changes, slightly shortened, version to
be published in MNRA
Universality of power law correlations in gravitational clustering
We present an analysis of different sets of gravitational N-body simulations,
all describing the dynamics of discrete particles with a small initial velocity
dispersion. They encompass very different initial particle configurations,
different numerical algorithms for the computation of the force, with or
without the space expansion of cosmological models. Despite these differences
we find in all cases that the non-linear clustering which results is
essentially the same, with a well-defined simple power-law behaviour in the
two-point correlations in the range from a few times the lower cut-off in the
gravitational force to the scale at which fluctuations are of order one. We
argue, presenting quantitative evidence, that this apparently universal
behaviour can be understood by the domination of the small scale contribution
to the gravitational force, coming initially from nearest neighbor particles.Comment: 7 pages, latex, 3 postscript figures. Revised version to be published
in Europhysics Letters. Contains additional analysis showing more directly
the central role of nearest neighbour interactions in the development of
power-law correlation
Gravitational dynamics of an infinite shuffled lattice: early time evolution and universality of non-linear correlations
In two recent articles a detailed study has been presented of the out of
equilibrium dynamics of an infinite system of self-gravitating points initially
located on a randomly perturbed lattice. In this article we extend the
treatment of the early time phase during which strong non-linear correlations
first develop, prior to the onset of ``self-similar'' scaling in the two point
correlation function. We establish more directly, using appropriate
modifications of the numerical integration, that the development of these
correlations can be well described by an approximation of the evolution in two
phases: a first perturbative phase in which particles' displacements are small
compared to the lattice spacing, and a subsequent phase in which particles
interact only with their nearest neighbor. For the range of initial amplitudes
considered we show that the first phase can be well approximated as a
transformation of the perturbed lattice configuration into a Poisson
distribution at the relevant scales. This appears to explain the
``universality'' of the spatial dependence of the asymptotic non-linear
clustering observed from both shuffled lattice and Poisson initial conditions.Comment: 11 pages, 11 figures, shortened introductory sections and other minor
modifications, version to appear in Phys. Rev.
Gravitational evolution of a perturbed lattice and its fluid limit
We apply a simple linearization, well known in solid state physics, to
approximate the evolution at early times of cosmological N-body simulations of
gravity. In the limit that the initial perturbations, applied to an infinite
perfect lattice, are at wavelengths much greater than the lattice spacing
the evolution is exactly that of a pressureless self-gravitating fluid treated
in the analagous (Lagrangian) linearization, with the Zeldovich approximation
as a sub-class of asymptotic solutions. Our less restricted approximation
allows one to trace the evolution of the discrete distribution until the time
when particles approach one another (i.e. ``shell crossing''). We calculate
modifications of the fluid evolution, explicitly dependent on i.e.
discreteness effects in the N body simulations. We note that these effects
become increasingly important as the initial red-shift is increased at fixed
. The possible advantages of using a body centred cubic, rather than simple
cubic, lattice are pointed out.Comment: 4 pages, 2 figures, version with minor modifications, accepted for
publication in Phys. Rev. Let
Gravitational Dynamics of an Infinite Shuffled Lattice: Particle Coarse-grainings, Non-linear Clustering and the Continuum Limit
We study the evolution under their self-gravity of infinite ``shuffled
lattice'' particle distributions, focussing specifically on the comparison of
this evolution with that of ``daughter'' particle distributions, defined by a
simple coarse-graining procedure. We consider both the case that such
coarse-grainings are performed (i) on the initial conditions, and (ii) at a
finite time with a specific additional prescription. In numerical simulations
we observe that, to a first approximation, these coarse-grainings represent
well the evolution of the two-point correlation properties over a significant
range of scales. We note, in particular, that the form of the two-point
correlation function in the original system, when it is evolving in the
asymptotic ``self-similar'' regime, may be reproduced well in a daughter
coarse-grained system in which the dynamics are still dominated by two-body
(nearest neighbor) interactions. Using analytical results on the early time
evolution of these systems, however, we show that small observed differences
between the evolved system and its coarse-grainings at the initial time will in
fact diverge as the ratio of the coarse-graining scale to the original
inter-particle distance increases. The second coarse-graining studied,
performed at a finite time in a specified manner, circumvents this problem. It
also makes more physically transparent why gravitational dynamics from these
initial conditions tends toward a ``self-similar'' evolution. We finally
discuss the precise definition of a limit in which a continuum (specifically
Vlasov-like) description of the observed linear and non-linear evolution should
be applicable.Comment: 21 pages, 8 eps figures, 2 jpeg figures (available in high resolution
at http://pil.phys.uniroma1.it/~sylos/PRD_dec_2006/
Generation of Primordial Cosmological Perturbations from Statistical Mechanical Models
The initial conditions describing seed fluctuations for the formation of
structure in standard cosmological models, i.e.the Harrison-Zeldovich
distribution, have very characteristic ``super-homogeneous'' properties: they
are statistically translation invariant, isotropic, and the variance of the
mass fluctuations in a region of volume V grows slower than V. We discuss the
geometrical construction of distributions of points in with similar
properties encountered in tiling and in statistical physics, e.g. the Gibbs
distribution of a one-component system of charged particles in a uniform
background (OCP). Modifications of the OCP can produce equilibrium correlations
of the kind assumed in the cosmological context. We then describe how such
systems can be used for the generation of initial conditions in gravitational
-body simulations.Comment: 7 pages, 3 figures, final version with minor modifications, to appear
in PR
Initial conditions, Discreteness and non-linear structure formation in cosmology
In this lecture we address three different but related aspects of the initial
continuous fluctuation field in standard cosmological models. Firstly we
discuss the properties of the so-called Harrison-Zeldovich like spectra. This
power spectrum is a fundamental feature of all current standard cosmological
models. In a simple classification of all stationary stochastic processes into
three categories, we highlight with the name ``super-homogeneous'' the
properties of the class to which models like this, with , belong. In
statistical physics language they are well described as glass-like. Secondly,
the initial continuous density field with such small amplitude correlated
Gaussian fluctuations must be discretised in order to set up the initial
particle distribution used in gravitational N-body simulations. We discuss the
main issues related to the effects of discretisation, particularly concerning
the effect of particle induced fluctuations on the statistical properties of
the initial conditions and on the dynamical evolution of gravitational
clustering.Comment: 28 pages, 1 figure, to appear in Proceedings of 9th Course on
Astrofundamental Physics, International School D. Chalonge, Kluwer, eds N.G.
Sanchez and Y.M. Pariiski, uses crckapb.st pages, 3 figure, ro appear in
Proceedings of 9th Course on Astrofundamental Physics, International School
D. Chalonge, Kluwer, Eds. N.G. Sanchez and Y.M. Pariiski, uses crckapb.st
Linear perturbative theory of the discrete cosmological N-body problem
We present a perturbative treatment of the evolution under their mutual
self-gravity of particles displaced off an infinite perfect lattice, both for a
static space and for a homogeneously expanding space as in cosmological N-body
simulations. The treatment, analogous to that of perturbations to a crystal in
solid state physics, can be seen as a discrete (i.e. particle) generalization
of the perturbative solution in the Lagrangian formalism of a self-gravitating
fluid. Working to linear order, we show explicitly that this fluid evolution is
recovered in the limit that the initial perturbations are restricted to modes
of wavelength much larger than the lattice spacing. The full spectrum of
eigenvalues of the simple cubic lattice contains both oscillatory modes and
unstable modes which grow slightly faster than in the fluid limit. A detailed
comparison of our perturbative treatment, at linear order, with full numerical
simulations is presented, for two very different classes of initial
perturbation spectra. We find that the range of validity is similar to that of
the perturbative fluid approximation (i.e. up to close to ``shell-crossing''),
but that the accuracy in tracing the evolution is superior. The formalism
provides a powerful tool to systematically calculate discreteness effects at
early times in cosmological N-body simulations.Comment: 25 pages, 21 figure