One dimensional versions of dissipationless cosmological N-body simulations have been shown to share many qualitative behaviours of the three dimensional problem. Their interest lies in the fact that they can resolve a much greater range of time and length scales, and admit exact numerical integration. We use such models here to study how non-linear clustering depends on initial conditions and cosmology. More specifically, we consider a family of models which, like the three dimensional EdS model, lead for power-law initial conditions to self-similar clustering characterized in the strongly nonlinear regime by power-law behaviour of the two point correlation function. We study how the corresponding exponent γ depends on the initial conditions, characterized by the exponent n of the power spectrum of initial fluctuations, and on a single parameter κ controlling the rate of expansion. The space of initial conditions/cosmology divides very clearly into two parts: (1) a region in which γ depends strongly on both n and κ and where it agrees very well with a simple generalisation of the so-called stable clustering hypothesis in three dimensions, and (2) a region in which γ is more or less independent of both the spectrum and the expansion of the universe. The boundary in (n, κ) space dividing the "stable clustering" region from the "universal" region is very well approximated by a "critical" value of the predicted stable clustering exponent itself. We explain how this division of the (n, κ) space can be understood as a simple physical criterion which might indeed be expected to control the validity of the stable clustering hypothesis. We compare and contrast our findings to results in three dimensions, and discuss in particular the light they may throw on the question of "universality" of non-linear clustering in this context.

We describe how a simple class out-of-equilibrium, rotating, and asymmetrical mass distributions evolve under their self-gravity to produce a quasi-planar spiral structure surrounding a virialized core, qualitatively resembling a spiral galaxy. The spiral structure is transient, but can survive tens of dynamical times, and further reproduces qualitatively noted features of spiral galaxies such as the predominance of trailing two-armed spirals and large pitch angles. As our models are highly idealized, a detailed comparison with observations is not appropriate, but generic features of the velocity distributions can be identified to be the potential observational signatures of such a mechanism. Indeed, the mechanism leads generically to a characteristic transition from predominantly rotational motion, in a region outside the core, to radial ballistic motion in the outermost parts. Such radial motions are excluded in our Galaxy up to 15 kpc, but could be detected at larger scales in the future by GAIA. We explore the apparent motions seen by external observers of the velocity distributions of our toy galaxies, and find that it is difficult to distinguish them from those of a rotating disk with sub-dominant radial motions at levels typically inferred from observations. These simple models illustrate the possibility that the observed apparent motions of spiral galaxies might be explained by non-trivial nonstationary mass and velocity distributions without invoking a dark matter halo or modification of Newtonian gravity. In this scenario the observed phenomenological relation between the centripetal and gravitational acceleration of the visible baryonic mass could have a simple explanation.

An isolated, initially cold and ellipsoidal cloud of self-gravitating particles represents a relatively simple system to study the effects of the deviations from spherical symmetry in the mechanism of violent relaxation. Initial deviations from spherical symmetry are shown to play a dynamical role that is equivalent to that of density fluctuations in the case of an initially spherical cloud. Indeed, these deviations control the amount of particles energy change and thus determine the properties of the final energy distribution, particularly the appearance of two species of particles: bound and free. Ejection of mass and energy from the system together with the formation of a density profile decaying as ρ(r) ∼ r −4 and a Keplerian radial velocity dispersion profile, are the prominent features similar to those observed after the violent relaxation of spherical clouds. In addition, we find that ejected particles are characterized by highly nonspherical shapes, whose features can be traced in the initial deviations from spherical symmetry that are amplified during the dynamical evolution: particles can indeed form anisotropic configurations, like bars and/or disks, even though the initial cloud was very close to spherical.

The evolution of self-gravitating systems, and long-range interacting systems more generally, from initial configurations far from dynamical equilibrium is often described as a simple two phase process: a first phase of violent relaxation bringing it to a quasi-stationary state in a few dynamical times, followed by a slow adiabatic evolution driven by collisional processes. In this context the complex spatial structure evident, for example, in spiral galaxies is understood either in terms of instabilities of quasi-stationary states, or a result of dissipative non-gravitational interactions. We illustrate here, using numerical simulations, that purely self-gravitating systems evolving from quite simple initial configurations can in fact give rise easily to structures of this kind of which the lifetime can be large compared to the dynamical characteristic time, but short compared to the collisional relaxation time scale. More specifically, for a broad range of non-spherical and non-uniform rotating initial conditions, gravitational relaxation gives rise quite generically to long-lived non-stationary structures of a rich variety, characterized by spiral-like arms, bars and even ring-like structures in special cases. These structures are a feature of the intrinsically out-of-equilibrium nature of the system's collapse, associated with a part of the system's mass while the bulk is well virialized. They are characterized by predominantly radial motions in their outermost parts, but also incorporate an extended flattened region which rotates coherently about a well virialized core of triaxial shape with an approximately isotropic velocity dispersion. We characterize the kinematical and dynamical properties of these complex velocity fields and we briefly discuss the possible relevance of these simple toy models to the observed structure of real galaxies emphasizing the difference between dissipative and dissipationless disc formation.PACS numbers: 05.10-a, 05.90.+m,98.62.Hr