Two Body Relaxation in Simulated Cosmological Haloes

This paper aims at quantifying discreetness effects, born of finite particle number, on the dynamics of dark matter haloes forming in the context of cosmological simulations. By generalising the standard calculation of two body relaxation to the case when the size and mass distribution are variable, and parametrising the time evolution using established empirical relations, we find that the dynamics of a million particle halo is noise-dominated within the inner percent of the final virial radius. Far larger particle numbers (~ 10^8) are required for the RMS perturbations to the velocity to drop to the 10 % level there. The radial scaling of the relaxation time is simple and strong: t_relax ~ r^2, implying that numbers >> 10^8 are required to faithfully model the very inner regions; artificial relaxation may thus constitute an important factor, contributing to the contradictory claims concerning the persistence of a power law density cusp to the very centre. The cores of substructure haloes can be many relaxation times old. Since relaxation first causes their expansion before recontraction occurs, it may render them either more difficult or easier to disrupt, depending on their orbital parameters. It may thus modify the characteristics of the subhalo distribution and effects of interactions with the parent. We derive simple closed form formulas for the characteristic relaxation times, as well as for the weak N-scaling reported by Diemand et al. when the main contribution comes from relaxing subhaloes (abridged).Comment: 11 Pages, 7 figs, Monthly Notices styl

From cusps to cores: a stochastic model

The cold dark matter model of structure formation faces apparent problems on galactic scales. Several threads point to excessive halo concentration, including central densities that rise too steeply with decreasing radius. Yet, random fluctuations in the gaseous component can 'heat' the centres of haloes, decreasing their densities. We present a theoretical model deriving this effect from first principles: stochastic variations in the gas density are converted into potential fluctuations that act on the dark matter; the associated force correlation function is calculated and the corresponding stochastic equation solved. Assuming a power law spectrum of fluctuations with maximal and minimal cutoff scales, we derive the velocity dispersion imparted to the halo particles and the relevant relaxation time. We further perform numerical simulations, with fluctuations realised as a Gaussian random field, which confirm the formation of a core within a timescale comparable to that derived analytically. Non-radial collective modes enhance the energy transport process that erases the cusp, though the parametrisations of the analytical model persist. In our model, the dominant contribution to the dynamical coupling driving the cusp-core transformation comes from the largest scale fluctuations. Yet, the efficiency of the transformation is independent of the value of the largest scale and depends weakly (linearly) on the power law exponent; it effectively depends on two parameters: the gas mass fraction and the normalisation of the power spectrum. This suggests that cusp-core transformations observed in hydrodynamic simulations of galaxy formation may be understood and parametrised in simple terms, the physical and numerical complexities of the various implementations notwithstanding.Comment: Minor revisions to match version to appear in MNRAS; Section~2.3 largely rewritten for clarit

Regular and chaotic motion in softened gravitational systems

The stability of the dynamical trajectories of softened spherical gravitational systems is examined, both in the case of the full $N$-body problem and that of trajectories moving in the gravitational field of non-interacting background particles. In the latter case, for $N \ge 10000$, some trajectories, even if unstable, had exceedingly long diffusion times, which correlated with the characteristic e-folding timescale of the instability. For trajectories of $N \approx 100000$ systems this timescale could be arbitrarily large --- and thus appear to correspond to regular orbits. For centrally concentrated systems, low angular momentum trajectories were found to be systematically more unstable. This phenomenon is analogous to the well known case of trajectories in generic centrally concentrated non-spherical smooth systems, where eccentric trajectories are found to be chaotic. The exponentiation times also correlate with the conservation of the angular momenta along the trajectories. For $N$ up to a few hundred, the instability timescales of $N$-body systems and their variation with particle number are similar to those of the most chaotic trajectories in inhomogeneous non-interacting systems. For larger $N$ (up to a few thousand) the values of the these timescales were found to saturate, increasing significantly more slowly with $N$. We attribute this to collective effects in the fully self-gravitating problem, which are apparent in the time-variations of the time dependent Liapunov exponents. The results presented here go some way towards resolving the long standing apparent paradoxes concerning the local instability of trajectories of gravitational systems (abridged).Comment: 16 pages, 11 figures, Monthly Notices styl