Collisionless Systems & Dark Matter Halos

N-body simulations of structure formation in the universe are reaching the point at which galaxy scale objects can be resolved. These structures have proven to have remarkably similar characteristics. Their density profiles appear to be well described by an analytical function that has become known as the NFW profile, after the work of Navarro, Frenk, & White. Halos with this profile have densities which are inversely proportional to radial distance close to the center (logarithmic slope = -1), a transition region where the logarithmic slope changes, and an outer region where the density is inversely proportional to the cube of the radial distance (logarithmic slope = -3). The exact details of the profile shape are still being debated, but the fact that simulations give rise to basically universal shapes seems clear.

What we are interested in answering is, why do N-body simulations produce this kind of profile? Is there some underlying physical mechanism governing collisionless systems that drives the halos to have this form? Are these halos the result of hierarchical structure formation? Or, is is simply the result of cosmological initial conditions?

Since N-body simulations are, by definition, complicated procedures, it is difficult to untangle different causes and effects within them. So, we are taking a step back and utilizing a simpler, more controllable technique to create halos. Ryden & Gunn laid out a method of determining the structure of a halo based on analytical expressions. We are utilizing a slightly modified form of this method which gives us a great deal of control over specific changes to conditions. This control has led us to several interesting insights regarding the origin of halo structure. Again, following a simplified approach, we have also used analytical arguments to try to understand the phase-space structure of simulated halos. The surprisingly power-law radial profile of the quantity density divided by velocity dispersion cubed has been investigated in several works.

We are also investigating a "thermodynamic" approach to this problem; are there global quantities that determine the structure of self-gravitating equilibria? We have begun to make our own N-body models that we can compare to analytical models derived from, for example, nonextensive thermodynamic considerations. The thermodynamics of one-dimensional self-gravitating collisionless systems can be tackled analytically, and we continue to use new techniques to study these simple systems for insight into general principles.

As mentioned above, N-body simulations are complex pieces of machinery. We have looked in detail at one cog in the N-body simulation process, gravitational softening. Either to maintain nearly-collisionless conditions or to ease integration conditions (or both), most N-body codes used to investigate dark matter halos use gravitational softening. Basically, point masses do not strictly interact through r-2 forces, but instead particles closer than a softening length are subject to some constant maximum force. This is commonly attributed to N-body particles not being true point masses but rather unresolved clusters of mass. Our work investigates the behavior of such clusters and compares their motions with those of point masses interacting through softened forces. Watch an animation of such a comparison. The clusters of particles are identical in the three portions of the animation, but the softening length of the point mass interaction is different in each section. Cluster centers-of-mass locations are marked by 'x' symbols, while the softened point mass locations are marked by diamonds. We have compiled more movies and results from a variety of initial conditions.