The inner structure of  CDM haloes -- I. A numerical convergence study

We present a comprehensive set of convergence tests which explore the role of various numerical parameters on the equilibrium structure of a simulated dark matter halo. We report on results obtained with two independent, state-of-the-art, multi-stepping, parallel <it>N</it>-body codes: <scp>pkdgrav</scp> and <scp>gadget</scp>. We find that convergent mass profiles can be obtained for suitable choices of the gravitational softening, time-step, force accuracy, initial redshift, and particle number. For softenings chosen so that particle discreteness effects are negligible, convergence in the circular velocity is obtained at radii where the following conditions are satisfied: (i) the time-step is much shorter than the local orbital time-scale; (ii) accelerations do not exceed a characteristic acceleration imprinted by the gravitational softening; and (iii) enough particles are enclosed so that the collisional relaxation time-scale is longer than the age of the Universe. Convergence also requires sufficiently high initial redshift and accurate force computations. Poor spatial, time, or force resolution leads generally to systems with artificially low central density, but may also result in the formation of artificially dense central cusps. We have explored several adaptive time-stepping choices and we have obtained the best results when individual time-steps are chosen according to the local acceleration and the gravitational softening (Δ<it>t</it><inf><it>i</it></inf>∝ (ϵ/<it>a</it><inf><it>i</it></inf>)1/2), although further experimentation may yield better and more efficient criteria. The most stringent requirement for convergence is typically that imposed on the particle number by the collisional relaxation criterion. This implies that, in order to estimate accurate circular velocities at radii where the density contrast may reach ∼106, the region must enclose of the order of 3000 particles (or more than a few times 106 within the virial radius). Applying these criteria to a galaxy-sized ΛCDM halo, we find that the spherically averaged density profile becomes progressively shallower from the virial radius inwards, reaching a logarithmic slope shallower than −1.2 at the innermost resolved point, ∼0.005 <it>r</it><inf>200</inf>, with little evidence for convergence to a power-law behaviour in the inner regions.