The Inner Structure of ΛCDM Halos 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 results obtained with two independent, state-of-the-art, multi-stepping, parallel N–body codes: PKDGRAV and GADGET. We find that convergent mass profiles can be obtained for suitable choices of the gravitational softening, timestep, 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 timestep is much shorter than the local orbital timescale; (ii) accelerations do not exceed a characteristic acceleration imprinted by the gravitational softening; and (iii) enough particles are enclosed so that the collisional relaxation timescale 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 obtained best results when individual timesteps are chosen according to the local acceleration and the gravitational softening (∆ti ∝ (ǫ/ai) ), 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, which implies that in order to estimate accurate circular velocities at radii where the density contrast may reach ∼ 10, the region must enclose of order 3000 particles (or more than a few times 10 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, r ∼ 0.005 r200, with little evidence for convergence to a power-law behaviour in the inner regions.