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

Power, C.; Navarro, J.F.; Jenkins, A.R.; Frenk, C.S.; White, S.D.M.; Springel, V.; Stadel, J.; Quinn, T.

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


C. Power

J.F. Navarro

S.D.M. White

V. Springel

J. Stadel

T. Quinn


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 N-body codes: pkdgrav and gadget. 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 (Δti (ε/ai)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 r200, with little evidence for convergence to a power-law behaviour in the inner regions.


Power, C., Navarro, J., Jenkins, A., Frenk, C., White, S., Springel, V., …Quinn, T. (2003). The inner structure of ΛCDM haloes - I. A numerical convergence study. Monthly Notices of the Royal Astronomical Society, 338(1), 14-34.

Journal Article Type Article
Publication Date 2003-01
Deposit Date Nov 22, 2006
Publicly Available Date Aug 11, 2014
Journal Monthly Notices of the Royal Astronomical Society
Print ISSN 0035-8711
Electronic ISSN 1365-2966
Publisher Royal Astronomical Society
Peer Reviewed Peer Reviewed
Volume 338
Issue 1
Pages 14-34
Keywords Gravitation, Cosmology, Theory, Dark matter.


Published Journal Article (781 Kb)

Copyright Statement
This article has been accepted for publication in Monthly notices of the Royal Astronomical Society © 2003 The Authors Published on behalf of Royal Astronomical Society. All rights reserved.

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