tokens: parameter in N -body simulations. Numerous studies have argued that minimizing the scatter in certain statistical descriptions of the matter field (i.e. power spectrum or two-point correlation function) among an ensemble of N -body realizations requires a softening length ε that is at least as large as the mean inter-particle separation l = L box / N 1/3 (e.g. Melott et al. 1997 ; Splinter et al. 1998 ; Romeo et al. 2008 ; Joyce, Marcos Baertschiger 2009 ). Here, L box is the (periodic) box size of the cosmological simulation and N is the total number of particles used. Most cosmological simulations, though, are run with much smaller softening lengths, typically of the order of ε/ l ∼ 0.01–0.02. For example, the Millennium simulation (Springel et al. 2005 ) has ε/ l ∼ 0.022, while the MDPL and SMDPL simulations of Klypin et al. ( 2016 ) adopt ε/ l ∼ 0.019 and ε/ l ∼ 0.014, respectively. Whether such simulations produce reliable results on scales below the (comoving) mean inter-particle separation is still a matter of fierce, ongoing debate (see e.g. Knebe et al. 2000 ; Melott 2007 ; Joyce et al. 2009 ; Power et al. 2016 ; Benhaiem, Joyce Sylos Labini 2017 , and references therein). Several studies have focused on the optimal softening length for simulating individual objects (i.e. individual DM haloes or galaxies). For example, Dehnen ( 2001 ), extending studies by Merritt ( 1996 ) and Athanassoula et al. ( 2000 ), suggests that one should aim to minimize force errors. This can be achieved by optimizing the softening length such that it simultaneously minimizes the force bias (which increases with increasing ε) and the force noise (which decreases with increasing ε). Based on such considerations, Dehnen ( 2001 ) advocates that a collisionless system with a Hernquist ( 1990 ) density profile (which has the same r −1 -cusp as a NFW profile) should be simulated with a Plummer softening , where r 0 is the scale radius of the Hernquist (or NFW) profile and N 5 = N /10 5 with N the number of particles used to model the system in question. Power et al. ( 2003 ) use a different criterion to optimize the softening length: by minimizing the impact of two-body effects on the (central) density profiles of virialized DM haloes in cosmological simulations, they advocate a much larger softening length of , where c is the NFW halo concentration parameter. Note that both of these criteria have the optimal softening length depend on the size of the system as well as the number of particles that is used to represent the system. If we adopt a concentration–mass relation of the form c ∝ M −0.1 (e.g. Dutton Macciò 2014 ) and use that the virial radii of DM haloes scale according to r vir ∝ M 1/3 ∝ N 1/3 , we find that the optimal softening length depends on halo mass according to ε opt ∝ M α , with α = +0.2 and −0.17 for the criteria of Dehnen and Power et al., respectively. This scaling is sufficiently weak so that the softening is close to optimal for haloes
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