tokens: accuracy and precision of the density and the potential depends on the choice of the parametrized density model used. Both the Gamma and the Einasto profiles produce nearly unbiased density-potential pairs when compared to observables, while the Einasto profiles are most sensitive to how the halo centers are defined. As noted in the Introduction, Serra et al. ( 2011 ) attribute the lower numerically integrated potential (Equation ( 3 )) compared to the NFW potential (Equation ( 4 b )) to mass outside the cluster that is not accounted for by the NFW profile shape. We show that the opposite is true. The NFW model density profile overpredicts the true density profile outside r 200 and thus overpredicts the true gravitational potential via the Poisson equation by 10%–15%. 3.3. Mass and Redshift Dependence Next, we examine how the edge varies as a function of halo mass and redshift. In this case, we use a new sub-set of the Millennium simulation with the 100 most massive halos halos smaller than 1 × 10 14 M . The minimum mass of this new subset is M , i.e., a factor of 10 smaller than the previous sample. We measure the spherically averaged density profiles and infer the escape edges via the potential from Equation ( 6 ). The edges are measured using the same algorithm as applied to the more massive subset studied previously. We find no statistical or systematic difference between the high mass and low mass halo data sets. The density profile predicts the escape edge via the Poisson equation to the same level of accuracy and precision for over two orders of magnitude in cluster halo mass. We study the low mass clusters at four different simulation snapshot outputs, corresponding to , and 0.75. We keep the 100 most massive halos in each snapshot as provided by the “millimil” subset of the Millennium data. At , the deceleration parameter goes from negative to positive. As the cluster density profiles evolve with redshift, we fit the profiles separately for each halo at each redshift. Instead of physical coordinates, we use radial coordinates with respect to the r 200 of each cluster for the profiles, due to the fact that the cluster sizes also evolve with redshift. In Figure 4 we show that within r 200 , the edge can be accurately predicted from the density and Poisson equation to z = 0.75. However, outside the virial radius, the edge is increasingly underpredicted compared to the model, as a function of increasing redshift. We find the same result when using the mean background density as opposed to the critical background density when defining the cluster masses and radii. Figure 4. The variation in the fractional difference between the measured v esc edge and the predictions from the density profile from redshift 0 to 0.75. The light (dark) gray bands represent the 67% (90%) scatter of the individual halos. We can explain this by the growing influence of the halo gravitational potential well on the dynamics of the infall regions around the
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