Logarithmic derivative of the average density profiles, and R_spl as function of mass, redshift, and theory of gravity. Upper panels: average profiles of d log ρ_shell/d log r for the samples of simulated halos in Table 1. The left, middle, and right panels show the average profiles for the low-, intermediate-, and high-mass samples, respectively, at the lowest and highest redshifts of each sample, as listed in the insets. The solid (dash-dotted) curves are the profiles in the ΛCDM (fR4) model. The vertical solid (dash-dotted) thick black lines show the location of R_spl at z = 0 in the ΛCDM (fR4) model. The pair of vertical blue (green) solid and dash-dotted lines shows R_spl at z = 1.0 (z = 0.3) for the ΛCDM and fR4 models, respectively. Bottom panels: the thin solid (dash-dotted) black curves show the radial derivative of the average mass profile in ΛCDM (fR4) at z = 0; the vertical thin solid (dash-dotted) black lines show ς_avg of the low-, intermediate-, and high-mass samples at z = 0 in the ΛCDM (fR4) models (Table 2, first row, values in brackets). When extended to the upper panels, these thin solid and dash-dotted black lines cross the average density profiles at dlog ρ_shell/d log r ≃ −2, as expected from Eq. (1).