Comparison between simulated and analytic density profiles for different redshifts. Upper panel: solid lines depict the median density profile at different redshifts of our MDPL2 sample in spherical shells of radius R/R_ta; shaded regions represent the 1σ spread of the density within each shell; dashed lines represent the analytical prediction for the outer portions of the mode profile discussed in Sect. 3. Middle panel: Logarithmic slope of the density profiles, with dashed lines illustrating the logarithmic slope of the analytical prediction. Lower panel: ratio of logarithmic slope in median simulated profile over the one of the analytically predicted profile. The analytic slope deviates from the one in simulations for R/R_ta ≲ 1.