Measuring the splashback feature: Dependence on halo properties and history

In this study, we define the novel splashback depth $\mathcal{D}$ and width $\mathcal{W}$ to examine how the splashback features of dark matter haloes are affected by the physical properties of haloes themselves. We use the largest simulation run in the hydrodynamic MillenniumTNG project. By stacking haloes in bins of halo mass, redshift, mass-dependent properties such as peak height and concentration, and halo formation history, we measure the shape of the logarithmic slope of the density profile of dark matter haloes. Our results show that the splashback depth has a strong dependence on the halo mass which follows a power law $\mathcal{D}\propto\left(\log_{10}M\right)^{2.8}$. Properties with strong correlation with halo mass demonstrate similar dependence. The splashback width has the strongest dependence on halo peak height and follows a power law $\mathcal{W}\proptoν^{-0.87}$. We provide the fitting functions of the splashback depth and width in terms of halo mass, redshift, peak height, concentrations and halo formation time. The depth and width are therefore considered to be a long term memory tracker of haloes since they depend more on accumulative physical properties, e.g., halo mass, peak height and halo formation time. They are shaped primarily by the halo's assembly history, which exerts a stronger influence on the inner density profile than short-term dynamical processes. In contrast, the splashback features have little dependence on the short term factors such as halo mass accretion rate and most recent major merger time. The splashback depth and width can therefore be used to complement information gained from quantities like the point of steepest slope or truncation radius to characterise the halo's history and inner structure.