Measuring the splashback feature: Dependence on halo properties and history

TL;DR

The paper defines splashback depth $\mathcal{D}$ and width $\mathcal{W}$ to quantify halo boundaries and uses MillenniumTNG hydrodynamic simulations to study their dependence on halo mass, redshift, peak height, concentration, and formation history. Density profiles are stacked and fitted with a two-component model to extract $\mathcal{D}$ and $\mathcal{W}$, with uncertainties estimated by bootstrap. Key results show $\mathcal{D} \propto (\log_{10} M)^{2.8}$ and $\mathcal{W} \propto \nu^{-0.87}$, indicating deeper and narrower features for more massive haloes and higher peak heights, while both features are mainly determined by long-term halo assembly. Concentration and formation time jointly modulate the features, with older, more concentrated haloes exhibiting shallower and wider splashbacks, and peak height providing a compact link between mass, redshift, and width. Across hydrodynamic and DM-only runs, the fits remain robust, supporting the interpretation that $\mathcal{D}$ and $\mathcal{W}$ encode halo history and inner structure with practical observational relevance.

Abstract

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.

Figures (14)

• Figure 1: The upper panel is the sample density profile of the dark matter halo, and the lower panel is the logarithmic slope of the density profile. The data points of the density profile come from the MTNG hydrodynamic simulations. This example is for the mass range $10^{14} \leq M/{\rm M}_{\odot}<10^{14.5}M_\odot$. The vertical black dashed line indicates the location of the steepest slope $R_{\rm st}$ of the stacked haloes. The solid blue line is the fitted density profile and logarithmic slope using Equation \ref{['eq:profile']} and Equation \ref{['eq:gradient']}, respectively. In the lower panel, the horizontal green and the vertical red lines indicate the width and the depth of the splashback features defined in Section \ref{['sec:Rsp']}.
• Figure 2: Depth (Left) and width (Right) of splashback features as a function of halo mass for the dark matter mass at $z = 0$. We stack the density profiles for haloes with $\log\left(M_{200\text{m}}/M_\odot\right)$ in 13–13.5, 13.5–14, 14–14.5, 14.5–15, and 15–15.5 and compute the splashback features of the median profile. The depth increases with mass while the width decreases. This indicates a more pronounced splashback feature for larger haloes. Notably, this is not just an enlargement of the feature, in which both the width and depth would increase, but the decreasing width indicates a sharper decrease in the density profile.
• Figure 3: Top left: Depth of splashback features as a function of halo mass for the dark matter mass at $z < 2$. The depth linearly increases with the log of the mass. Top right: Depth of splashback features as a function of redshift for the dark matter halo mass with $13\leq\log\left(M_{200\text{m}/M_\odot}\right)\leq 15.5$. There is also a slight increasing trend with redshift, although this is much less pronounced than the trend with depth. Bottom left: width of splashback features as a function of halo mass for the dark matter mass at $z < 2$. The width decreases with mass, indicating that the splashback feature is narrower for larger mass haloes. Bottom right: Width of splashback features as a function of redshift for the dark matter halo mass with $13\leq\log\left(M_{200\text{m}/M_\odot}\right)\leq 15.5$. There is a decreasing trend with redshift, indicating that the width is more sensitive to cosmic evolution than the depth. The dashed lines refer to the power law fitting in Equations \ref{['eq:massD']} and \ref{['eq:massW']}.
• Figure 4: Depth (Top) and width (Bottom) of splashback features as a function of halo accretion rate for the dark matter mass at z = 0. The points show the data from the simulation. The dashed line represents the power law fitting while the dotted line in the upper panel represents the linear fit. The depth increases with accretion rate while the width decreases, making the splashback feature significantly narrower at higher accretion rates.
• Figure 5: Relationships between width ($\mathcal{W}$) and depth ($\mathcal{D}$) of the splashback feature as a function of redshift ($z$) and accretion rate ($\Gamma$). When haloes are stacked by accretion rate, the width increases with time. However, the depth is uncorrelated with redshift and both width and depth appear only weakly dependent on accretion rate. In all panels, points show the data from fits in the simulation, with colour indicating either redshift (left) or accretion rate (right). Haloes are stacked into accretion rate bins between $\Gamma=0$ and $\Gamma=6$ and the median profile is fitted. Top left: Depth of the splashback features as a function of accretion rate for various redshifts. Top right: Depth of the splashback feature as a function of redshift for various accretion rates. There is no clear trend of the depth and width as a function of accretion rate. Bottom left: Width of the splashback features as a function of accretion rate for various redshifts. Bottom right: Width of the splashback feature as a function of redshift for various accretion rates. The black dashed line is fitted with the redshift only, while the black solid line is fitted using the scale factor. There is not much of a trend for the depth and width in terms of halo accretion rate, so we fit a function only in terms of redshift. There is less dependence on redshift when separated by accretion rate compared to the mass bins, as shown in Figure \ref{['fig:mass_depth_z']}, although there is still some dependence, with the depth decreasing significantly between redshifts $0<z<1$ and with the width peaking at redshift $z\sim1$.