Splashback Width in Dark Matter Halos

• Splashback width is the quantitative measure of the radial breadth over which the density profile steepens, marking the boundary between orbiting and infalling material.
• Recent research introduces an explicit parameter 𝒲 that scales with halo peak height, mass, and redshift, offering insights into long-term assembly history.
• Alternative formulations capture width via angular spread, transition function curvature, or diffusion coefficients, highlighting effects of asphericity and environmental disturbances.

Splashback width denotes the radial breadth, sharpness, or ambiguity of the splashback feature at the outskirts of dark-matter halos and galaxy clusters. The underlying splashback feature is the steepening associated with recently accreted material reaching first apocenter, i.e. the boundary between the inner multistream region and the outer infall region (Adhikari et al., 2014). In the literature, however, “width” is not a single universal observable. In many studies it is inferred indirectly from the steepness of the logarithmic slope, from transition-function parameters, or from the angular spread of a nonspherical splashback surface; only more recent work defines an explicit splashback width variable, $\mathcal{W}$, as a property of the slope-profile valley itself (Yu et al., 29 Jul 2025). Alternative formulations recast width as boundary ambiguity in threshold space through a diffusion coefficient, or as directional scatter in a fully aspherical boundary distribution (Ryu et al., 2022, Sen et al., 20 Feb 2026).

1. Conceptual status and basic definitions

The canonical splashback radius is the radius where the logarithmic slope of the density profile is steepest, and it is identified with the outermost caustic or first apocenter after collapse (Adhikari et al., 2014). In this sense, splashback width concerns not the location of that minimum, but how abruptly the profile descends into and recovers from it.

Early analytic and simulation-based papers emphasized that the feature is a very steep density drop over a narrow radial range without introducing a formal width parameter. In the spherical-collapse treatment of Adhikari et al., the relevant observable is the splashback radius or its associated overdensity, while the apparent width is discussed qualitatively in terms of caustic smearing by triaxiality, substructure, and the spread in apocenters (Adhikari et al., 2014). The improved spherical infall model of Del Popolo et al. likewise identifies splashback with the outermost caustic and the steepening of the logarithmic density slope, but does not fit a separate thickness parameter (Popolo et al., 2022).

A recurrent source of confusion is that splashback width is not synonymous with splashback radius. The 2025 MillenniumTNG analysis makes this distinction explicit by defining the splashback feature as a valley-shaped region in $\mathrm{d}\log\rho/\mathrm{d}\log r$: the splashback radius is the minimum of the logarithmic slope, the splashback depth $\mathcal{D}$ measures how deep that minimum is, and the splashback width $\mathcal{W}$ measures how broad the valley is (Yu et al., 29 Jul 2025). In that paper, $\mathcal{W}$ is defined as the width of the logarithmic-slope valley at the midpoint of the depth.

2. Width before explicit parameterization: steepness, transition functions, and radial span

Before $\mathcal{W}$ was introduced, most studies operationalized width through the sharpness of the slope transition. In the nonspherical-halo analysis of Shin et al., the splashback transition is sharper in the velocity-based alignment momentum correlation than in the density-based alignment correlation. The steepest slope of the conventional density correlation reaches about $\gamma \simeq -4.5$, while the momentum correlation reaches $\gamma_p \to -5.5$; the density-derived splashback radius is $\sim 3.5\%$ smaller than the momentum-derived one, with correlation coefficient $0.605$ (Okumura et al., 2018). That paper explicitly states that it does not define a formal splashback width parameter, so width is effectively inferred from the steepness of the logarithmic slope.

The same indirect logic appears in semi-analytic spherical infall modeling. Del Popolo et al. describe splashback as a sharp steepening over a narrow radial interval and report that for high-$\mathrm{d}\log\rho/\mathrm{d}\log r$0 halos the logarithmic slope changes from $\mathrm{d}\log\rho/\mathrm{d}\log r$1 to $\mathrm{d}\log\rho/\mathrm{d}\log r$2 over a radius range restricted to $\mathrm{d}\log\rho/\mathrm{d}\log r$3 rescaled radius, with the radius of steepest slope around $\mathrm{d}\log\rho/\mathrm{d}\log r$4–1.2 and a post-splashback flattening to approximately $\mathrm{d}\log\rho/\mathrm{d}\log r$5 at $\mathrm{d}\log\rho/\mathrm{d}\log r$6 (Popolo et al., 2022). This provides a radial-span notion of width without a standalone fit parameter.

Observational profile modeling has usually encoded width in transition functions. In the LoCuSS analysis, the profile is modeled as

$\mathrm{d}\log\rho/\mathrm{d}\log r$7

where $\mathrm{d}\log\rho/\mathrm{d}\log r$8 controls the shape of the transition, $\mathrm{d}\log\rho/\mathrm{d}\log r$9 its depth or strength, and $\mathcal{D}$0 the transition radius (Bianconi et al., 2020). That study does not report an explicit splashback width, but the full sample shows a dip in the projected logarithmic slope around $\mathcal{D}$1 Mpc, extending from $\mathcal{D}$2 to $\mathcal{D}$3 Mpc, with peak steepness $\mathcal{D}$4 at $\mathcal{D}$5 relative to neighboring radii (Bianconi et al., 2020). The closest width-like result is therefore the extent of the steepening region together with lower bounds $\mathcal{D}$6 and $\mathcal{D}$7, which indicate a sharp transition.

3. Explicit splashback width $\mathcal{D}$8 and its scaling relations

The most direct formalization of splashback width is the definition introduced in the MillenniumTNG study “Measuring the splashback feature: Dependence on halo properties and history” (Yu et al., 29 Jul 2025). Haloes are stacked in bins of mass, redshift, peak height, concentration, accretion rate, and formation time; the median density profile is fitted with an analytic halo density model; the logarithmic derivative of the fitted profile is then used to identify the splashback valley. Width is measured from the shape of that fitted slope valley rather than from particle apocenters directly.

The principal reported scaling is with peak height: $\mathcal{D}$9 This is described as the strongest and cleanest dependence for $\mathcal{W}$0, with relatively little redshift scatter once written in terms of $\mathcal{W}$1 (Yu et al., 29 Jul 2025). The equivalent mass-redshift form is

$\mathcal{W}$2

and the paper also gives a direct fit

$\mathcal{W}$3

The same functional form remains valid with the updated Diemer (2023) density profile and in dark-matter-only runs (Yu et al., 29 Jul 2025).

In that analysis, width decreases with halo mass and decreases with redshift at fixed mass. At $\mathcal{W}$4, $\mathcal{W}$5 decreases from about $\mathcal{W}$6 to $\mathcal{W}$7 across the explored mass range. The width increases with concentration, with fit

$\mathcal{W}$8

and increases with halo formation time, with direct $\mathcal{W}$9 fit

$\mathcal{W}$0

By contrast, the dependence on short-term dynamical variables is weak: at $\mathcal{W}$1 a power-law trend with accretion rate,

$\mathcal{W}$2

is present, but it becomes much weaker at higher redshift, and the width shows little to no clear trend with most recent major merger time (Yu et al., 29 Jul 2025).

The paper’s interpretation is that $\mathcal{W}$3, like the splashback depth, is a long-term memory tracker of halo assembly. Smaller $\mathcal{W}$4 corresponds to a sharper, cleaner halo boundary; larger $\mathcal{W}$5 corresponds to a more smeared-out transition between orbiting and infalling material (Yu et al., 29 Jul 2025).

4. Nonsphericity, shell geometry, and directional spread

A distinct line of work treats splashback width as a consequence of halo asphericity. Mansfield et al. introduced SHELLFISH to identify the splashback shell in individual halos from a single density snapshot, defining the one-number splashback radius as the volume-equivalent radius

$\mathcal{W}$6

The shell itself is the full 3D surface of first apocenters rather than an imposed spherical boundary (Mansfield et al., 2016). That study does not define a unique scalar thickness, but width is encoded in the angular variation of the shell radius, in the spread between visually identified steepening regions, and in the distinction between short-type, long-type, and featureless-type radial profiles. The shells are reported to have non-ellipsoidal oval shapes, and the splashback radii measured by SHELLFISH are $\mathcal{W}$7 larger than those estimated from stacked density profiles at high accretion rates, with the latter biased low by high-mass subhalos (Mansfield et al., 2016).

Shin et al. showed that nonsphericity also imprints a directional dependence on the observed steepening. Using angle-binned density and alignment momentum correlations, they found that along the halo major axis the slope is shallower, while perpendicular to the major axis the slope is steeper; the splashback radius is larger along the major axis and smaller perpendicular to it (Okumura et al., 2018). Their interpretation is that spherical averaging smears the feature, and that the broader or shallower appearance along the major axis is a consequence of the varying boundary in that direction.

The most explicit directional-spread formulation appears in the IllustrisTNG study of splashback and accretion shocks. There the splashback boundary is not defined by the density-slope minimum but by the minimum in the logarithmic derivative of the radial velocity dispersion profile within each angular wedge, yielding a directional radius $\mathcal{W}$8 (Sen et al., 20 Feb 2026). Width is then characterized statistically from the full angular distribution $\mathcal{W}$9, summarized by its median, scatter $\mathcal{W}$0, and skewness $\mathcal{W}$1. In that sense, splashback width is the directional spread of a non-spherical multistreaming boundary. The paper finds that the splashback surface is more regular and less scattered than the shock surface, and that the gas–dark-matter boundary offset is robustly $\mathcal{W}$2–2 depending on boundary definition, with the largest offsets along void directions (Sen et al., 20 Feb 2026).

5. Observational inference, projection effects, and tracer dependence

Observed splashback width is usually inferred from projected profiles, which introduces systematic broadening and model dependence. In the DES/SPT/ACT analysis of splashback around SZ-selected clusters, the profile is modeled with an inner Einasto-like component, a transition function, and an outer infall component. That paper states explicitly that the splashback feature appears as a narrow minimum in the logarithmic derivative of the profile, and that the third derivative of the profile at splashback effectively measures the width of this minimum (Shin et al., 2018). Using this curvature-based proxy, the measured third-derivative $\mathcal{W}$3 ranges are $\mathcal{W}$4 for SPT and $\mathcal{W}$5 for ACT, compared with $\mathcal{W}$6 for simulation particles; optically selected redMaPPer clusters show a much larger range, $\mathcal{W}$7, corresponding to a narrower and sharper transition region in slope space (Shin et al., 2018).

The Planck-SZ cluster analysis similarly does not fit a standalone width, but describes the drop as “very sharp” and occurring within a factor of 2 in radius (Zuercher et al., 2018). Its 3D splashback radius is $\mathcal{W}$8, and the steepening is to slopes steeper than $\mathcal{W}$9 (Zuercher et al., 2018). By contrast, the redMaPPer-oriented critique of assembly bias and splashback argues that apparent width, depth, and location in projected cluster samples can be badly distorted by projection, contamination by foreground or background groups, and the cluster-identification radius $\gamma \simeq -4.5$0; in 3D the true splashback minimum is smoother and less extreme than in the projected result (Busch et al., 2017).

Galaxy-based cluster analyses often absorb width into a truncation function. In the SDSS-based mass-estimation study, the projected profile is modeled as

$\gamma \simeq -4.5$1

where $\gamma \simeq -4.5$2 is the transition scale and $\gamma \simeq -4.5$3 controls how sharp the transition is: larger $\gamma \simeq -4.5$4 means a steeper, narrower truncation, and smaller $\gamma \simeq -4.5$5 means a broader, smoother transition (Gabriel-Silva et al., 9 Jun 2025). That paper measures the splashback radius from the minimum in the logarithmic slope of the surface density within $\gamma \simeq -4.5$6, reports observed projected radii smaller than dark-matter expectations with $\gamma \simeq -4.5$7, and emphasizes that projection can broaden and shift the feature (Gabriel-Silva et al., 9 Jun 2025).

Tracer population matters. In the C-EAGLE analysis of stellar splashback, the stellar or intracluster-light caustic lies at almost the same radius as the dark-matter splashback but has a steeper typical slope, $\gamma \simeq -4.5$8, compared with $\gamma \simeq -4.5$9 for dark matter (Deason et al., 2020). Yet the stellar feature is more vulnerable to broadening or washout by substructure, bound stellar material, non-diffuse debris, and projection. Projected caustics are typically slightly smaller, $\gamma_p \to -5.5$0, and angular-median profiles recover sharper edges than mean profiles (Deason et al., 2020).

Dynamical friction provides another tracer-dependent effect. In isolated-cluster orbit integrations based on IllustrisTNG halos, dynamical friction can reduce measured galaxy-number splashback radii by up to roughly $\gamma_p \to -5.5$1 in low-mass clusters and can widen the distribution of radii at $\gamma_p \to -5.5$2, matching wider bootstrapped $\gamma_p \to -5.5$3 distributions; for $\gamma_p \to -5.5$4, however, the effect is not significant (O'Shea et al., 2024). This constrains interpretations in which a broadened or inward-shifted galaxy splashback feature is attributed solely to dynamical friction.

6. Alternative width notions: boundary ambiguity, environment, and physical interpretation

Some papers recast splashback width as stochastic boundary ambiguity rather than radial thickness. In the splashback mass-function formalism of Lee et al., the relevant quantity is the diffusion coefficient $\gamma_p \to -5.5$5, defined as the ratio of the variance of the splashback density threshold to the mass variance (Ryu et al., 2022). Larger $\gamma_p \to -5.5$6 means a more disturbed environment, a more stochastic threshold, and therefore a less sharply localized splashback boundary. The redshift evolution is fitted as

$\gamma_p \to -5.5$7

with $\gamma_p \to -5.5$8 for $\gamma_p \to -5.5$9 and $\sim 3.5\%$0 for $\sim 3.5\%$1 over $\sim 3.5\%$2 (Ryu et al., 2022). In this formulation, width is a threshold-space broadening rather than a geometric $\sim 3.5\%$3.

The physical interpretation of width across methodologies is consistent in one respect: broadening is associated with a less distinct separation between orbiting and infalling material. Shin et al. attribute the greater sharpness of the momentum-space splashback to the fact that splashback is fundamentally a phase-space feature; the density statistic is more contaminated by the two-halo term and nonlinear, scale-dependent bias, which smear the steepening (Okumura et al., 2018). The MillenniumTNG width analysis similarly argues that a wider valley reflects a more smeared-out boundary, while a narrower valley indicates a more distinct halo edge shaped by long-term assembly history rather than short-term dynamical transients (Yu et al., 29 Jul 2025).

Taken together, the literature supports three technically distinct but related meanings of splashback width. First, it can denote the radial breadth of the valley in the logarithmic slope profile, formalized by $\sim 3.5\%$4 (Yu et al., 29 Jul 2025). Second, it can denote the angular spread of a genuinely aspherical splashback shell or surface, quantified by directional radii and their dispersion (Mansfield et al., 2016, Sen et al., 20 Feb 2026). Third, it can denote ambiguity in the location of the splashback boundary, encoded statistically by transition-function curvature or by a diffusion coefficient in threshold space (Shin et al., 2018, Ryu et al., 2022). The absence of a single universal definition is therefore not a contradiction but a reflection of the fact that splashback is simultaneously a caustic, a profile transition, and a nonspherical boundary.