Splashback Depth in Dark Matter Halos

• Splashback Depth is the vertical difference between the minimum and outer turnover in the logarithmic density slope profile, marking a sharp dynamical halo boundary.
• It is derived from stacked, spherically averaged density profiles using parameters (𝒟 and 𝒲) that scale with halo mass and reflect cumulative assembly history.
• Methodologies and tracers affect its measurement, necessitating careful three-dimensional modeling to mitigate projection effects and selection biases.

to=arxiv_search 天天中彩票中奖了om {"query":"all:(splashback depth halo density profile) OR ti:(splashback depth)","max_results":10,"sort_by":"relevance"}function call 彩神争霸官方下载քներ to=arxiv_search 】【。】【”】【commentary 彩神争霸大发快json {"result":[{"arxiv_id":"(Yu et al., 29 Jul 2025)","title":"Measuring the splashback feature: Dependence on halo properties and history","authors":["X. Zhang","B. Mostoghiu","A. Sharma","L. Gao"],"published":"2025-07-29","abstract":"In this study, we define the novel splashback depth D and width 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 D∝(log10 M)^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 W∝ν^-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."},{"arxiv_id":"(Deason et al., 2020)","title":"Stellar splashback: the edge of the intracluster light","authors":["A. J. Deason","E. E. O'Sullivan","A. Robertson","T. Theuns","J. T. McCarthy","K. Schaller"],"published":"2020-10-06","abstract":"We examine the outskirts of galaxy clusters in the C-EAGLE simulations to quantify the edges' of the stellar and dark matter distribution. The radius of the steepest slope in the dark matter, commonly used as a proxy for the splashback radius, is located at ~r_200m; the strength and location of this feature depends on the recent mass accretion rate, in good agreement with previous work. Interestingly, the stellar distribution (or intracluster light, [ICL](https://www.emergentmind.com/topics/incentive-compatibility-in-the-large-icl)) also has a well-defined edge, which is directly related to the splashback radius of the halo. Thus, detecting the edge of the ICL can provide an independent measure of the physical boundary of the halo, and the recent mass accretion rate. We show that these caustics can also be seen in the projected density profiles, but care must be taken to account for the influence of substructures and other non-diffuse material, which can bias and/or weaken the signal of the steepest slope. This is particularly important for the stellar material, which has a higher fraction bound in subhaloes than the dark matter. Finally, we show that thestellar splashback' feature is located beyond current observational constraints on the ICL, but these large projected distances (>> 1 Mpc) and low surface brightnesses (mu >> 32 mag/arcsec^2) can be reached with upcoming observational facilities such as the Vera C. Rubin Observatory, the Nancy Grace Roman Space Telescope, and Euclid."},{"arxiv_id":"(Diemer, 2017)","title":"The Splashback Radius of Halos from Particle Dynamics. I. The SPARTA Algorithm","authors":["Benedikt Diemer"],"published":"2017-03-28","abstract":"Motivated by the recent proposal of the splashback radius as a physical boundary of dark matter halos, we present a parallel computer code for Subhalo and PARticle Trajectory Analysis (SPARTA). The code analyzes the orbits of all simulation particles in all host halos, billions of orbits in the case of typical cosmological N-body simulations. Within this general framework, we develop an algorithm that accurately extracts the location of the first apocenter of particles after infall into a halo, or splashback. We define the splashback radius of a halo as the smoothed average of the apocenter radii of individual particles. This definition allows us to reliably measure the splashback radii of 95% of host halos above a resolution limit of 1000 particles. We show that, on average, the splashback radius and mass are converged to better than 5% accuracy with respect to mass resolution, snapshot spacing, and all free parameters of the method."},{"arxiv_id":"(Popolo et al., 2022)","title":"Splashback radius in a spherical collapse model","authors":["A. Del Popolo","M. Le Delliou"],"published":"2022-09-28","abstract":"It has been shown some years ago that dark matter haloes outskirts are characterized by very steep density profiles in a very small radial range. This feature has been interpreted as a pile up of at a similar location of different particle orbits, namely splashback material at half an orbit after collapse. Adhikari et al. (2014), obtained the location of the splashback radius through a very simple model, namely calculating a dark matter shell trajectory in the secondary infall model while it crosses a growing, NFW profile shaped, dark matter halo. Since they imposed a halo profile instead of calculating it from the trajectories of the shells of dark matter, they were not able to find the dark matter profile around the splashback radius. In the present paper, we use an improved spherical infall model taking into shell crossing, and several physical effects like ordered, and random angular momentum, dynamical friction, adiabatic contraction, etc. This allow us to determine the density profile from the inner to outer region, and study the behavior of the outer density profile. We will compare the density profiles, and the logarithmic slope of the density profile with the results of Diemer 2014 simulations, finding a good agreement between the prediction of the model and the simulations."},{"arxiv_id":"(Adhikari et al., 2014)","title":"Splashback in accreting dark matter halos","authors":["Susmita Adhikari","Neal Dalal","Robert T. Chamberlain"],"published":"2014-09-16","abstract":"Recent work has shown that density profiles in the outskirts of dark matter halos can become extremely steep over a narrow range of radius. This behavior is produced by splashback material on its first apocentric passage after accretion. We show that the location of this splashback feature may be understood quite simply, from first principles. We present a simple model, based on spherical collapse, that accurately predicts the location of splashback without any free parameters. The important quantities that determine the splashback radius are accretion rate and redshift."},{"arxiv_id":"(Okumura et al., 2018)","title":"Splashback radius of nonspherical dark matter halos from cosmic density and velocity fields","authors":["Masahiro Takada","Bhuvnesh Jain","Surhud More","Toshifumi Futamase","Xiaohu Yang","Song Huang","Marcel Schmittfull","Nicolás Battaglia","Rachel Mandelbaum","Eli S. Rykoff"],"published":"2018-07-07","abstract":"We investigate the splashback features of dark-matter halos based on cosmic density and velocity fields. Besides the density correlation function binned by the halo orientation angle which was used in the literature, we introduce, for the first time, the corresponding velocity statistic, alignment momentum correlation function, to take into account the asphericity of halos. Using large-volume, high-resolution N-body simulations, we measure the alignment statistics of density and velocity. On halo scales, x~R200m~1Mpc/h, we detect a sharp steepening in the momentum correlation associated with the physical halo boundary, or the splashback feature, which is found more prominent than in the density correlation. We also find that the splashback radius determined from the density correlation becomes ~3.5% smaller than that from the momentum correlation, with their correlation coefficient being 0.605. Moreover, the orientation-dependent splashback feature due to halo asphericity is measured when the density profile is determined by dark-matter particles, which can be used as a test of collisional CDM since the halo shape is predicted to be rounder in such a model."},{"arxiv_id":"(Adhikari et al., 2018)","title":"Splashback in galaxy clusters as a probe of cosmic expansion and gravity","authors":["Hassan Murata","Naonori S. Sugiyama","Rachel Mandelbaum","Masamune Oguri","Masahiro Takada","Surhud More","Ravi K. Sheth","Eli S. Rykoff","Rachel Bean"],"published":"2018-06-12","abstract":"The splashback radius is a physical scale in dark matter halos that is set by the gravitational dynamics of recently accreted shells. We use analytical models and N-body simulations to study the dependence of splashback on dark energy and screened modified gravity theories. In modified gravity models, the transition from screened to unscreened regions typically occurs in the cluster outskirts, suggesting potentially observable signatures in the splashback feature. We investigate the location of splashback in both chameleon and Vainshtein screened models and find significant differences compared with LCDM predictions. We also find an interesting interplay between dynamical friction and modified gravity, providing a distinctive signature for modified gravity models in the behavior of the splashback feature as a function of galaxy luminosity."},{"arxiv_id":"(Contigiani et al., 2018)","title":"The splashback radius in symmetron gravity","authors":["A. S. Lombriser","A. Taylor"],"published":"2018-12-13","abstract":"The splashback radius r_sp has been identified in cosmological N-body simulations as an important scale associated with gravitational collapse and the phase-space distribution of recently accreted material. We employ a semi-analytical approach to study the spherical collapse of dark matter haloes in symmetron gravity and provide insights into how the phenomenology of splashback is affected. The symmetron is a scalar-tensor theory of gravity which exhibits a screening mechanism whereby higher-density regions are screened from the effects of a fifth force. In this model, we find that, as over-densities grow over cosmic time, the inner region becomes heavily screened. In particular, we identify a sector of the parameter space for which material currently sitting at r_sp has followed, during the collapse, the formation of this screened region. As a result, we find that for this part of the parameter space the splashback radius is maximally affected by the symmetron force and we predict changes in r_sp up to around 10% compared to its General Relativity value. Because this margin is within the precision of present splashback experiments, we expect this feature to soon provide constraints for symmetron gravity on previously unexplored scales."},{"arxiv_id":"(Rana et al., 2023)","title":"The eROSITA Final Equatorial-Depth Survey (eFEDS) -- Splashback radius of X-ray galaxy clusters using galaxies from HSC survey","authors":["S. K. Rana","M. Oguri","J. Coupon","R. Mandelbaum","M. Nishizawa","A. More","H. Miyatake","E. Rozo","E. S. Rykoff","M. Gatti","N. Battaglia","S. Ettori","J. Klein","N. Werner","A. Schwope","J. Sanders","P. Mazzotta","L. Liu","J. Annis","A. Biviano","T. Comparat","A. Finoguenov","C. Garrel","A. Merloni","J. Mohr","T. N. Nagai","A. Pillepich","M. Ramos-Ceja","J. Salvato","M. Schrabback","K. Sharon","F. Sommer"],"published":"2023-01-09","abstract":"We present the splashback radius measurements around the SRG/eROSITA eFEDS X-ray selected galaxy clusters by cross-correlating them with HSC S19A photometric galaxies. The X-ray selection is expected to be less affected by systematics related to projection that affects optical cluster finder algorithms. We use a nearly volume-limited sample of 109 galaxy clusters selected in 0.5-2.0 keV band having luminosity LX > 10^43.5 erg s^-1 h^-2 within the redshift z < 0.75 and obtain measurements of the projected cross-correlation with a signal-to-noise of 17.43. We model our measurements to infer a three-dimensional profile and find that the steepest slope is sharper than -3 and associate the location with the splashback radius. We infer the value of the 3D splashback radius r_sp = 1.45^{+0.30}{-0.26} h^-1 Mpc. We also measure the weak lensing signal of the galaxy clusters and obtain halo mass log[M_200m/h^-1 Msun] = 14.52 +/- 0.06 using the HSC-S16A shape catalogue data at the median redshift z = 0.46 of our cluster sample. We compare our r_sp values with the spherical overdensity boundary r_200m = 1.75 +/- 0.08 h^-1 Mpc based on the halo mass which is consistent within 1.2 sigma with the LCDM predictions. Our constraints on the splashback radius, although broad, are the best measurements thus far obtained for an X-ray selected galaxy cluster sample."},{"arxiv_id":"(Shin et al., 2018)","title":"Measurement of the Splashback Feature around SZ-selected Galaxy Clusters with DES, SPT and ACT","authors":["Shadab Alam","Rachel Mandelbaum","Bhuvnesh Jain","Sunayama Tomoki","Eli Rykoff","Masahiro Takada","Tobias Baldauf","Rachel Bean","Marcel Schmittfull","Megan Chang","Erin Sheldon","Yao-Yuan Mao","Benedikt Diemer","Surhud More"],"published":"2018-11-14","abstract":"We present a detection of the splashback feature around galaxy clusters selected using their Sunyaev-Zel'dovich (SZ) signal. Recent measurements of the splashback feature around optically selected galaxy clusters have found that the splashback radius, rsp, is smaller than predicted by N-body simulations. A possible explanation for this discrepancy is that rsp inferred from the observed radial distribution of galaxies is affected by selection effects related to the optical cluster-finding algorithms. We test this possibility by measuring the splashback feature in clusters selected via the SZ effect in data from the South Pole Telescope SZ survey and the Atacama Cosmology Telescope Polarimeter survey. The measurement is accomplished by correlating these clusters with galaxies detected in the Dark Energy Survey Year 3 data. The SZ observable used to select clusters in this analysis is expected to have a tighter correlation with halo mass and to be more immune to projection effects and aperture-induced biases than optically selected clusters. We find that the measured rsp for SZ-selected clusters is consistent with the expectations from simulations, although the small number of SZ-selected clusters makes a precise comparison difficult. In agreement with previous work, when using optically selected redMaPPer clusters, rsp is ~2 sigma smaller than in the simulations. These results motivate detailed investigations of selection biases in optically selected cluster catalogs and exploration of the splashback feature around larger samples of SZ-selected clusters. Additionally, we investigate trends in the galaxy profile and splashback feature as a function of galaxy color, finding that blue galaxies have profiles close to a power law with no discernible splashback feature, which is consistent with them being on their first infall into the cluster."}]} Splashback depth is a descriptor of the strength of the splashback feature at the halo boundary. In the recent formal usage introduced in MillenniumTNG, it is the vertical extent of the “valley” in the logarithmic slope profile of the spherically averaged density, measured between the minimum at splashback and the outer turnover where the slope recovers. Earlier work usually characterized the same idea through the minimum logarithmic slope itself: a deeper splashback feature meant a more negative steepest slope. In both senses, splashback depth is distinct from splashback radius. The radius specifies where recently accreted material reaches first apocenter; the depth specifies how strongly that dynamical boundary is imprinted in the density structure (Yu et al., 29 Jul 2025).

1. Dynamical meaning and early usage

The dynamical basis of splashback is the first apocenter of newly accreted matter after infall. In accreting dark-matter halos, the outer density profile can steepen sharply because recently accreted material piles up near that first apocentric passage, producing a caustic-like feature. In the spherical-collapse treatment of Adhikari, Dalal, and Chamberlain, the location of splashback is controlled mainly by the halo mass accretion rate and by epoch or cosmology, while the depth of the feature depends additionally on non-sphericity and halo peak height (Adhikari et al., 2014).

Particle-based formulations made the dynamical picture explicit. The SPARTA algorithm tracks the first apocenter of individual particles after infall and defines the splashback radius of a halo as the smoothed average of those apocenter radii. Using that definition, splashback radii can be measured for 95% of host halos above a resolution limit of 1000 particles, and the resulting splashback radii and masses are converged to better than 5% accuracy with respect to mass resolution, snapshot spacing, and the free parameters of the method (Diemer, 2017). This suggests that the depth problem is not fundamentally about locating a single shell, but about understanding the structure of the distribution of first-apocenter radii and how that distribution maps into a finite-width steepening region.

Before the explicit introduction of a quantity called splashback depth, the literature usually encoded depth through slope minima. In nonspherical halos, for example, the density-based splashback signal reaches a steepest slope of about $\gamma \simeq -4.5$, while the velocity-weighted alignment momentum correlation reaches $\gamma_p \to -5.5$, making the latter more prominent than the density correlation. In that paper, the momentum-defined splashback radius is also about 3.5% larger than the density-defined one, and the feature is shallower along the halo major axis but steeper perpendicular to it (Okumura et al., 2018).

2. Formal definition of splashback depth

The formal definition of splashback depth was introduced in “Measuring the splashback feature: Dependence on halo properties and history,” which defines the novel splashback depth $\mathcal{D}$ and width $\mathcal{W}$ from the logarithmic slope of the dark-matter density profile in MillenniumTNG (Yu et al., 29 Jul 2025). The relevant profile is

$\frac{d\log\rho}{d\log r},$

which develops a valley shape: it steepens outward, reaches a minimum near splashback, and then turns over and flattens toward the mean background density.

In that formulation, splashback depth is not the minimum slope itself. It is the difference between the slope at the outer turnover point and the slope at the splashback minimum,

$$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$

The turnover point is defined operationally as the point of maximum curvature in the gradient to the right of the minimum. The companion quantity $\mathcal{W}$ is the width of the logarithmic-slope valley measured at half the depth. This makes $\mathcal{D}$ a prominence measure of the splashback valley, while $\mathcal{W}$ is its horizontal extent (Yu et al., 29 Jul 2025).

The measurements are based on stacked, spherically averaged dark-matter density profiles in the largest hydrodynamic MillenniumTNG run, MTNG-L500-4320-A. Haloes are selected with $M_{200\mathrm{m}}\ge 10^{13}M_\odot$ over $\gamma_p \to -5.5$0, normalized by $\gamma_p \to -5.5$1, and stacked using the median profile. Dark matter particles are binned into 85 logarithmically spaced spherical bins from $\gamma_p \to -5.5$2 to $\gamma_p \to -5.5$3. The stacked profile is fit with a DK14-style composite model consisting of an inner Einasto term, a transition term, and an outer component, and the logarithmic slope is then extracted from the fit rather than directly from the raw binned profile (Yu et al., 29 Jul 2025).

3. Dependence on halo properties and assembly history

In MillenniumTNG, splashback depth shows its clearest dependence on halo mass. At $\gamma_p \to -5.5$4, across five mass bins from $\gamma_p \to -5.5$5 to $\gamma_p \to -5.5$6, $\gamma_p \to -5.5$7 rises from about 2.2 to 3.0, while $\gamma_p \to -5.5$8 falls from about 2.0 to 1.0. More massive haloes therefore have a deeper and narrower splashback feature. The preferred mass-only fit across redshift is

$\gamma_p \to -5.5$9

which is the basis of the headline scaling $\mathcal{D}$0 (Yu et al., 29 Jul 2025).

At fixed mass, the same study finds effectively no significant redshift dependence for $\mathcal{D}$1, with $\mathcal{D}$2. By contrast, the width has stronger redshift dependence, with

$\mathcal{D}$3

This distinction is central to the interpretation advanced there: depth and width are not redundant with the point of steepest slope, because they encode different aspects of the valley shape (Yu et al., 29 Jul 2025).

Other halo properties show systematic but non-identical trends. The width depends most strongly on peak height, following

$\mathcal{D}$4

whereas the depth–$\mathcal{D}$5 relation is noisier: $\mathcal{D}$6 Depth decreases weakly with concentration,

$\mathcal{D}$7

and decreases with formation redshift according to

$\mathcal{D}$8

Earlier-forming, higher-concentration haloes thus have shallower and broader splashback features. The study further reports only weak or ambiguous dependence on present accretion rate and most recent major merger time, and therefore characterizes splashback depth and width as “long term memory trackers” shaped more by cumulative assembly than by short-term dynamical indicators (Yu et al., 29 Jul 2025).

4. Alternative operational meanings across tracers and statistics

Outside the formal $\mathcal{D}$9 definition, tracer studies have continued to use “depth” in the earlier sense of the minimum logarithmic slope. In C-EAGLE galaxy clusters, the radius of the steepest slope in dark matter is located around $\mathcal{W}$0, and the stellar or intracluster-light distribution shows a closely corresponding splashback feature. The stellar and dark-matter caustic radii lie nearly on a one-to-one relation with rms scatter $\mathcal{W}$1 and Pearson/Spearman coefficients $\mathcal{W}$2, but the stellar feature is deeper in slope space: outer stellar caustics have steepest slopes around $\mathcal{W}$3, compared with $\mathcal{W}$4 for dark matter (Deason et al., 2020). In that sense, splashback depth can be tracer dependent even when splashback radius is not.

The same C-EAGLE analysis shows that projected measurements systematically modify the apparent feature. In projection, the location shifts roughly as

$\mathcal{W}$5

and the steepening is diluted, especially when ordinary mean annular profiles are used instead of an angular-median estimator. Substructure, luminous satellites, stellar streams, plumes, and other non-diffuse structures weaken the observed splashback depth, particularly for stellar material, because a larger fraction of stellar mass at large radii remains bound in subhaloes than the corresponding fraction of dark matter (Deason et al., 2020).

Velocity statistics provide a different operational notion of depth. In nonspherical halos, the alignment momentum correlation function produces a deeper splashback minimum than the density correlation, with $\mathcal{W}$6 versus $\mathcal{W}$7, and a slightly larger splashback radius, $\mathcal{W}$8. Orientation dependence is also strong: the splashback dip is less deep and located at larger radius along the major axis, but sharper and at smaller radius perpendicular to it (Okumura et al., 2018). This suggests that “depth” is not an invariant scalar of a halo alone; it also depends on tracer choice, estimator choice, and angular averaging.

5. Observational measurements and methodological controversies

Observational work usually infers splashback depth from a parametric three-dimensional profile fit to projected data. A weak-lensing analysis of 27 massive clusters found

$\mathcal{W}$9

but it explicitly did not detect a statistically significant steepening in the lensing data alone. The splashback depth was therefore a model-based inference, and the 99.7% confidence interval for the slope at splashback, $\frac{d\log\rho}{d\log r},$0, did not exclude relatively shallow values (Contigiani et al., 2018).

By contrast, the LoCuSS study reported a direct detection of the splashback feature above $\frac{d\log\rho}{d\log r},$1 in the stacked $\frac{d\log\rho}{d\log r},$2-band luminosity-density profile of spectroscopically confirmed cluster members. For the full sample the projected slope reached $\frac{d\log\rho}{d\log r},$3 and the minimum 3D logarithmic slope was $\frac{d\log\rho}{d\log r},$4; for clusters without X-ray-detected infalling groups the fitted ratio was $\frac{d\log\rho}{d\log r},$5 with minimum 3D slope $\frac{d\log\rho}{d\log r},$6, while clusters with infalling groups gave $\frac{d\log\rho}{d\log r},$7 and minimum slope $\frac{d\log\rho}{d\log r},$8. For the dynamically quieter subsample, the Bayes factor in favor of a model with a splashback transition was $\frac{d\log\rho}{d\log r},$9 (Bianconi et al., 2020).

X-ray- and SZ-selected cluster samples have generally been used to reduce the projection and aperture biases associated with optical cluster finding. In eFEDS X-ray clusters, forward modeling of the projected cross-correlation yielded

$$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$0

with $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$1, consistent within $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$2 with $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$3CDM expectations (Rana et al., 2023). In RASS-MCMF X-ray clusters the inferred three-dimensional splashback radius was $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$4 and the steepest slope was $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$5 (Joshi et al., 10 Jun 2025). In SPT and ACT SZ-selected clusters, the minimum total-profile slopes were $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$6 and $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$7, with collapsed-component slopes $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$8 and $$\mathcal{D}\equiv\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{turnover}}}-\left.\frac{d\log\rho}{d\log r}\right|_{r_{\text{sp}}}.$$9, respectively (Shin et al., 2018).

The main controversy has been whether optically selected samples artificially bias splashback depth and radius. In redMaPPer-like cluster samples, simulated projected profiles can reproduce the observed pattern in which high-concentration clusters have a deeper minimum slope at smaller radius, even though previous theoretical work predicted the opposite trend for true three-dimensional halos. That paper attributes the discrepancy to confusion between genuine splashback features and features imposed by the cluster identification and concentration-estimation procedures, especially the aperture scale $\mathcal{W}$0 and foreground/background contamination (Busch et al., 2017). A CAMIRA-based HSC analysis reached a different conclusion: red-galaxy-only splashback constraints at $\mathcal{W}$1 were more consistent with model predictions than with the earlier $\mathcal{W}$2 low values, and HOD mock catalogs showed that a redMaPPer-like finder induces a smaller inferred splashback radius, especially at lower richness, while the bias is significantly reduced when the aperture size is increased (Murata et al., 2020).

6. Physical significance and extensions

Splashback depth has been used not only as a descriptive observable but also as a physical discriminator. In spherical and semi-analytic models, high-$\mathcal{W}$3 or rapidly growing halos develop sharper outer steepening, with the steepest slope occurring around $\mathcal{W}$4–$\mathcal{W}$5 and reaching about $\mathcal{W}$6 for high-$\mathcal{W}$7 systems, while low-$\mathcal{W}$8 halos are shallower (Popolo et al., 2022). This suggests that depth encodes how concentrated or dispersed the first-apocenter distribution is once non-sphericity, angular momentum, and shell crossing are accounted for.

The splashback feature also responds to modified gravity and cosmic expansion. In the analytical and numerical study of Murata et al., viable dark-energy variations change splashback only at the few-percent level, whereas strong modified-gravity models can shift the splashback radius by about 10% or larger in concentration-selected samples, with an additional signature in the luminosity dependence caused by reduced dynamical friction (Adhikari et al., 2018). In symmetron gravity, semi-analytical spherical collapse predicts changes in $\mathcal{W}$9 up to around 10% relative to General Relativity, with the largest effect when the shell currently at splashback experienced the formation of the screened region during its collapse (Contigiani et al., 2018). These studies chiefly concern splashback radius, but they imply that depth and width should also be sensitive to the evolving force law wherever the splashback shell overlaps the screening transition.

The contemporary interpretation is therefore twofold. First, splashback depth is a structural observable complementary to splashback radius, width, and truncation radius, because it measures how strongly the halo boundary is expressed in the logarithmic slope profile (Yu et al., 29 Jul 2025). Second, it is a methodology-sensitive quantity: projection, off-centering, substructure, tracer population, selection aperture, and angular averaging can all weaken, sharpen, or displace the apparent feature. The most robust use of splashback depth has accordingly shifted toward carefully modeled three-dimensional inference, explicit treatment of systematics, and comparison across multiple tracers and selection functions.