S-type Voids in the Cosmic Web

• S-type voids are cosmic underdensities with overdense shells, classifying them as void-in-cloud systems influenced by surrounding collapse.
• They exhibit distinct dynamics, including mutual convergence and non-zero galaxy bias profiles, differentiating them from R-type voids.
• Their identification through integrated radial density profiles informs studies of void clustering, galaxy bias, and dark-sector phenomenology.

Searching arXiv for recent and foundational papers on S-type voids and related cosmic void classifications. tool call: arxiv_search({"query":"S-type voids cosmic voids void-in-cloud galaxy bias profile of cosmic voids", "max_results": 10, "sort_by": "relevance"}) S-type voids are an environmental class of cosmic voids defined by the presence of an overdense large-scale surrounding region or, equivalently in the earlier profile-based literature, by an overdense shell around an underdense interior. In the hierarchical void picture, they correspond to the void-in-cloud regime: voids embedded in overdense environments and typically influenced by surrounding collapse, in contrast to R-type or void-in-void systems embedded in underdense regions and typically expanding. This classification has become important because S-type voids differ from R-type voids not only in their radial density structure, but also in their dynamics, clustering, relation to superstructures, and the large-scale galaxy bias of the galaxies they host (Montero-Dorta et al., 20 Apr 2025, Lambas et al., 2015).

1. Definition and taxonomic status

In the void-environment literature, S-type voids are the population associated with overdense surroundings. Earlier work operationalized this through the integrated radial density contrast profile: R-type voids have continuously rising integrated radial density profiles, whereas S-type voids exhibit an overdense shell. In this language, S-type voids are shell-bearing, steeply rising-profile systems in the void-in-cloud regime (Lambas et al., 2015, Lares et al., 2017).

A later formulation makes the environmental criterion explicit through the maximum overdensity in the void’s outer neighborhood. In that scheme, the relevant variable is $\Delta_{\rm max}$ measured in the radial range $[2R_{\rm void}, 3R_{\rm void}]$. A void is classified as S-type if $\Delta_{\rm max}\ge 0$, meaning that it is embedded in a large-scale overdense surrounding region; it is R-type if $\Delta_{\rm max}<0$, meaning that it sits inside a large-scale underdense environment. The associated physical picture is that S-type voids tend to be surrounded by contracting regions, while R-type voids are surrounded by expanding regions (Montero-Dorta et al., 20 Apr 2025).

The terminological point is consequential. In this research area, “S-type” denotes an environmental/dynamical class, not a shape class. This is underscored by the fact that catalog-construction work such as DESIVAST does not define S-type or R-type subclasses at all, even when it provides detailed void geometries and boundary metadata (Rincon et al., 2024).

Aspect	S-type voids	R-type voids
Hierarchical mode	void-in-cloud	void-in-void
Environment	large-scale overdense surroundings	large-scale underdense surroundings
Radial-profile signature	overdense shell; steeply rising integrated profile	continuously or gently rising integrated profile
Sign of $\Delta_{\rm max}$	$\Delta_{\rm max}\ge 0$	$\Delta_{\rm max}<0$
Qualitative dynamics	contraction, compression, convergence	expansion, separation, recession

2. Identification criteria and profile diagnostics

The underlying void catalogs in this literature are commonly constructed from spherical underdensities or watershed zones, and the S/R split is then imposed through the surrounding density profile. In simulation and SDSS-based analyses of void motions and clustering, the void-finding condition is

$\Delta(r) < -0.9,$

used to select the largest underdense sphere around a candidate center. Within those catalogs, the S-type/R-type distinction is then assigned from the behavior of $\Delta(r)$ outside the void: S-type voids are those with a compensating overdense shell, while R-type voids rise toward the mean density without such a shell (Lambas et al., 2015).

The bias-profile analysis in IllustrisTNG uses a spherical-overdensity identification scheme and then classifies voids environmentally through $\Delta_{\rm max}$ in $[2R_{\rm void}, 3R_{\rm void}]$0. This makes explicit that the S-type label is not determined only by the interior emptiness of the void, but by the outer neighborhood. The classification therefore encodes the large-scale context of the void rather than only the density deficit at its center (Montero-Dorta et al., 20 Apr 2025).

A recurring implication is that void taxonomy is partly algorithm-dependent. DESIVAST, for example, constructs catalogs with VoidFinder and with the watershed-based $[2R_{\rm void}, 3R_{\rm void}]$1 pipeline, distinguishes interior, edge, and near-edge voids, and emphasizes that the catalogs are suitable for studying variation in galaxy properties with cosmic environment and for cosmological studies, but it does not provide a formal S-type/R-type classification. This suggests that any S-type labeling of modern survey void catalogs must be added through an external environmental diagnostic rather than assumed to be intrinsic to the catalog definition (Rincon et al., 2024).

3. Environmental dynamics and large-scale flows

The dynamical signature of S-type voids is convergence rather than free expansion. In the simulation and SDSS-DR7 analysis of coherent void motions, S-type voids are the observational/simulation proxy for void-in-cloud systems, and the central result is that S-type voids systematically approach each other, whereas R-type voids are mutually receding. Using

$[2R_{\rm void}, 3R_{\rm void}]$2

with $[2R_{\rm void}, 3R_{\rm void}]$3 the angle between them, approaching configurations satisfy $[2R_{\rm void}, 3R_{\rm void}]$4, and the scalar pairwise measure

$[2R_{\rm void}, 3R_{\rm void}]$5

is negative for approaching pairs and positive for receding pairs. The pairwise velocity distribution is bimodal, and the origin of the bimodality is the void large-scale environment (Lambas et al., 2015).

The same study finds that void bulk motions are non-negligible, with coincident core and shell velocity estimates in the range $[2R_{\rm void}, 3R_{\rm void}]$6–$[2R_{\rm void}, 3R_{\rm void}]$7, and that the systematic relative velocities associated with the S/R split are of order $[2R_{\rm void}, 3R_{\rm void}]$8–$[2R_{\rm void}, 3R_{\rm void}]$9 with coherence lengths up to $\Delta_{\rm max}\ge 0$0. The observational analysis, based on the SDSS-DR7 Main Galaxy Sample and a peculiar velocity field reconstructed from linear theory, reproduces the same environmental split: S-type trends match the simulation and remain approaching, while R-type trends remain receding (Lambas et al., 2015).

The linkage to nearby overdense structures sharpens this interpretation. Using Future Virialized Structures (FVSs) as proxies for superstructures, S-type voids are found to be positively correlated with FVSs, whereas R-type voids are anticorrelated. The correlation is strongest below about $\Delta_{\rm max}\ge 0$1 and weakens beyond that scale. Voids close to FVSs show infall regardless of type, but the effect is stronger and more coherent for S-type voids because they tend to lie closer to the superstructures; the infall signal remains visible out to about $\Delta_{\rm max}\ge 0$2 in the combined sample, with the strong S-type signal most evident within roughly $\Delta_{\rm max}\ge 0$3 (Lares et al., 2017).

These results support a common physical reading. S-type voids are underdense interiors embedded in collapsing or overcompensated environments; their kinematics are therefore shaped by the surrounding overdensity, which channels nearby voids toward each other and toward neighboring superstructures. A plausible implication is that S-type voids are best understood as components of the cosmic web whose dynamics are externally regulated to a greater degree than those of R-type voids.

4. Large-scale galaxy bias in S-type voids

A recent development is the direct measurement of the large-scale galaxy bias profile inside voids. In IllustrisTNG, the large-scale linear bias is not estimated from the conventional ratio of correlation functions or power spectra alone, even though the usual large-scale definition is of the form

$\Delta_{\rm max}\ge 0$4

Instead, the analysis assigns each galaxy an individual effective bias from the large-scale Fourier-space relation between the galaxy and dark-matter density fields and then averages over the selected population. Modes up to $\Delta_{\rm max}\ge 0$5 are used, where the galaxy-to-dark-matter power-spectrum ratio is approximately constant, so the inferred quantity is explicitly a large-scale linear bias rather than a local overdensity ratio (Montero-Dorta et al., 20 Apr 2025).

The principal empirical result is that the bias profile inside voids rises outward when distance is normalized by the void radius. For the full void sample, galaxies near the center have the lowest bias, and the bias generally increases toward the void wall. The normalized coordinate $\Delta_{\rm max}\ge 0$6 reveals this trend more clearly than physical distance in $\Delta_{\rm max}\ge 0$7, especially for larger voids (Montero-Dorta et al., 20 Apr 2025).

For S-type voids specifically, the stacked bias profile is higher at all normalized radii than for R-type voids and is also slightly steeper. Within S-type voids, the average bias is positive on average, $\Delta_{\rm max}\ge 0$8, and it tends to rise outward with $\Delta_{\rm max}\ge 0$9. By contrast, R-type voids have $\Delta_{\rm max}<0$0. The sign difference is tied to environment: the mean bias inside a void correlates strongly with $\Delta_{\rm max}<0$1, and since S-type voids satisfy $\Delta_{\rm max}<0$2 by definition, they occupy the high-bias side of that relation (Montero-Dorta et al., 20 Apr 2025).

Void size and environment are entangled in this behavior. The average bias for the entire galaxy population within voids rises with the density of the surrounding environment and consequently decreases with increasing void size. The paper reports that S-type voids are slightly smaller on average than R-type voids, with average $\Delta_{\rm max}<0$3 of $\Delta_{\rm max}<0$4 for S-type voids and $\Delta_{\rm max}<0$5 for R-type voids. The difference in bias is not primarily a halo-mass effect, because the host halo mass profiles of S-type and R-type voids are very similar; the interpretation advanced is therefore that the S-type/R-type split in bias is a secondary bias effect, analogous to assembly bias (Montero-Dorta et al., 20 Apr 2025).

5. Clustering, clumps, and collective organization

S-type voids are also a distinct population in void clustering statistics. In the analysis of void clustering and void clumps, the two-point autocorrelation function of void centers is measured with pair separation normalized by the sum of void radii,

$\Delta_{\rm max}<0$6

so that $\Delta_{\rm max}<0$7 corresponds to voids in contact. Both R-type and S-type voids cluster more strongly than the full void sample, demonstrating that voids of the same evolutionary/environmental type preferentially gather together. S-type voids show enhanced clustering, although R-type voids exhibit the strongest and most significant clustering (Lares et al., 2017).

This enhanced same-type clustering motivates the construction of void clumps through a friend-of-friends or percolation algorithm. Voids are linked if their centers are separated by less than the mean void separation,

$\Delta_{\rm max}<0$8

where $\Delta_{\rm max}<0$9 is the number density of the void sample. The geometry of these clumps is characterized through the maximum separation $\Delta_{\rm max}$0, the minimal spanning tree (MST) length, and the elongation parameter defined as $\Delta_{\rm max}$1. R-type and S-type clumps have broadly similar sizes and shapes, and there is no compelling evidence that S-type clumps are geometrically special beyond their environment (Lares et al., 2017).

The dynamics of clumps preserve the S-type signature of convergence. S-type void pairs favor negative $\Delta_{\rm max}$2, meaning that the voids are approaching each other, while R-type pairs favor positive $\Delta_{\rm max}$3, meaning recession. In the galaxy velocity field around two-void clumps, galaxies flow inward around S-type clumps, consistent with collapse, whereas around R-type clumps they flow outward from the pair center. At the same time, the Minimal Enclosing Sphere of a clump has surprisingly low bulk motion: for both R-type and S-type clumps, the enclosing regions move at less than half the speed of randomly placed spheres with the same radii distribution. The implication is that the salient dynamics of S-type clumps are internal convergence and collapse rather than large net translation (Lares et al., 2017).

6. Modeling, phenomenology, and interpretive boundaries

S-type voids have also been used as targeted probes of dark-sector phenomenology. In the decaying-dark-matter study, S-type voids are identified as small or compensated voids with a compensating overdense shell, and that shell structure is precisely what makes them more sensitive than R-type voids to decay-induced momentum transfer. In the two-dust-fluid covariant model, the interaction is written as

$\Delta_{\rm max}$4

with the acceleration term

$\Delta_{\rm max}$5

The main result is that decaying dark matter can generate an additional ditch-like underdensity adjacent to the boundary of an S-type void, a feature absent in the non-decaying LTB case and much weaker in R-type voids (Lester et al., 2021).

The predicted present-day imprint is therefore not primarily a large change in the overall expansion profile, but a shape change in the density profile near the compensating shell, accompanied by enhanced shear $\Delta_{\rm max}$6 and electric Weyl scalar $\Delta_{\rm max}$7. Because individual voids contain few galaxies, the proposed observational route is statistical, for example through stacked voids or gravitational lensing. The paper treats this as a proof of concept and suggests that lack of the reported feature could constrain the decay scenario in terms of $\Delta_{\rm max}$8 and $\Delta_{\rm max}$9 km/s (Lester et al., 2021).

The broader theoretical context places limits on how literally one should interpret idealized void classifications. Work on void abundance modeling shows that the standard spherical-evolution approximation is a useful organizing principle but not exact for realistic void catalogs: ZOBOV voids are highly aspherical, the effective initial threshold exhibits substantial scatter and scale dependence, and the measured volume function is highly sensitive to subvoid treatment. This does not invalidate the environmental S-type/R-type classification, but it does imply that any attempt to connect S-type voids to analytic abundance models must account for asphericity, hierarchy, sampling dependence, and tracer bias (Achitouv et al., 2013).

A common misconception is to conflate S-type voids with “spheroidal” voids. Exact prolate and oblate spheroidal spacetime models do exist for cosmic voids embedded in a Robertson–Walker background, but that work does not explicitly use the S-type label in the environmental sense. In the standard void literature surveyed here, S-type means void-in-cloud, shell-bearing, overdense-environment void, not merely a void with non-spherical shape. That distinction is essential for interpreting observational catalogs, dynamical studies, and galaxy-bias measurements consistently (Boero et al., 2016, Rincon et al., 2024).