Stellar-to-Subhalo Mass Relation

• The stellar-to-subhalo mass relation is a mapping that connects a satellite galaxy's stellar mass with the dark-matter mass of its hosting subhalo, emphasizing pre-stripping metrics.
• Abundance matching, hydrodynamical simulations, and weak-lensing studies indicate that using accretion or peak mass measures yields a more monotonic and accurate relation.
• Empirical tests reveal significant dark matter stripping in subhalos post-infall, impacting satellite clustering and the inferred stellar mass functions.

The stellar-to-subhalo mass relation (SSMR) is the relation between the stellar mass of a satellite galaxy and the dark-matter mass assigned to the subhalo that hosts it. It is a specialized branch of the broader galaxy–halo connection, but it is not reducible to the central-galaxy stellar-to-halo mass relation unless the subhalo variable is defined so as to track the satellite’s pre-stripping potential well. Across abundance-matching models, hydrodynamical simulations, weak-lensing measurements, and recent SHAM generalizations, the central issue is not only whether the mapping is monotonic, but which subhalo mass or proxy makes that monotonicity physically meaningful and statistically accurate (Simha et al., 2010, Rodriguez-Puebla et al., 2012).

1. Definition and conceptual scope

In the satellite-specific literature, the central and satellite relations are explicitly separated. A central stellar–halo mass relation is written as $M_*(M_{\rm h})$, where $M_{\rm h}$ is the mass of a distinct host halo, while the satellite stellar–subhalo mass relation is written as $m_*(m_{\rm sub})$, where $m_{\rm sub}$ is the mass of the embedded subhalo hosting a satellite galaxy. The notation is not merely cosmetic: distinct halos continue to accrete mass, whereas subhalos generally lose dark matter after infall through tidal stripping, so the satellite mapping is intrinsically sensitive to the mass definition (Rodriguez-Puebla et al., 2012).

Several non-equivalent subhalo variables appear in the literature. Present-day bound or observed subhalo mass, often denoted $m_{\rm sub}^{\rm obs}$, is the mass measured after stripping. Accretion-time or infall mass, $m_{\rm sub}^{\rm acc}$ or $M_{\rm acc}$, is the mass at first incorporation into a larger halo. Maximum historical mass, $M_{\rm max}$ or $M_{\rm peak}$, is the largest mass ever attained while the object was still effectively a central. In cluster weak-lensing work, a more local definition has also been used: $m_{\rm bg}$, the mass within the radius where the subhalo density equals the host-cluster background density, intended to approximate the mass still bound to the subhalo and to match what local-overdensity subhalo finders recover in simulations (Sifón et al., 2017). These alternatives are physically distinct and yield different inferred SSMRs.

A persistent source of confusion is the assumption that satellites and centrals obey the same stellar-mass relation. Some abundance-matching frameworks treat the central relation as applicable to satellites once the horizontal axis is reinterpreted as accretion mass rather than present-day mass (Behroozi et al., 2010). Satellite-only inferences, however, show that this identification is at best approximate and depends strongly on the subhalo definition, especially when one compares to present-day surviving satellites rather than to their pre-infall states (Rodriguez-Puebla et al., 2013).

2. Abundance matching and the choice of subhalo variable

The classic abundance-matching statement is rank based. In one common form,

$M_{\rm h}$0

where $M_{\rm h}$1 is the cumulative number density of galaxies above stellar mass $M_{\rm h}$2, and $M_{\rm h}$3 is the cumulative number density of halos or subhalos above the chosen dark-matter variable $M_{\rm h}$4. In hydrodynamical tests of SHAM, the preferred definition is

$M_{\rm h}$5

where $M_{\rm h}$6 is the epoch when a halo first enters the virial radius of a more massive halo (Simha et al., 2010). This is effectively an infall or pre-accretion mass prescription.

Satellite-only abundance matching makes the same point in a more explicit probabilistic language. If $M_{\rm h}$7 denotes the probability that a subhalo of mass $M_{\rm h}$8 hosts a satellite of stellar mass $M_{\rm h}$9, then the satellite stellar mass function can be written as

$m_*(m_{\rm sub})$0

with the corresponding conditional stellar mass functions obtained by convolving $m_*(m_{\rm sub})$1 with the subhalo conditional mass function inside host halos (Rodriguez-Puebla et al., 2012). This formulation makes clear that an SSMR is not determined by the total galaxy stellar mass function alone; it requires either a decomposition into centrals and satellites or auxiliary constraints such as conditional mass functions or clustering.

The same logic underlies early SHAM calibrations that used

$m_*(m_{\rm sub})$2

arguing that present-day stripped subhalo mass is a poor tracer of stellar mass for satellites, while the maximum past mass remains more tightly connected to the depth of the potential well in which the galaxy formed (0903.4682). Later uncertainty analyses kept the same basic structure: the total mass function used in abundance matching includes both host halos and subhalos, but satellites are associated with subhalo accretion mass $m_*(m_{\rm sub})$3, not with present bound mass (Behroozi et al., 2010).

3. Hydrodynamical tests and the case for pre-stripping masses

Direct tests in cosmological SPH simulations show that the stellar-mass–subhalo-mass mapping is nearly monotonic only if satellites are assigned the mass of their parent halo at the epoch when they became satellites, not the stripped subhalo mass measured at the observation epoch. In both no-wind and momentum-driven-wind simulations, the median relation for central and satellite galaxies is nearly identical once satellites are placed on the $m_*(m_{\rm sub})$4 plane using $m_*(m_{\rm sub})$5. With this choice, the 68% scatter in SHAM-assigned versus true stellar mass is $m_*(m_{\rm sub})$6 dex in the no-wind run and $m_*(m_{\rm sub})$7 dex in the wind run at $m_*(m_{\rm sub})$8, rising to $m_*(m_{\rm sub})$9 and $m_{\rm sub}$0 dex by $m_{\rm sub}$1 (Simha et al., 2010).

The same simulations also show why present-day satellite mass fails. Dark matter in subhalos is efficiently stripped after infall, while the stellar component is more tightly bound near the center of the potential. Using $m_{\rm sub}$2 subhalo mass therefore depresses the inferred stellar masses of satellites in dense environments, underpredicts the galaxy occupation of high-mass halos, suppresses inner radial profiles, and degrades small-scale clustering. In the matched dark-matter-only simulations, the preferred infall-mass SHAM recovers the two-point correlation function of the no-wind SPH galaxies to better than 10% over $m_{\rm sub}$3, whereas the present-day-mass variant performs significantly worse (Simha et al., 2010).

An earlier N-body SHAM study reached a closely related conclusion from clustering rather than from hydro truth. There, using the maximum historical subhalo mass $m_{\rm sub}$4 reproduced the observed SDSS stellar-mass-dependent projected correlation functions well, whereas using present-day subhalo mass could not reproduce the small-scale signal; inclusion of orphan satellites was also required to recover the one-halo term (0903.4682). The conceptual convergence of these results established a standard SHAM prescription: for satellites, the physically relevant variable is a pre-stripping quantity such as $m_{\rm sub}$5, $m_{\rm sub}$6, or $m_{\rm sub}$7, rather than the current bound mass (Behroozi et al., 2010).

This does not imply that an infall-mass description is exact. Even in the preferred hydrodynamical SHAM implementation, the main failure mode is a small but non-negligible population of satellites that lose substantial stellar mass after infall while their host subhalos survive. That effect causes overpopulation of massive halos, particularly in the wind simulation, where halo occupations are overpredicted by $m_{\rm sub}$8–$m_{\rm sub}$9 dex and small-scale clustering can be overpredicted by up to a factor of $m_{\rm sub}^{\rm obs}$0 (Simha et al., 2010).

4. Separate satellite relations and direct empirical constraints

Satellite-only abundance-matching analyses show that the SSMR is not, in general, the same as the central stellar–halo mass relation. When the observed galaxy stellar mass function is decomposed into centrals and satellites and satellites are matched to subhalos separately, forcing $m_{\rm sub}^{\rm obs}$1 underpredicts the satellite abundance, conditional mass functions, and small-scale clustering, especially if $m_{\rm sub}^{\rm obs}$2 is defined at the present epoch (Rodriguez-Puebla et al., 2012). Constraining the satellite relation directly yields a systematically different mapping: at fixed stellar mass, satellites occupy less massive dark structures than centrals, with the discrepancy growing toward lower stellar masses.

The contrast is particularly strong for present-day subhalo mass. One satellite-only study reports that at $m_{\rm sub}^{\rm obs}$3, a satellite resides in a present-day subhalo about four times less massive than the halo of a central galaxy of the same stellar mass. Comparing the accretion-time and observation-time SSMRs implies that for $m_{\rm sub}^{\rm obs}$4, the dark mass of satellites decreased on average by 60–65% relative to their masses at accretion (Rodriguez-Puebla et al., 2012). A subsequent joint fit to stellar mass functions and clustering found essentially the same qualitative result and constrained the intrinsic scatters of the central-halo and satellite-subhalo relations to be nearly equal,

$m_{\rm sub}^{\rm obs}$5

while still concluding that the central and satellite mass relations are distinct, especially for present-day subhalo mass (Rodriguez-Puebla et al., 2013).

Direct lensing measurements provide an observational calibration of the SSMR in clusters. For spectroscopically confirmed and red-sequence cluster satellites, weak-lensing measurements of the bound subhalo mass $m_{\rm sub}^{\rm obs}$6 yield

$m_{\rm sub}^{\rm obs}$7

where $m_{\rm sub}^{\rm obs}$8 is defined within the radius at which subhalo density matches the cluster background density (Sifón et al., 2017). This directly measured cluster-satellite relation implies that at $m_{\rm sub}^{\rm obs}$9, subhalo masses are roughly 50% of those of central galaxies, and that this fraction decreases at higher stellar mass (Sifón et al., 2017).

Cluster weak-lensing analyses also show strong environmental modulation of $m_{\rm sub}^{\rm acc}$0. In redMaPPer clusters, stacked lensing around satellites gives $m_{\rm sub}^{\rm acc}$1 at projected cluster-centric radius $m_{\rm sub}^{\rm acc}$2, rising to $m_{\rm sub}^{\rm acc}$3 at $m_{\rm sub}^{\rm acc}$4, consistent with stronger stripping at smaller radii (Li et al., 2015). In Illustris cluster satellites, the same qualitative picture appears in the simulation itself: the satellite SHMR is shifted toward lower present-day dark-matter masses than the central relation, with an average stripping-offset parameter $m_{\rm sub}^{\rm acc}$5, subhaloes starting to lose dark matter inside $m_{\rm sub}^{\rm acc}$6, and up to 80% of dark matter stripped during infall (Niemiec et al., 2018).

A related high-mass diagnostic is the stellar mass-gap between the central and the most massive surviving satellite. In group and cluster halos, abundance matching with $m_{\rm sub}^{\rm acc}$7 shows that the flattening of the stellar-mass–halo-mass relation compresses large subhalo mass ratios into much smaller stellar mass ratios. As a result, stellar mass-gap is only a noisy, mass-dependent proxy for halo/subhalo mass-gap, especially at the high-mass end (Deason et al., 2013).

5. Scatter, systematics, and departures from universality

The SSMR is not exact even when the mass definition is well chosen. In hydrodynamical simulations, the strongest outliers around the nearly monotonic satellite relation arise from satellites, especially those that have lost stellar mass after infall. Those outliers show little dependence on final host halo mass or on accretion epoch $m_{\rm sub}^{\rm acc}$8, and are instead linked primarily to post-infall stellar stripping or disruption (Simha et al., 2010). This makes clear that even an infall-mass SSMR is partly a statement about the survivorship and subsequent evolution of satellites, not only about their pre-infall structure.

Uncertainty budgets are also asymmetric. For the global stellar mass–halo mass relation, the dominant uncertainties at $m_{\rm sub}^{\rm acc}$9 come from stellar mass systematics and from scatter at fixed halo mass, while subhalo/substructure uncertainty has a comparatively small effect because satellites are $M_{\rm acc}$0 of galaxies by number. The same analysis nevertheless treats satellites as attached to subhalo accretion mass $M_{\rm acc}$1, which means that any SSMR interpretation inherits the uncertainty associated with post-infall stellar evolution (Behroozi et al., 2010). This distinction matters because present-day satellite stellar mass and present-day stripped subhalo mass are measured at the same epoch, while accretion-mass constructions mix quantities defined at different times.

Broader work on the galaxy–halo connection suggests that a universal one-parameter mapping may be incomplete even before the satellite-specific complications are added. Hydrodynamical zoom simulations show that central galaxies in large-scale overdense regions have systematically larger $M_{\rm acc}$2 than centrals of the same halo mass in underdense regions over $M_{\rm acc}$3, and matching local density within 2 physical Mpc at $M_{\rm acc}$4 does not remove the difference (Tonnesen et al., 2015). This is a host-halo result rather than a direct SSMR calibration, but it suggests that any mass-only mapping for satellites may also miss assembly-history information. A related morphology-dependent study finds that the host-halo stellar-to-halo mass relation differs strongly between early and late types at high mass, again for central halos rather than subhalos (Posti et al., 2021). These results do not directly define an SSMR, but they caution against interpreting satellite stellar mass as a function of a single dark-matter variable without regard to environment or formation history.

6. Applications, generalizations, and current directions

In semi-empirical merger models, the SSMR is often implemented as an infall-time extension of the central relation rather than as an independently fit present-day satellite law. DECODE, for example, assigns stellar masses to all parent halos and to all subhalos of any order at the time of infall using the same redshift-dependent stellar mass–halo mass relation as for centrals, and then freezes satellite stellar mass after infall in its baseline implementation (2208.00014). In that framework, the high-mass slope and redshift evolution of the infall-time mapping dominate predictions for satellite abundances, major-merger rates, brightest-cluster-galaxy growth, and the fraction of massive ellipticals (2208.00014). This operational choice is not equivalent to a present-day SSMR, but it is an important applied use of satellite abundance matching.

Recent SHAM work has also moved beyond subhalo mass as the primary ordering variable. One study finds that the common choice $M_{\rm acc}$5 can be improved upon by a composite proxy

$M_{\rm acc}$6

which combines the 90th percentile of a subhalo’s $M_{\rm acc}$7 history with the 60th percentile of the absolute dark-matter mass-variation rate. In IllustrisTNG300, this three-parameter proxy improves stellar-mass prediction by 15% relative to the earlier $M_{\rm acc}$8 proxy over $M_{\rm acc}$9–2 and also improves clustering performance (Chuang et al., 2022). In this sense, part of the modern “stellar-to-subhalo mass relation” literature is no longer about $M_{\rm max}$0 alone, but about finding the subhalo history variable that best preserves the stellar-mass ranking.

At lower halo masses, strong-lensing forecasts indicate a prospective extension of SSMR measurements into the dwarf regime. Simulations of Euclid-like imaging show sensitivity to $M_{\rm max}$1 subhalos at $M_{\rm max}$2, and a future sample of 48 subhalos with $M_{\rm max}$3 detection significance would constrain the relation at that mass scale with $M_{\rm max}$4 uncertainty of 0.045 dex, provided the subhalo and lens light are modeled simultaneously (Wang et al., 27 Jan 2025). In the setup of that study, the objects are lens-plane subhalos hosting very faint satellite-like galaxies, so the forecast is effectively for a low-mass stellar-to-subhalo constraint rather than for a central-halo SHMR (Wang et al., 27 Jan 2025).

Taken together, these results define the SSMR as a family of relations conditioned by the choice of subhalo variable and by the physical question being asked. If the goal is to reproduce satellite clustering or to attach galaxies to dark-matter-only simulations, pre-stripping quantities such as $M_{\rm max}$5, $M_{\rm max}$6, or $M_{\rm max}$7 are strongly preferred (0903.4682, Simha et al., 2010). If the goal is to describe the current dynamical state of satellites in groups and clusters, present-day bound-mass relations measured by weak lensing or by hydrodynamical simulations are the relevant objects, and they show substantial offsets from the central relation because subhalos have already been stripped (Sifón et al., 2017, Niemiec et al., 2018). The modern literature therefore treats the stellar-to-subhalo mass relation not as a unique curve, but as a satellite-specific mapping whose meaning depends on whether one is modeling formation, infall, survival, or present-day observability.