Stellar Mass–Halo Mass Relation

• The stellar mass–halo mass relation is an empirical link that defines how a galaxy's stellar mass correlates with its dark matter halo mass, highlighting feedback and star-formation efficiency.
• Researchers use methods including abundance matching, halo occupation, and dynamic modeling to quantify this relation across dwarf to cluster scales.
• The relation peaks near a characteristic halo mass (≈10^12 M⊙), with steep low-mass and shallow high-mass slopes influenced by feedback, environment, and assembly history.

The stellar mass–halo mass relation, usually abbreviated SHMR or SMHM, is the empirical and theoretical relation linking a galaxy’s stellar mass $M_\star$ to the mass of its host dark matter halo, $M_{\rm halo}$. In its most generic form it is written as $M_\star = f(M_{\rm halo})$, or as a stellar mass fraction $M_\star/M_{\rm halo}$; some studies further normalize by the cosmic baryon fraction and define $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$ as a time-averaged star-formation efficiency (Brook et al., 2013, Posti et al., 2021). The relation is a central ingredient of abundance matching, halo occupation modeling, semi-analytic models, and hydrodynamical simulations, but it is not operationally unique: published work variously uses $M_{200}$, $M_{500}$, or $M_{\rm 200c}^{\rm DMO}$, and the stellar component may refer to a central galaxy, a brightest cluster galaxy (BCG), or the total stellar mass of all member galaxies within $r_{200}$ (Wang et al., 25 Nov 2025, Patel et al., 2015, Kravtsov et al., 2014).

1. Definitions and parameterizations

A common phenomenological form is the double power law adopted in semi-analytic and empirical work,

$$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$

where $M_{\rm halo}$0 is the normalization, $M_{\rm halo}$1 is the characteristic halo mass, $M_{\rm halo}$2 governs the low-mass branch, and $M_{\rm halo}$3 the high-mass branch (Mitchell et al., 2015). In the GALFORM reference model, a single non-evolving fit over $M_{\rm halo}$4 gives $M_{\rm halo}$5, $M_{\rm halo}$6, $M_{\rm halo}$7, and $M_{\rm halo}$8, implying $M_{\rm halo}$9 at low masses and $M_\star = f(M_{\rm halo})$0 at high masses (Mitchell et al., 2015).

Other studies adopt regime-specific parameterizations. Local Group dwarfs over $M_\star = f(M_{\rm halo})$1 were fitted with a single power law $M_\star = f(M_{\rm halo})$2 with $M_\star = f(M_{\rm halo})$3, steepening to $M_\star = f(M_{\rm halo})$4 if the completeness limit is pushed to $M_\star = f(M_{\rm halo})$5 (Brook et al., 2013). At the cluster scale, BCG analyses often fit log-linear relations such as $M_\star = f(M_{\rm halo})$6, emphasizing the sub-linear growth of the central galaxy relative to the halo (Erfanianfar et al., 2019).

The relation is also definition-dependent. In the COLIBRE analysis, the fiducial $M_\star = f(M_{\rm halo})$7 SMHM is the median relation between the stellar mass of the central within a 50 physical kpc aperture and the halo mass $M_\star = f(M_{\rm halo})$8 of the matched dark-matter-only halo (Wang et al., 25 Nov 2025). In low-mass X-ray groups, by contrast, $M_\star = f(M_{\rm halo})$9 denotes the aggregate stellar mass of all group members within $M_\star/M_{\rm halo}$0, corrected to a common lower stellar-mass limit of $M_\star/M_{\rm halo}$1 (Patel et al., 2015). At the high-mass end, some cluster studies include diffuse light smoothly connected to the BCG profile as part of $M_\star/M_{\rm halo}$2, while others explicitly exclude most intracluster light; this distinction materially affects the normalization of the inferred relation (Kravtsov et al., 2014, Toptun et al., 10 Feb 2026).

2. Inference methods

The dominant empirical strategy is abundance matching: one rank-orders galaxies by stellar mass and halos by mass, then imposes a one-to-one monotonic mapping between the cumulative stellar mass function and the halo mass function (Brook et al., 2013, Shankar et al., 2014). Parametric abundance matching was used in COSMOS-UltraVISTA to derive the SHMR from $M_\star/M_{\rm halo}$3 to $M_\star/M_{\rm halo}$4, exploiting a large area and highly complete stellar masses in the COSMOS field (Legrand et al., 2018). At the dwarf scale, Brook et al. applied the same logic inside a Local Group-sized volume, matching the observed Local Group stellar mass function within 1.8 Mpc to a simulated halo mass function that is well described by a single power law (Brook et al., 2013).

Clustering-based methods infer halo masses statistically through halo occupation modeling. In VUDS, the projected two-point function $M_\star/M_{\rm halo}$5 for $M_\star/M_{\rm halo}$6 galaxies at $M_\star/M_{\rm halo}$7 was fitted with a three-parameter HOD model,

$M_\star/M_{\rm halo}$8

so that $M_\star/M_{\rm halo}$9 provides the characteristic halo mass for a stellar-mass-threshold sample (Durkalec et al., 2014). BOSS clustering has also been used to constrain the scatter of the high-mass SMHM relation by comparing the large-scale bias $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$0 of samples ranked by stellar mass to abundance-matching models with different $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$1 (Tinker et al., 2016).

A second class of methods infers halo masses dynamically for individual systems. For massive early-type galaxies, halo masses have been derived from the phase-space distribution of globular cluster systems using an action-angle distribution function embedded in an NFW potential, with a prior on concentration from the $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$2CDM concentration–mass relation (Posti et al., 2021). For isolated field dwarfs, HI rotation curves were modeled with the coreNFW profile, which accounts for feedback-driven cusp–core transformations and yields direct estimates of $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$3 over $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$4 (Read et al., 2016).

At group and cluster masses, X-ray methods are especially important. Halo masses may be inferred from $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$5 scaling relations (Patel et al., 2015, Erfanianfar et al., 2019), from X-ray temperatures (Kravtsov et al., 2014), or, more recently, from stacked eROSITA spectra whose average gas temperatures are converted to $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$6 and then to $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$7 via the $$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$8–$$f_\star \equiv (M_\star/M_{\rm h})/(\Omega_b/\Omega_m)$$9 relation (Toptun et al., 10 Feb 2026). These X-ray-based measurements are complementary to weak lensing, satellite kinematics, and group catalogs.

3. Mean shape and redshift evolution

Across many determinations, the stellar fraction rises with halo mass at low masses, peaks near a characteristic halo mass of order $M_{200}$0, and declines again toward group and cluster scales. In COSMOS-UltraVISTA, the ratio of stellar-to-halo mass content peaks at $M_{200}$1 at $M_{200}$2, increases to $M_{200}$3 at $M_{200}$4, and then remains flat up to $M_{200}$5 (Legrand et al., 2018). For early-type galaxies inferred from globular-cluster dynamics, the baryon-fraction-normalized stellar fraction peaks at $M_{200}$6 and $M_{200}$7, then declines at higher masses (Posti et al., 2021). An X-ray-based SHMR for central galaxies across $M_{200}$8–$M_{200}$9 likewise finds the relation peaking near $M_{500}$0, followed by a declining central stellar fraction at higher masses (Toptun et al., 10 Feb 2026).

Direct high-redshift constraints broadly support the same picture. In VUDS, galaxies with $M_{500}$1 to $M_{500}$2 occupy halos with $M_{500}$3 to $M_{500}$4, implying SMHR values from $M_{500}$5 to $M_{500}$6; the corresponding integrated star-formation efficiency is $M_{500}$7–$M_{500}$8 for lower-mass galaxies and $M_{500}$9 for galaxies with median stellar mass $M_{\rm 200c}^{\rm DMO}$0 at $M_{\rm 200c}^{\rm DMO}$1 (Durkalec et al., 2014). In the GALFORM semi-analytic model, the standard implementation of supernova feedback and gas reincorporation predicts only weak evolution of the median SHM relation over $M_{\rm 200c}^{\rm DMO}$2, with nearly constant $M_{\rm 200c}^{\rm DMO}$3, $M_{\rm 200c}^{\rm DMO}$4, and low-mass slope (Mitchell et al., 2015).

At the high-mass end, BCG-focused studies find similarly slow evolution. For X-ray-selected groups and clusters over $M_{\rm 200c}^{\rm DMO}$5, the central-galaxy stellar mass–halo mass relation is a simple power law in log space, and the low- and high-redshift fits agree within their uncertainties; no notable redshift evolution is detected since $M_{\rm 200c}^{\rm DMO}$6 (Erfanianfar et al., 2019). A plausible implication is that, in this regime, central stellar mass growth broadly tracks halo growth, even though the stellar mass fraction declines with increasing halo mass.

4. Mass-regime dependence

The dwarf regime remains the sharpest test of the low-mass branch. For Local Group galaxies with $M_{\rm 200c}^{\rm DMO}$7, Brook et al. find a steep power-law SHMR with slope $M_{\rm 200c}^{\rm DMO}$8, significantly steeper than most abundance-matching extrapolations based on large surveys (Brook et al., 2013). For isolated field dwarfs, rotation-curve modeling with coreNFW yields a monotonic $M_{\rm 200c}^{\rm DMO}$9–$r_{200}$0 relation with little scatter, and abundance matching based on the SDSS field stellar mass function agrees with the directly inferred dwarf relation down to $r_{200}$1, or to $r_{200}$2 if the SDSS stellar mass function is extrapolated as a power law below $r_{200}$3 (Read et al., 2016).

In low-mass X-ray groups at $r_{200}$4, the relevant observable is often the total stellar mass in member galaxies rather than the central galaxy alone. Patel et al. measure

$r_{200}$5

for groups, with an observed scatter of $r_{200}$6 dex, and find that stars comprise $r_{200}$7–$r_{200}$8 of the total mass for halos with $r_{200}$9 (Patel et al., 2015). This sub-unity slope means that $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$0 decreases toward higher halo masses.

At cluster scales, the central-galaxy relation is distinctly shallow. For BCGs, the relation $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$1 at $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$2 and $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$3 at $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$4, with $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$5, shows that BCG stellar mass increases sub-linearly with halo mass (Erfanianfar et al., 2019). A deeper photometric re-analysis of nearby clusters finds $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$6, $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$7, and $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$8, again implying a declining stellar fraction with increasing cluster mass (Kravtsov et al., 2014). This regime is also particularly sensitive to aperture and surface-brightness systematics: older abundance-matching calibrations based on standard SDSS photometry were argued to underestimate BCG stellar masses by factors of $$\frac{M_\star}{M_{\mathrm{H}}} = 2N \left[ \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{-\beta} + \left(\frac{M_{\mathrm{H}}}{M_{1}}\right)^{\gamma} \right]^{-1},$$9–4 (Kravtsov et al., 2014).

5. Scatter and secondary dependencies

A recurrent result is that the scatter in stellar mass at fixed halo mass is small, typically $M_{\rm halo}$00 dex, but the physical origin of that scatter is not unique. From BOSS clustering of galaxies with $M_{\rm halo}$01, the preferred scatter is $M_{\rm halo}$02, including measurement error, and repeated spectra imply an upper limit of $M_{\rm halo}$03 dex on the intrinsic scatter (Tinker et al., 2016). In a merger-tree analysis anchored to the Behroozi et al. relation, hierarchical assembly alone produces $M_{\rm halo}$04 dex scatter at $M_{\rm halo}$05, while adding an intrinsic in-situ scatter of $M_{\rm halo}$06 dex yields an approximately constant $M_{\rm halo}$07 dex scatter from $M_{\rm halo}$08 to $M_{\rm halo}$09 at $M_{\rm halo}$10; this suggests that the observed flatness of the scatter with halo mass is largely a coincidence of two different growth channels (Gu et al., 2016).

Secondary halo properties measurably structure that scatter. In COLIBRE, at fixed $M_{\rm halo}$11, stellar mass correlates positively with halo concentration, with Spearman $M_{\rm halo}$12 near $M_{\rm halo}$13 and $M_{\rm halo}$14 near $M_{\rm halo}$15; the correlation weakens but remains positive above $M_{\rm halo}$16 (Wang et al., 25 Nov 2025). Concentration also correlates with stellar age, but stellar age itself correlates only weakly with stellar mass at fixed halo mass, whereas stellar metallicity correlates strongly with both concentration and stellar mass; moreover, the concentration–metallicity correlation persists at fixed stellar mass and halo mass with mean $M_{\rm halo}$17. This supports a potential-depth interpretation in which deeper halos suppress feedback-driven outflows and retain more baryons and metals (Wang et al., 25 Nov 2025).

Morphology, color, and size also act as second parameters. Dynamical modeling of early types and rotation-curve modeling of late types indicate that the SHMR is not universal across morphology: early types follow the familiar peaked relation, while late types show a monotonically rising $M_{\rm halo}$18 over the mass range probed, with the two branches differing by a factor of $M_{\rm halo}$19 in $M_{\rm halo}$20 at $M_{\rm halo}$21 (Posti et al., 2021). At fixed very high stellar mass $M_{\rm halo}$22 and $M_{\rm halo}$23, PAC measurements show that the most compact galaxies with Sérsic index $M_{\rm halo}$24 have halo masses around $M_{\rm halo}$25 times larger than disk-like systems with $M_{\rm halo}$26; red galaxies inhabit halos $M_{\rm halo}$27 times more massive than blue galaxies, and large galaxies halos $M_{\rm halo}$28 times more massive than small galaxies (Xu et al., 2021). Environment matters as well: in hydrodynamical simulations, central galaxies in large-scale overdense regions have larger $M_{\rm halo}$29 than counterparts in underdense regions at the same halo mass, and this persists even when the local density within 2 Mpc at $M_{\rm halo}$30 is matched (Tonnesen et al., 2015).

6. Physical interpretation, controversies, and open directions

The low-mass branch is widely interpreted as feedback- and reionization-regulated. In field dwarfs, the decline of $M_{\rm halo}$31 toward low halo mass is consistent with efficient supernova-driven outflows, with sufficient supernova energy to create dark-matter cores down to $M_{\rm halo}$32 if star formation has persisted for long enough (Read et al., 2016). In fully cosmological SPH simulations with metal-line cooling, dust and self-shielding, H$M_{\rm halo}$33-based star formation, and supernova-driven outflows, the present-day SHM relation agrees well with abundance matching over $M_{\rm halo}$34 to $M_{\rm halo}$35; the agreement improves once stellar masses are measured from synthetic photometry and halo masses are compared to dark-matter-only runs, because photometric stellar masses can underestimate true stellar masses and dark-matter-only runs overestimate halo masses when baryon loss is ignored (Munshi et al., 2012).

The high-mass decline is usually attributed to AGN feedback, long cooling times, and ex-situ assembly. In the eROSITA-based SHMR, the decline of $M_{\rm halo}$36 at group and cluster scales is interpreted as the result of AGN feedback, reduced cooling efficiency, and the increasing dominance of ex-situ assembly while halos continue to grow through mergers and accretion (Toptun et al., 10 Feb 2026). GALFORM makes this logic explicit: weak evolution of the median SHM relation over $M_{\rm halo}$37 follows if gas reincorporation efficiency is effectively constant and the average efficiency of supernova feedback is approximately constant at fixed halo mass, while AGN feedback introduces a break whose location evolves only modestly (Mitchell et al., 2015). This suggests that the near-stationarity of the median relation is a constraint on gas cycling as much as on star formation itself.

Several controversies remain methodological rather than purely physical. The first is the treatment of extended light in massive galaxies: improved surface-brightness photometry and Sérsic-based fits can raise BCG stellar masses enough to steepen the inferred high-mass SMHM slope and reduce the apparent tension between direct cluster measurements and abundance matching (Kravtsov et al., 2014, Shankar et al., 2014). The second is environmental non-universality: abundance matching based on the field stellar mass function produces a dwarf-scale SHMR consistent with isolated field rotation curves, whereas using the stellar mass function of nearby groups recovers the familiar missing-satellites and too-big-to-fail problems; this indicates that those problems are confined to group environments and arise from galaxy-formation physics rather than exotic cosmology (Read et al., 2016). The third is that morphology-blind or environment-blind parameterizations can absorb real secondary structure into an apparently larger intrinsic scatter; one estimate suggests that ignoring the late-type branch could inflate the inferred intrinsic scatter by $M_{\rm halo}$38 dex at $M_{\rm halo}$39 (Posti et al., 2021).

Future progress in the literature represented here is correspondingly twofold. Deeper Local Group surveys are expected to push completeness to $M_{\rm halo}$40 or below and tighten the low-mass slope (Brook et al., 2013); larger samples of massive galaxies are needed to test the redshift dependence of the characteristic halo mass and galaxy-formation efficiency at $M_{\rm halo}$41 (Legrand et al., 2018). On the group and cluster side, deeper X-ray data and larger stacked samples should enable SHMR measurements conditioned on morphology, star-formation rate, concentration, or luminosity gap, directly probing the second-parameter structure that is already evident in simulations and high-mass observations (Toptun et al., 10 Feb 2026).