Stellar–Halo Mass Relation

• The Stellar-to-Halo Mass Relation is the empirical mapping between a galaxy’s stellar mass and its host dark matter halo mass, reflecting the roles of gas accretion, star formation, and feedback.
• It is parameterized using flexible double power-law functions and calibrated with methods such as abundance matching, weak lensing, and satellite statistics to constrain galaxy formation models.
• Variations with redshift, environment, and galaxy type reveal key insights into feedback regulation, assembly bias, and the evolution of star formation efficiency.

The stellar-to-halo mass relation (SHMR) is the principal empirical mapping between the stellar mass of a galaxy and the mass of its host dark matter halo. It encapsulates the integrated impact of gas accretion, star formation, and feedback on baryonic assembly within cosmic structure formation, serving as a cornerstone of both galaxy evolution theory and interpretation of cosmological survey data. Quantitative characterization of the SHMR across halo mass, redshift, morphological type, and environment is fundamental to constraining models of galaxy formation and to generating realistic mock catalogs for large-scale structure analyses.

1. Parametric Forms and Empirical Calibration

The SHMR is most commonly parameterized via a flexible double power-law or a smoothed, five-parameter function encoding the observed efficiency peak and high/low-mass slopes. A canonical form is (Girelli et al., 2020, Munshi et al., 2012):

$$\frac{M_*}{M_h} = \frac{2\,N}{(M_h/M_1)^{-\beta} + (M_h/M_1)^{\gamma}}$$

where:

• $M_*$ is stellar mass (typically total or within some aperture),
• $M_h$ is halo virial mass (e.g., $M_{200}$ or $M_{\rm peak}$ for satellites),
• $N$ is the normalization (peak efficiency),
• $M_1$ is the pivot halo mass,
• $\beta$ and $\gamma$ are low- and high-mass slopes.

More recent refinements introduce smooth transitions and curvature parameters to better capture the turnover near the efficiency peak (Zacharegkas et al., 27 Jun 2025): $$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$ with $M_*$0 defined to ensure a continuous and differentiable transition between regimes.

Empirical calibration relies on abundance matching of observed stellar mass functions (SMFs) to simulated or analytic halo mass functions (HMFs) per epoch, under the assumption of monotonic correspondence and with inclusion of log-normal scatter. This approach is validated and complemented by independent constraints from weak lensing, satellite kinematics, group catalogs, and strong lensing (Girelli et al., 2020, Zacharegkas et al., 27 Jun 2025).

2. Evolution With Redshift and Mass Scale

The SHMR exhibits a non-monotonic, bell-shaped dependence of $M_*$1 on halo mass, peaking at $M_*$2 at $M_*$3 and declining toward both higher and lower masses (Girelli et al., 2020, Legrand et al., 2018). The normalization and characteristic mass evolve with cosmic time:

• At $M_*$4, the peak efficiency $M_*$5 at $M_*$6 (Girelli et al., 2020).
• The peak moves to higher $M_*$7 with increasing redshift, reaching $M_*$8 ($M_*$9) at $M_h$0.
• The peak star-formation efficiency, $M_h$1, remains between 30–35% over $M_h$2–4 (Girelli et al., 2020).
• The low-mass slope, $M_h$3, is typically in the range 1–2.3, and is subject to significant steepening in local group dwarfs ($M_h$4–3.1) (Danieli et al., 2022, Brook et al., 2013).
• The high-mass slope, $M_h$5, is $M_h$6–0.7, indicating gentle decline in efficiency in group and cluster-scale halos (Munshi et al., 2012, Kravtsov et al., 2014, Zacharegkas et al., 27 Jun 2025).

Abundance-matching models and lensing measurements show overall agreement, with scatter of $M_h$7–0.25 dex at fixed $M_h$8 (Uitert et al., 2016, Patel et al., 2015).

3. Physical Drivers and Feedback Regulation

The non-monotonic shape of the SHMR emerges from the mass dependence of feedback and gas accretion processes:

• Below $M_h$9, supernova feedback and photoionization heating inhibit baryon retention and star formation, leading to $M_{200}$0 with $M_{200}$1.
• Above $M_{200}$2, AGN feedback and long cooling times restrict star formation, flattening the high-mass slope ($M_{200}$3).
• In ultra-low-mass halos ($M_{200}$4), reionization imposes a sharp threshold below which galaxy formation is stochastically suppressed, resulting in a "quenching-driven tightening" of the SHMR scatter (O'Leary et al., 2023).

Star-formation efficiency is thus highest near $M_{200}$5 and falls off in both directions. This trend is robust across empirical, semi-analytic, and high-resolution hydrodynamical calibration, though the precise normalization is sensitive to IMF assumptions and stellar mass estimation methodology (Kravtsov et al., 2014).

4. Environmental and Secondary Dependencies

While the SHMR is primarily governed by halo mass, secondary dependencies—"assembly bias" and environmental effects—are now empirically established:

• At fixed $M_{200}$6, central galaxy $M_{200}$7 correlates strongly with proxies of halo formation time (e.g., large-scale density $M_{200}$8), with variations up to $M_{200}$9 dex in $M_{\rm peak}$0 associated with $M_{\rm peak}$1–$M_{\rm peak}$2 Gyr differences in assembly epoch (Oyarzún et al., 2024).
• Hydrodynamical cosmological simulations show up to $M_{\rm peak}$3 higher $M_{\rm peak}$4 in $M_{\rm peak}$5 Mpc overdensities than in voids, at fixed $M_{\rm peak}$6 in the $M_{\rm peak}$7–$M_{\rm peak}$8 range, driven by earlier halo formation, enhanced filamentary accretion, and increased "early neighbor" interactions (Tonnesen et al., 2015).
• The SHMR shape for central galaxies does not vary strongly with local group richness (as defined by group multiplicity or satellite number) (Uitert et al., 2016), but strong assembly bias imparts significant systematic errors if not accounted for in HOD and abundance-matching frameworks.

Satellite galaxies follow an SHMR systematically shifted from that of centrals, consistent with $M_{\rm peak}$9–$N$0 dark matter stripping after cluster infall; this shift is largely independent of stellar mass at $N$1–$N$2 (Niemiec et al., 2018, Niemiec et al., 2017).

5. Methodologies: Clustering, Lensing, and Forward Modeling

A spectrum of methodologies underpins SHMR determination:

• Abundance matching (AM): Rank-ordering the observed SMF to the cumulative HMF with specified scatter; evolutionary extensions model all redshifts via evolving fit coefficients (0903.4682, Girelli et al., 2020).
• Weak lensing: Stacking background shear around foreground galaxies/groups provides a direct, unbiased probe of average halo mass as a function of stellar mass (Uitert et al., 2016, Patel et al., 2015, Zacharegkas et al., 27 Jun 2025).
• Group catalogs and satellite statistics: SHMR inferred for groups via sum of satellite and central stellar masses and robust halo mass proxies (X-ray, velocity dispersion, lensing) (Patel et al., 2015, Kravtsov et al., 2014).
• Strong lensing at the dwarf scale: Subhalo masses of $N$3 (and lower in future deep data) measured without bias via arc perturbations in galaxy-galaxy lenses, directly constraining the SHMR at otherwise inaccessible mass scales (Wang et al., 27 Jan 2025).
• Dynamical: Early- and late-type differences resolved by action-based dynamical modeling of globular cluster or H I kinematics, showing genuine bifurcation by morphology at fixed $N$4 (Posti et al., 2021).

Forward (simulated) modeling is adopted for satellite–halo connections in the ultra-faint regime, leveraging completeness and photometric modeling in wide-area Local Volume surveys (Danieli et al., 2022).

6. Morphology and Satellite/Central Dichotomy

The SHMR is not universal across galaxy types:

• Early-type (spheroidal) galaxies exhibit the characteristic inverted-U shape, peaking near Milky Way-scale halos and declining at higher $N$5 (Posti et al., 2021).
• Late-type (disk) galaxies display a monotonic rise in $N$6 with increasing halo mass, with no observed turnover up to the brightest systems, attributed to enhanced baryon retention regulated by stellar feedback.
• Satellite galaxies in clusters show a mass-independent $N$7–$N$8 dex deficit in $N$9 at fixed $M_1$0 compared to centrals, a direct consequence of tidal stripping (Niemiec et al., 2018, Danieli et al., 2022).
• The low-mass slope of the satellite SHMR is tightly constrained: in the Local Volume, $M_1$1 with $M_1$2 down to $M_1$3 and with very low intrinsic scatter; both constant-scatter and growing-scatter variants fit the data (Danieli et al., 2022).

7. Future Prospects and Systematic Uncertainties

Upcoming surveys (LSST, Euclid, Roman) will dramatically refine the SHMR:

• Galaxy–galaxy strong lensing will enable $M_1$4 dex constraints at $M_1$5 (Wang et al., 27 Jan 2025).
• Deep, wide imaging coupled with robust group, satellite, and environmental characterization will further constrain assembly bias effects (Zacharegkas et al., 27 Jun 2025).
• At higher redshift ($M_1$6), clustering-based HOD modeling provides direct SHMR constraints for $M_1$7–$M_1$8 at $M_1$9, indicating star-formation efficiencies as high as 15–20\% in actively assembling galaxies (Durkalec et al., 2014).

Key systematics include:

• Photometric biases in extended light profiles yield up to factor $\beta$0–$\beta$1 errors in bright-end $\beta$2, shifting the inferred SHMR normalization at cluster scale (Kravtsov et al., 2014).
• IMF assumptions and stellar population modeling introduce $\beta$3–$\beta$4 dex uncertainties.
• Incompleteness, environmental bias, and feedback modeling remain limiting factors at the faint end and high redshift.

Systematic inclusion of environmental and assembly variables is now necessary to model the full scatter and normalization of the stellar–halo connection (Oyarzún et al., 2024, Tonnesen et al., 2015).

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Selected Parametric Fits to SHMR in Recent Literature

Reference	$\beta$5 ($\beta$6)	Peak $\beta$7	Low-mass $\beta$8	High-mass $\beta$9	Notes
Moster+12/13 (Munshi et al., 2012)	$\gamma$0	$\gamma$1	$\gamma$2	$\gamma$3	$\gamma$4; scatter $\gamma$5 dex
Girelli+19 (Girelli et al., 2020)	$\gamma$6 (z=0)–$\gamma$7(z=4)	$\gamma$8–$\gamma$9	$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$0–$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$1	$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$2–$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$3	Explicit $$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$4 evolution in all parameters
van Uitert+16 (Uitert et al., 2016)	$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$5	—	$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$6	$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$7	GAMA+KiDS; joint lensing+SMF fit
Kravtsov+14 (Kravtsov et al., 2014)	Cluster regime	—	—	$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$8	$$\log M_*(M_h) = \log(\epsilon M_1) + f[\log(M_h/M_1)] - f(0)$$9, scatter $M_*$00 dex
ELVES (Danieli et al., 2022)	Satellite regime	—	$M_*$01	—	Applies to $M_*$02, $M_*$03 dex

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The SHMR framework now incorporates redshift, environment, further secondary dependencies (assembly bias), and morphological type. Its ongoing refinement and empirical anchoring directly dictate constraints on galaxy formation feedback physics and underpin the precision cosmological interpretation of large-scale structure surveys.