Stellar–Halo Mass Relationship

• The stellar mass–halo mass relationship is defined as the statistical link between a galaxy’s stellar mass and its dark matter halo mass, typically peaking near 10^12 M☉.
• Methodologies such as abundance matching, clustering measurements, and halo occupation distribution modeling quantify the SHMR while accounting for intrinsic scatter from feedback processes.
• This relationship informs our understanding of galaxy formation, assembly bias, and cosmological structure evolution by constraining star-formation efficiency and feedback effects.

The stellar mass–halo mass relationship (SMHM or SHMR) describes the statistical mapping between a galaxy’s stellar mass, $M_*$, and the mass of its host dark matter halo, $M_h$. This relationship encodes the integrated efficiency with which baryons are converted into stars across cosmic time and is central to contemporary models of galaxy formation and cosmological structure evolution. A precise SHMR has become critical for connecting empirical galaxy properties to theoretical predictions for mass assembly, clustering, feedback, and environmental effects.

1. Functional Formulation and Parameterization

A central result of SHMR studies is its strong non-linearity as a function of halo mass. It is empirically well described by a doubly-broken (double power-law) form:

$$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$

where:

• $(m/M)_0$ is the peak value of the stellar-to-halo mass ratio,
• $M_1$ is the characteristic halo mass where stellar-mass assembly efficiency peaks,
• $\beta$ governs the slope at the low-mass end, and
• $\gamma$ characterizes the high-mass decline (0903.4682).

An alternative, often used for abundance matching, parameterizes the inverse mapping or uses composite functions to capture the additional curvature at high masses (Behroozi et al., 2010). For central galaxies, satellite and scatter corrections are incorporated using conditional mass functions (CMFs) and log-normal or Schechter components.

Typical values at $z \sim 0$ are $M_1 \sim 10^{12} M_\odot$ (where $M_*/M_h$ peaks at 0.02–0.04, i.e., 2–4%), $M_h$0, and $M_h$1. The SHMR steepens dramatically below $M_h$2: $M_h$3 for low-mass haloes, flattening above $M_h$4 due to feedback and quenching effects.

2. Methodologies: Abundance Matching, Clustering, and HOD

The primary empirical tools for measuring the SHMR are:

• Abundance matching, which equates the cumulative stellar mass function $M_h$5 to the cumulative halo or subhalo mass function $M_h$6, optionally convolved with scatter. This yields a monotonic mapping from $M_h$7 to $M_h$8 (Behroozi et al., 2010).
• Clustering measurements (e.g., two-point correlation or bias vs. $M_h$9), which provide halo mass constraints at fixed $$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$0 via halo occupation distribution (HOD) modeling (0903.4682, Wake et al., 2010, Golden-Marx et al., 2017).
• Halo Occupation Distribution modeling directly fits the probability for halos of a given mass to host galaxies above a stellar mass threshold. The mean occupation is often parameterized with softened step functions for centrals and power-laws for satellites (Wake et al., 2010, Zaidi et al., 2024).
• Conditional mass functions (CMFs) and satellite kinematics are employed to further refine central/satellite distinctions and to compare star formation histories.

The incorporation of scatter—both intrinsic and due to measurement uncertainties—is essential; observed scatter in $$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$1 at fixed $$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$2 is typically $$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$3–$$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$4 dex for massive galaxies (Gu et al., 2016, Tinker et al., 2016).

3. Physical Interpretation and Evolution

The bulk of baryonic matter within halos is not converted efficiently into stars. Dark matter halos with $$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$5 have maximal star formation efficiency ($$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$6), with only $$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$720% of baryons as stars; both lower and higher mass halos are less efficient, a direct consequence of stellar and AGN feedback (supernovae dominate suppression in low-mass halos; AGN in high-mass regimes) (Behroozi et al., 2010, Kravtsov et al., 2014).

Redshift evolution in the SHMR is modest below $$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$8: $$\frac{M_*(M_h)}{M_h} = 2 (m/M)_0 \left[ \left( \frac{M_h}{M_1} \right)^{-\beta} + \left( \frac{M_h}{M_1} \right)^{\gamma} \right]^{-1}$$9 and $(m/M)_0$0 increase with redshift, meaning that at higher $(m/M)_0$1 halos must be more massive to reach the same $(m/M)_0$2, and the integrated efficiency declines (0903.4682, Wake et al., 2010, Shankar et al., 2014, Zaidi et al., 2024). At $(m/M)_0$3, the characteristic peak shifts slightly toward higher masses, but with greater uncertainty due to limited dynamic range and stronger systematics (Zaidi et al., 2024).

Morphology, color, and size are additional parameters that modulate the SHMR at fixed $(m/M)_0$4: more compact (high Sersic index) and redder galaxies reside in significantly more massive halos—factors of 2.3–5.5 times higher than large disks or blue galaxies—highlighting assembly and quenching bias effects (Xu et al., 2021).

Variations by environment (overdense/underdense regions) drive non-universality: at $(m/M)_0$5, central galaxies in overdensities have higher $(m/M)_0$6 than those in voids, even when controlling for local density, as a result of assembly bias and increased early gas accretion (Tonnesen et al., 2015).

4. Cluster and Low-Mass Regimes

In galaxy clusters ($(m/M)_0$7), the SHMR is shallow, with $(m/M)_0$8 for BCGs and $(m/M)_0$9 for all cluster galaxies (Kravtsov et al., 2014, Erfanianfar et al., 2019). Total stellar fractions decrease steadily with $M_1$0 but are suppressed by only factors of 3–5 relative to the peak efficiency in $M_1$1 halos—significantly less than previously estimated due to advances in photometry and IMF assumptions.

At the low-mass end, modern photometric mass estimators and Local Group studies find a steep power-law SHMR: $M_1$2 for $M_1$3 (Zaritsky et al., 2022), or $M_1$4 in the range $M_1$5 (Brook et al., 2013). Scatter remains at $M_1$60.3 dex and the relation is consistent across field and cluster populations given NFW potentials and standard assumptions.

For the faintest Local Group dwarfs, some systems show “excess” inner dark mass, potentially indicating the presence of intermediate-mass black holes at masses far above standard BH–halo scaling relations (Zaritsky et al., 2022).

5. Scatter, Assembly Bias, and Secondary Parameters

The observed scatter is remarkably constant ($M_1$7 dex) across a broad halo mass range; theoretical modeling demonstrates this is due to a “coincidence” in which in-situ star formation dominates scatter at low masses, and ex-situ hierarchical assembly (mergers) at high masses. The two contributions sum to a (nearly) constant total variance (Gu et al., 2016). Scatter is further reducible by accounting for secondary parameters such as the cluster magnitude gap, which traces BCG merger history and reduces the intrinsic $M_1$8 from $M_1$90.16 dex to $\beta$0 dex for fixed halo mass (Golden-Marx et al., 2017).

Morphology introduces further complexity—early-type galaxies exhibit a peaked, then declining SHMR (“inverted-U”) with $\beta$1, while late-types show a monotonic increase—differences that encode distinct formation routes (merger-driven spheroids vs. regulated disc growth) and feedback processes (Posti et al., 2021).

6. Applications, Comparisons, and Survey Context

SHMR calibrations are foundational for the calibration of semi-analytic and hydrodynamical galaxy formation models: the observed normalization and slope force strong constraints on feedback feedback prescriptions for SNe and AGN (Mitchell et al., 2015). Accurate SHMRs enable cosmological inferences from weak lensing, clustering, and group catalogs, and are critical for interpreting baryonic effects, satellite fractions, and cold gas (HI) properties. Extending the parameterization to quantities such as HI mass or star-formation rate, as with the abundance matching approach, further strengthens constraints on halo formation, quenching, and assembly physics (Lu et al., 2024).

Comparisons amongst surveys, and between different environments (clusters, groups, field), affirm the global consistency of the SHMR but reveal modest but significant systematic offsets—especially when considering cosmic variance, photometric systematics, or structural parameters. Combined large-area surveys such as COSMOS+UDS, when employing continuity priors in HOD modeling, have successfully reduced systematic uncertainties and produced robust constraints out to $\beta$2 (Zaidi et al., 2024).

7. Implications and Future Directions

The SHMR encapsulates the interplay between hierarchical halo growth, feedback-driven self-regulation, environmental assembly bias, morphology, and star-formation quenching. Key open areas include:

• Reducing systematic uncertainties in stellar mass estimation (e.g., from photometry, IMF) and halo mass scaling relations.
• Refining high-redshift constraints with larger samples; at $\beta$3, data is currently limited by survey depth and dynamic range (Legrand et al., 2018, Zaidi et al., 2024).
• Better characterizing environmental effects and secondary dependencies (color, size, assembly history), and incorporating these into both empirical and theoretical models (Tonnesen et al., 2015, Xu et al., 2021).
• Extending measurements to ultra-low mass galaxies and constraining the role of non-standard components (e.g., IMBHs) (Zaritsky et al., 2022).
• Integrating multi-wavelength data (e.g., HI, SFR) for holistic baryonic accounting (Lu et al., 2024).

The SHMR is now established as the critical empirical tool bridging observable galaxy properties and the predictions of $\beta$4CDM structure formation, providing a stringent baseline for the next generation of cosmological and astrophysical modeling.