Stellar Velocity Dispersion Functions (VDFs)

• Stellar velocity dispersion functions are measures of the comoving number density of galaxies as a function of their central stellar velocity dispersion, serving as key dynamical diagnostics.
• They are determined using high-S/N spectroscopic surveys and photometric inference methods, with careful corrections to mitigate biases such as aperture effects and Eddington bias.
• Simulations and strong lensing analyses utilize VDFs to probe AGN feedback, halo assembly, and galaxy evolution, providing essential constraints on theoretical models.

Stellar velocity dispersion functions (VDFs) describe the comoving number density of galaxies as a function of their central stellar velocity dispersion, serving as critical dynamical diagnostics of galaxy structure, the mass distribution in dark matter halos, and the assembly histories of galaxies across cosmic time. VDFs are empirically constrained through direct spectroscopic surveys, strong lensing statistics, or photometric inference, and are fundamental to the calibration of empirical galaxy–halo connection models and for testing the predictive power of hydrodynamic simulations of galaxy formation.

1. Formal Definition and Analytic Forms

The stellar VDF, denoted Φ(σ), specifies the number density of galaxies per unit logarithmic velocity dispersion, typically in units of Mpc⁻³ dex⁻¹: $\Phi(\sigma) \equiv \frac{dN}{d\log\sigma}$ Alternatively, one can express the differential form as φ(σ) = dN/dσ = (1/\sigma\,\ln10) Φ(σ). The canonical analytic fit is a modified Schechter function, adapted for velocity dispersion: $$\Phi(\sigma)\,d\sigma = \Phi_* \left(\frac{\sigma}{\sigma_*}\right)^{\alpha} \frac{\exp[-(\sigma/\sigma_*)^{\beta}]}{\Gamma(\alpha/\beta)} \,\beta\,\frac{d\sigma}{\sigma}$$ where parameters {Φ*, σ*, α, β} control normalization, characteristic dispersion ("knee"), low-σ slope, and high-σ exponential cutoff, respectively (Montero-Dorta et al., 2016, Taylor et al., 2022). Alternative parameterizations (e.g., incorporating additional curvature terms) have been utilized for specialized studies (Chae, 2010).

2. Measurement Methodologies and Completeness

Observational measurement of the VDF requires accurate central stellar velocity dispersions. These are obtained directly from high-S/N spectroscopy (e.g., SDSS, BOSS, LEGA-C) and homogenized via aperture corrections to a consistent physical scale (e.g., 3 kpc or effective radius). Selection criteria—typically based on quiescent/star-forming demarcation (via D_n4000 or UVJ colors), magnitude limits (e.g., r < 17.77 in SDSS), and completeness thresholds in σ (to avoid Malmquist and L–σ relation selection biases)—are implemented to construct σ-complete samples (Sohn et al., 2017, Taylor et al., 2022, Hasan et al., 2019). Small measurement errors (e.g., 0.06 dex in σ) particularly affect the inferred high-σ tail via Eddington bias, motivating convolution of models with the measurement scatter (Bezanson et al., 2011).

Indirect approaches infer σ from photometric quantities—stellar mass, size, and Sérsic index—through a virial estimator calibrated against direct SDSS data: $$\sigma_{\mathrm{inf}} = \sqrt{\frac{G\,M_*}{0.557\,K_v(n)\,r_e}}$$ where the structural coefficient K_v(n) depends on the profile Sérsic index (Bezanson et al., 2011, Bezanson et al., 2012). These methods enable VDF measurement at higher redshift, subject to uncertainties in size measurements and underlying calibrations.

3. VDFs in the Local Universe: Field and Cluster Environments

In the local universe (z ≲ 0.1), multiple studies show that the field VDF of quiescent galaxies is nearly flat between log σ ≈ 2.0 and 2.3 (100–200 km s⁻¹), with Φ(σ) ≈ 5–7 × 10⁻³ Mpc⁻³ dex⁻¹, and falls off steeply beyond σ ≳ 200 km s⁻¹ (Sohn et al., 2017, Hasan et al., 2019). Complete samples demonstrate that earlier reports of a downturn at low σ were artifacts of magnitude-limited incompleteness. Morphological decompositions reveal that late-type galaxies dominate the low-σ regime and early-types the high-σ regime, with spectroscopic σ values systematically higher in rotating galaxies due to rotational broadening (Hasan et al., 2019).

Cluster VDFs, measured in systems such as Coma and A2029, show two key deviations from the field: a substantially flatter or even rising low-σ slope and a prominent high-σ excess (σ > 250 km s⁻¹). The high-σ excess is interpreted as the result of an overabundance of massive subhalos and brightest cluster galaxies formed via dry mergers and hierarchical assembly in dense environments (Sohn et al., 2016, Sohn et al., 2020, Sohn et al., 31 May 2024).

Representative VDF Parameters (Local, Quiescent/All Types)

Study/Type	σ_* (km/s)	α	β	Φ_* (Mpc⁻³ dex⁻¹)
Choi+2007, field ETG	161 ± 5	2.32 ± 0.10	2.67 ± 0.07	(2.7 ± 0.3)×10⁻³
Montero-Dorta+2016, z=0.55 RS	118.9 ± 12.4	6.75 ± 0.99	2.37 ± 0.14	(7.0 ± 1.1)×10⁻³
Bernardi+2010, z~0	161 ± 5	2.32 ± 0.10	2.67 ± 0.20	(8.0 ± 0.5)×10⁻³
Cluster: A2029+Coma	195 ± 28	0.0 ± 0.23	2.47 ± 0.63	(see text)
All-galaxy SDSS DR6	161 ± 5	2.32 ± 0.10	2.67 ± 0.20	(8.0 ± 0.5)×10⁻³

The flat low-σ slope (α ≈ 0) in clusters is contrasted with the much steeper field values (α = 2–6), highlighting the environmental dependence of dynamical galaxy populations (Sohn et al., 2016).

4. VDF Evolution with Redshift

Direct spectroscopic and strong-lensing analyses show minimal evolution in the shape or normalization of the high-σ (σ ≳ 250 km s⁻¹) tail of the VDF from z ≈ 1.5 to 0 (Bezanson et al., 2011, Bezanson et al., 2012, Taylor et al., 2022, Ferrami et al., 28 Oct 2024, Geng et al., 2021). At high σ, the number density of massive galaxies is already in place by z ≈ 1, with changes <0.2 dex, while the density of lower-σ (less massive) quiescent galaxies builds up by a factor ≈ 4, reflecting "downsizing" in galaxy quenching. The star-forming VDF is approximately constant, indicating that migrations in σ due to star-formation-driven growth compensate for galaxies quenching and moving off the star-forming sequence (Bezanson et al., 2012).

Strong lensing statistics, using the lens-redshift test, constrain the VDF to evolve with a slow decline in number density (factor ~2) and a mild increase (15–20%) in characteristic σ at z ~ 1 (Geng et al., 2021, Ferrami et al., 28 Oct 2024). These findings are consistent with ΛCDM hierarchical assembly and in tension with the strong “mass-downsizing” seen in photometric SMFs.

5. Simulation Predictions and Environmental Sensitivity

Hydrodynamic cosmological simulations (IllustrisTNG, EAGLE) predict VDFs that can be directly compared with observations by mimicking aperture-based σ measurements (Sohn et al., 31 May 2024, Choi et al., 4 Dec 2025). The field and cluster VDFs in these simulations reproduce the observed relative excess of high-σ galaxies in clusters, albeit requiring horizontal shifts to match absolute normalizations due to discrepancies in the simulated M_–σ_ relations and subhalo abundance.

The shape of the VDF, particularly its high-σ tail, is sensitive to AGN feedback strength in simulations. EAGLE runs with standard/enhanced feedback yield steeply declining VDF tails, while weak/no-AGN feedback leads to an excess of high-σ galaxies due to overconcentration of centrally formed stars. This highlights the VDF as a direct probe of baryonic physics—especially AGN feedback—in the coevolution of galaxies and their halos (Choi et al., 4 Dec 2025).

6. Physical Interpretation and Links to Halo Assembly

Central stellar velocity dispersion is a robust proxy for the depth of the gravitational potential, and thus is tightly correlated with total halo mass (σ* ∝ M*^0.3; σ_DM ∝ M_halo^0.3). Consequently, the VDF constitutes a photometry-independent tracer of the subhalo velocity-dispersion (or mass) function (Sohn et al., 2017, Montero-Dorta et al., 2016, Sohn et al., 2016). The parallel evolution of the VDF and dark matter halo mass function (HMF) found through abundance matching implies that the growth of stellar σ closely tracks that of the host halo, constraining the baryon conversion efficiency and the modeling of inner halo density profiles.

The observed stability of the high-σ VDF across redshift underpins the tightness of empirical scaling relations such as M_BH–σ and indicates that the most massive galaxies and black holes assembled early. In contrast, the growth of the low-σ quiescent VDF reflects ongoing quenching and morphological transformation at later epochs.

7. VDFs in the Milky Way and Local Volume

On Galaxy scales, stellar kinematic surveys (Gaia, APOGEE) directly measure multidimensional velocity DFs for thin disk, thick disk, and halo components. Population-decomposed DFs are well-approximated by multivariate Gaussians in cylindrical velocities, with the thin disk dominating the local VDF (~82%), followed by thick disk (~17%) and halo (~1.5%) (Anguiano et al., 2020). The vertical velocity-dispersion function ("vertical temperature distribution") in the Solar Neighborhood peaks at low dispersions (10–20 km s⁻¹) and declines steadily, consistent with SDSS/SEGUE measurements (Li et al., 2021).

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In summary, the stellar velocity dispersion function is a foundational dynamical census for galaxies, enabling stringent tests of theoretical models of galaxy and dark matter halo assembly, constraining feedback physics, and anchoring the empirical galaxy–halo connection across environments and cosmic time. Its observational determination—now reaching large volumes and high precision—continues to refine the quantitative understanding of galaxy formation physics, especially when synthesized with simulation predictions and strong-lensing constraints.