Fundamental Plane of Galaxies

• Fundamental Plane is an empirical scaling relation connecting effective radius, central velocity dispersion, and surface brightness of galaxies.
• It reveals systematic deviations from pure virial predictions, driven by variations in dark matter content and stellar population properties.
• Recent studies show the FP as a warped surface with redshift and mass dependencies, offering insights into galaxy formation and evolution.

The Fundamental Plane (FP) is an empirical, multi-dimensional scaling relation describing the structural and kinematic properties of dynamically hot, pressure-supported stellar systems, most notably early-type galaxies (ETGs), but also extending to galaxy clusters, groupings, and even dwarf galaxies. Defined in terms of effective radius, central velocity dispersion, and surface brightness (or related mass quantities), the FP is not only a diagnostic of equilibrium structure and stellar populations, but also a key to understanding the assembly, evolution, and dynamical states of galaxies and galaxy systems. While classical treatments posit the FP as a plane in log–observable space, recent work demonstrates that it is best described as a warped or curved surface whose intrinsic properties (“tilt,” scatter, projection curvature) encode the interplay of baryonic physics, dark matter, stellar population effects, and hierarchical assembly.

1. Canonical Formulation and Empirical Characterization

The classical FP is expressed as a trivariate relation among the logarithms of the effective (half-light) radius, $R_e$, the central velocity dispersion, $\sigma_0$, and the mean surface brightness within $R_e$, $\langle I_e \rangle$ (often transformed to mean surface brightness in mag arcsec$^{-2}$, $\mu_e$):

$$\log R_e = a\,\log\sigma_0 + b\,\log\langle I_e \rangle + c$$

or, in surface brightness terms,

$\log R_e = a\,\log\sigma_0 + b\,\mu_e + c$

Empirical coefficients for local, massive ETGs are typically $a \sim 1.2$–$1.5$, $b \sim -0.8$ (or $b\sim0.3$ if using $\mu_e$), with intrinsic scatter $\sim0.05$–$0.15$ dex in $\log R_e$ (Magoulas et al., 2012, Bernardi et al., 2020, D'Eugenio et al., 2021). The near-infrared FP from large samples (e.g., 6dFGS) follows similar scaling, e.g., $$R_e \propto \sigma_0^{1.52 \pm 0.03} I_e^{-0.89 \pm 0.01}$$, with measured scatter in $R_e$ about the FP of $\sim29\%$, decomposing into observational and intrinsic components (Magoulas et al., 2012).

For massive, quiescent galaxies out to $z\sim2$, the FP persists with only mild evolution in coefficients and zero point (Sande et al., 2014, Stockmann et al., 2020, Bezanson et al., 2013). However, both the normalization and slopes can be mass- and redshift-dependent, with lower-mass systems and high-redshift samples showing increased scatter and deviations in tilt.

2. Physical Origin: Virial Theorem, Tilt, and Surface Warping

The FP emerges fundamentally from the virial equilibrium of pressure-supported systems, constrained by the scalar virial theorem:

$M_\text{dyn} \propto \sigma_0^2 R_e / G$

and the definition of luminosity in terms of effective surface brightness,

$L = 2\pi R_e^2 \langle I_e \rangle.$

For structurally homologous galaxies with constant mass-to-light ratio (M/L), the theoretical expectation (the “virial plane”) is $a=2$, $b=-1$. Observationally, however, the FP exhibits a significant “tilt”—the coefficients systematically deviate from the virial predictions (Bernardi et al., 2020, D'Onofrio et al., 2021, D'Addona et al., 17 Mar 2025):

• In classical FP projections, $a < 2$ and $b > -1$ (absolute value), indicating weaker dependence on $\sigma_0$ and flatter slope in $\langle I_e \rangle$.
• This tilt results from a combination of stellar population variation (stellar $M/L$), structural/dynamical non-homology (e.g., Sèrsic index changes), and, crucially, smooth systematic variation in the dark matter fraction within $R_e$ (2207.13491, Bernardi et al., 2020, D'Addona et al., 17 Mar 2025, D'Eugenio et al., 2021, Yoon et al., 2022).

Recent work, explicitly fitting the FP across finely binned mass, surface brightness, or velocity dispersion subsamples, shows that the fundamental “plane” is actually a curved surface or warped manifold in $(\log R_e, \log\sigma_0, \mu_e)$-space (Yoon et al., 2022). The coefficients $(a, b, c)$ are not constants, but functions of intrinsic galaxy properties such as central velocity dispersion or surface brightness, with systematic, monotonic dependence across parameter space.

3. Origin of the FP Tilt: Stellar/DM Content and Hyperplane Extensions

The FP “tilt” encodes the underlying physical variation of the baryon and dark-matter content, as well as gradients in stellar population properties. Recent modifications of the standard FP include the “mass Fundamental Plane” (mass FP), where surface brightness is replaced with stellar mass surface density, $\Sigma_*$, yielding (Bezanson et al., 2013, D'Addona et al., 17 Mar 2025):

$$\log R_e = \alpha\,\log\sigma_0 + \beta\,\log\Sigma_* + \gamma$$

The best-fit mass FP coefficients for local quiescent galaxies are $\alpha\simeq1.63$, $\beta\simeq-0.84$, with little evidence for tilt relative to the virial expectation (homology holds to first order), and minimal zero-point evolution to $z\sim2$ (shifts $\lesssim0.04$ dex in $\log R_e$). The remaining small tilt at fixed stellar mass is attributed primarily to smooth variations in the dark matter content inside $R_e$ (2207.13491).

Moving beyond the classic 3D FP, 4D “hyperplanes” include additional physically meaningful axes: stellar mass $(M_*)$ or stellar fraction $(f_{e}^*)$. D’Addona et al. show that, for massive cluster ETGs, the inclusion of $f_{e}^*$ yields a hyperplane with slopes close to the virial expectation, and significantly reduced scatter—demonstrating that dark-matter fraction variations are the principal source of the classical FP tilt (D'Addona et al., 17 Mar 2025).

4. Time Evolution, Environmental Dependence, and Sample Selection Effects

The FP exhibits mild but systematic evolution with cosmic time and galaxy mass (D'Onofrio et al., 24 Apr 2024, Stockmann et al., 2020, Fritz et al., 2010, Sande et al., 2014):

• Massive ETGs show only weak zero-point drift to $z\sim 2$, with the observed evolution in the luminosity FP primarily tracing population aging ($M/L$ evolution), not dynamical structure (Bezanson et al., 2013, Sande et al., 2014, Stockmann et al., 2020).
• The FP slope steepens toward higher redshift and lower masses, indicative of “downsizing”: lower-mass galaxies quench later and exhibit younger luminous populations (Fritz et al., 2010).
• Intrinsic scatter about the FP remains low ($\sim0.04$–$0.15$ dex) across epochs when refitting the FP at each redshift and mass interval, but increases markedly if the local FP is naïvely extrapolated to high-z or low-mass systems (D'Onofrio et al., 24 Apr 2024).

Environmental effects—though often minor—can be detected: field galaxies are slightly larger than group/cluster counterparts at fixed velocity dispersion and surface brightness (offsets of $\sim5\%$), and S0/spiral bulges are systemically offset in the FP relative to ellipticals (Magoulas et al., 2012, Campbell et al., 2014).

The FP residuals also carry subtle information about local environment and large-scale structure. Notably, spatial correlations in FP residuals trace the density field to scales of $\gtrsim10$ Mpc, with central galaxies lying systematically above the FP and satellites below, plausibly reflecting differing mass-to-light ratios due to tidal or environmental effects (Joachimi et al., 2015).

5. Scatter, Residuals, and Connections to Stellar Populations

The thickness of the FP (orthogonal scatter) is exceptionally small for slow-rotator ellipticals (E-SRs), declining to $\sim0.02$–$0.05$ dex when limiting the sample to massive, morphologically pure systems (Bernardi et al., 2020):

Morphological Type	Intrinsic Scatter (dex)	rms$_\text{obs}$ (\%)
E-SRs (massive)	0.010	7
All ETGs+S0s	0.048	13
Dwarfs (faint)	0.12–0.15	30–35

FP residuals are most strongly correlated with stellar population age and $M_*/L$. For example, in the SAMI survey, the FP residual correlates at $8\sigma$ with log SSP age, and the residuals are weakly correlated with other structural parameters (Sèrsic index, ellipticity) (D'Eugenio et al., 2021). Quantitatively, variation in stellar population $M/L$ at fixed structural parameters accounts for $\sim75\%$ of the FP scatter, but only partially for the tilt. The remaining tilt arises from systematics in the dark-to-stellar mass ratio and/or IMF effects (D'Eugenio et al., 2021, 2207.13491).

The residuals also reflect the impact of galaxy-specific evolutionary pathways—dry mergers, minor accretion, and quenching—all of which can be described by variation in the FP coefficients and projective curvature (Yoon et al., 2022, D'Onofrio et al., 2021).

6. FP as a Warped Surface: Curvature and Projections

The classic interpretation of the FP as a “plane” is superseded by evidence of curvature—a warped, twisted 2D surface in log-space whose local normal vector shifts with position (Yoon et al., 2022, D'Onofrio et al., 2021):

• Subsamples binned by velocity dispersion ($\sigma_0$) or surface brightness ($\mu_e$) reveal systematic changes in FP coefficients, with $b$ rising and the zero-point $c$ falling as $\sigma_0$ increases.
• The local structure of the FP is best described by

$$\log R_e = a(\sigma_0)\,\log\sigma_0 + b(\sigma_0)\,\mu_e + c(\sigma_0)$$

where $(a,b,c)$ are continuous functions of galaxy physical properties.

• Classical projections, e.g., Faber–Jackson ($L$–$\sigma_0$) or Kormendy ($R_e$–$\mu_e$) relations, also show continuous curvature as one moves through the FP parameter space.

This “warping” results physically from differential merger histories: more massive ETGs, having experienced more dry minor mergers, exhibit more prominent curvature, with larger $R_e$, lower $\mu_e$, and stronger dependence of $R_e$ on $\mu_e$ (Yoon et al., 2022, D'Onofrio et al., 24 Apr 2024).

7. Extensions to Groups, Clusters, and Alternative Theories

The FP concept extends robustly to galaxy groups and clusters, with analogous scaling between size, dynamical velocity, and total luminosity or mass (Kopylova et al., 3 Sep 2024). For clusters,

$$\log R_e = 0.98\,\log\sigma - 0.56\,\log\langle I_e\rangle + 3.57$$

with observed scatter corresponding to $\sim16\%$ distance accuracy. The cluster FP slopes are remarkably close to those of galaxies, hinting at homologous dynamical structure across mass scales and providing a cosmological distance tool (Kopylova et al., 3 Sep 2024).

Alternative gravity theories, such as $f(R)$ gravity, have also been invoked to recover the observed FP tilt without dark matter, by modifying the Newtonian potential and connecting structural and kinematic radii; the observed FP coefficients can be recovered for certain parameter choices (Jovanović et al., 2016).

8. Implications for Cosmology, Simulation, and Galaxy Formation

The FP—by virtue of its thinness and universality—serves as both a high-precision distance indicator and a probe of cosmic structure, albeit with the caveat that residual correlations with large-scale environment can bias peculiar velocity and weak lensing measurements at the 10%–100% level if uncorrected (Joachimi et al., 2015).

State-of-the-art cosmological simulations (Illustris, EAGLE, TNG) reproduce the FP’s overall thinness and zero-point evolution, attribute the tilt mainly to smooth dark-matter fraction variations, and highlight the sensitivity of the FP coefficients to feedback, mass profile, and resolution effects (D'Onofrio et al., 24 Apr 2024, 2207.13491). The “mass FP” emerges early ($z\sim2$) and remains nearly stable in normalization and tilt, in contrast to dramatic changes in luminous size and morphology.

The FP thus encodes both the dynamical assembly (merger-driven size growth, dark-matter profile) and the integrated stellar-population evolution of galaxies; its observed warping, tilt, environmental dependencies, and minimal scatter together provide stringent constraints on the complex baryonic and dynamical processes that drive galaxy evolution across cosmic time.