Baryonic Scaling Model in Galaxies & Clusters

• Baryonic scaling model is an empirically grounded framework connecting baryonic mass, structure, and dynamics via precise power-law relations in galaxies and clusters.
• It employs methods like baryonification and two-fluid hydrodynamics to accurately reproduce rotation curves, scaling laws, and low-scatter mass–size relations.
• The model underpins cosmological inference and halo mass calibration by quantifying feedback efficiency, predicting power spectrum suppression, and refining dark matter interplay.

The baryonic scaling model encompasses a suite of empirically grounded, physically interpretable frameworks that connect baryonic mass, structure, and dynamics in galaxies and clusters through tractable laws, typically power-law relations with low intrinsic scatter. These models, operating at both the galactic and cosmological levels, relate observable quantities—baryonic mass, disk scale length, rotation velocity, central surface density, or mass fractions—to properties of the dark matter halo or modified gravity background, as well as the impact of baryonic feedback within haloes. The frameworks manifest as either phenomenological scaling relations (e.g., the baryonic Tully–Fisher relation, mass–size law), explicit dynamical models (e.g., baryonification algorithms, hydrodynamical models with dark matter–baryon heating), or parameterizations used in analysis of large-scale structure data.

1. Fundamental Scaling Relations and Mathematical Formulation

The baryonic scaling model is typified by a minimal set of functional forms, often power laws or piecewise power laws, that link baryonic observables. The most prominent are:

• Baryonic Tully–Fisher Relation (BTFR): $M_{\rm b} = A\,V^n$, with $n \approx 4$ for galaxies; $A$ and $n$ may evolve or transition at high masses, e.g., in clusters (Fortune, 2019, Marongwe et al., 25 Nov 2025).
• Mass–Size Relation: $R_\star \propto M_{\rm b}^{\alpha}$, with empirical $\alpha = 0.385$ for late-type disks (Wu, 2017).
• Central Surface Density Law (CSDR): $\Sigma_{\rm DM}\sim a_0/G$ in HSB galaxies and $\Sigma_{\rm DM}\propto\sqrt{a_0\Sigma_b/G}$ in LSB disks (Famaey et al., 2019).
• Composite Rotation Curves: $$v_{\rm rot}(r) = \left[\Upsilon_{\rm d}v_{\rm d}^2(r) + \eta v_{\rm HI}^2(r)\right]^{1/2}$$, with $\Upsilon_{\rm d}$ and $n \approx 4$0 fit per galaxy (Swaters et al., 2012).

In the cosmological context, baryonic scaling models are realized as modifications to the halo model, introducing baryonic mass fractions $n \approx 4$1 for each component (gas, stars, etc.) with mass-dependent parameterizations, and yielding relations such as $n \approx 4$2 or broken/double power laws (Miller et al., 28 Apr 2025). The models typically enforce mass conservation and constrain the partitioning among stars, gas, ejected baryons, and diffuse components (Baggen et al., 1 Sep 2025).

2. Physical Principles and Model Assumptions

Baryonic scaling models derive their predictive power from a compact set of physical or phenomenological arguments:

• Hydrodynamical Equilibrium (Two-Fluid Models): For example, BIDM theory posits dark matter as an ideal gas, in local thermal equilibrium on a Jeans time ($n \approx 4$3), with baryonic heating that scales inversely with DM density, generating the MDAR and related scaling laws (Famaey et al., 2019).
• Empirical Decomposition: Observed rotation curves are reproduced by scaling up the dynamically distinct contributions (stars, HI, sometimes bulge) with two parameters, marginalizing over the halo, or—alternatively—encoding the halo as a baryon-tracing component (e.g., via $n \approx 4$4) (Swaters et al., 2012).
• Mass Fraction Partitioning: Cosmological models divide total baryon content into stars (central/diffuse), bound gas, and ejected gas, parameterizing their fractions as functions of halo mass and calibrating to simulations and observations (e.g., SHMR, hydrostatic gas fraction, X-ray/SZ constraints) (Baggen et al., 1 Sep 2025, Schneider et al., 2015).
• Scaling Symmetries: Several models identify anisotropic, sometimes enhanced, scaling symmetries of the hydrodynamics+gravity system (e.g., $n \approx 4$5), which enforce the parametric dependencies observed in galaxy scaling relations (Famaey et al., 2019).

3. Key Predictions and Observational Manifestations

Baryonic scaling models successfully reproduce a diverse set of empirical relations and trends:

• BTFR Slope and Zero Point: Both pure-empirical and physically motivated models produce $n \approx 4$6, with low intrinsic scatter ($n \approx 4$7–$n \approx 4$8 dex). The BTFR extends seamlessly from dwarfs to massive disks; galaxy clusters may occupy an offset but parallel BTFR, naturally explained by evolutionary effects (Marongwe et al., 25 Nov 2025).
• Mass–Size Law: $n \approx 4$9 with a remarkably small, mass-independent scatter of $A$0 dex over $A$1–$A$2 (Wu, 2017). The scatter is below that expected from halo spin, illustrating the fine-tuning of baryon–halo coupling.
• Rotation Curve Reproduction: Scaling stellar and HI contributions retrieves the observed circular velocity profiles of both late- and early-type disks, matching both the inner rise and outer fall, with two or three parameters per galaxy [$A$3 for most systems; (Swaters et al., 2012)].
• Cluster Baryon Fractions: The baryonic fraction $A$4 follows a broken or double power law: flat (self-similar) at high halo masses, steeper at $A$5, where AGN/stellar feedback redistributes baryons to the outskirts and into undetected phases (ICL, WHIM) (Miller et al., 28 Apr 2025).
• Power Spectrum Suppression: Baryonification algorithms predict a suppression of $A$6 relative to DMO runs of 10–25% at $A$7, with the amplitude governed by the ejected gas fraction and the suppression scale set by the ejection radius (Aricò et al., 2019, Schneider et al., 2015).

4. Computational and Practical Methodologies

Baryonic scaling models provide computationally efficient, physically interpretable prescriptions for both galactic and cosmological analysis:

Methodology	Key Application	Tunable Parameters / Calibrates To
Phenomenological fit	Rotation curves, BTFR, mass–size	$A$8, $A$9; observed profiles
Baryonification	Power spectrum, bispectrum ($n$0/$n$1)	$n$2, $n$3, $n$4, $n$5, etc.; hydro/X-ray/SZ
Two-fluid hydro	MDAR, CSDR, BTFR in disc galaxies	$n$6, heating rate normalization
Halo model extension	Cluster $n$7, $n$8	$n$9, $R_\star \propto M_{\rm b}^{\alpha}$0, $R_\star \propto M_{\rm b}^{\alpha}$1

These models allow rapid emulation or forward modeling of observations, direct parameter inference (e.g., via MCMC fitting against weak lensing or satellite kinematics), and systematic calibration to state-of-the-art hydrodynamic simulations (Aricò et al., 2019, Burger et al., 23 Jun 2025).

5. Interpretational Context, Limitations, and Theoretical Connections

• Origin of Tight Scaling Laws: The surprisingly small intrinsic scatter in $R_\star \propto M_{\rm b}^{\alpha}$2, BTFR, and analogous cluster scaling laws implies either a deep coupling between baryons and the gravitational potential or finely tuned baryonic processes (e.g., feedback efficiency, angular momentum retention). Standard CDM models, when supplemented with empirical baryonic mass–halo mass relations, can recover the mean slopes, but typically overpredict the scatter unless feedback-driven outflow scaling is tightly controlled (Sales et al., 2016, Dutton, 2012).
• Ambiguity in Physical Origin: The $R_\star \propto M_{\rm b}^{\alpha}$3 parameter in the rotation curve fits can represent extra cold gas, a dark matter component that traces HI, or an empirical mapping to a flattened halo; the models themselves generally remain agnostic without further data (Swaters et al., 2012).
• Limits at the Extremes: Some early-type spirals and the outermost regions in low-density HI disks deviate from the simple scaling fits, possibly due to ionization, missing flux, or kinematic departures from equilibrium. In clusters, “missing baryons” ($R_\star \propto M_{\rm b}^{\alpha}$414% of the cosmic budget) likely reside in warm/hot phases or the intracluster light, largely invisible to current X-ray/SZ observations but predicted by hydro and scaling models (Miller et al., 28 Apr 2025).
• Unified Phenomenology vs. Microphysical Models: While the scaling models are agnostic as to the microphysics (dark matter vs. modified gravity vs. coupled DM–baryon heating), they provide a robust benchmark that any successful theory of galaxy and cluster formation must reproduce.

6. Cosmological and Astrophysical Applications

Baryonic scaling models are essential for:

• Cosmological Inference: Accurate modeling of baryonic suppression/enhancement in non-linear structure (power- and bispectra) is required for unbiased parameter extraction in Stage IV lensing and clustering surveys (Burger et al., 23 Jun 2025, Aricò et al., 2019).
• Halo Mass Calibration: Satellite kinematics and HOD modeling must incorporate baryonification corrections to avoid $R_\star \propto M_{\rm b}^{\alpha}$5–$R_\star \propto M_{\rm b}^{\alpha}$6 dex systematic errors at $R_\star \propto M_{\rm b}^{\alpha}$7 (Baggen et al., 1 Sep 2025).
• Landscape of Feedback/Ejection Efficiency: Quantitative trends in the ejected baryon fraction, break masses in $R_\star \propto M_{\rm b}^{\alpha}$8, and the cosmic evolution of normalization in the BTFR provide empirical constraints on the physics of feedback and gas cycling, as well as the timescale and mode of structure growth (Marongwe et al., 25 Nov 2025).

7. Future Prospects and Model Extensions

Advances in deep imaging and high-fidelity hydro simulations will sharpen both parameter calibration (e.g., for gas/ICL in the circumgalactic and intracluster medium) and model selectivity (e.g., via redshift evolution of the BTFR zero point and tightness). Generalizations incorporating modified gravity, explicit DM–baryon microphysics, or fully non-linear coupling to cosmic expansion can be evaluated against the baryonic scaling “backbone” established by these models (Famaey et al., 2019, Shenavar, 2021). Continued observation-simulation synergy, as exemplified by joint SPT–SZ/IllustrisTNG or Euclid FLAMINGO emulator analyses, will further situate the baryonic scaling paradigm as both an interpretive lens and a practical standard in astrophysics.