Resolving Galactic and Cluster Dynamics Without Dark Matter: Tsallis Entropy as the Unique Foundation of Emergent Gravity

Technical Summary: Resolving Galactic and Cluster Dynamics Without Dark Matter via Tsallis Entropy

Problem Statement
The paper addresses the persistent discrepancy between observed gravitational dynamics and predictions based on visible baryonic matter, a problem traditionally attributed to Dark Matter (DM). While the $\Lambda$CDM model posits non-baryonic DM, direct detection remains elusive, motivating alternative theories such as Modified Newtonian Dynamics (MOND) and Entropic Gravity. A central challenge in the entropic gravity paradigm (proposed by Verlinde and Jacobson) is identifying the correct entropy functional that governs spacetime microstates. Various generalizations of the Bekenstein-Hawking entropy—specifically Barrow, Tsallis, Kaniadakis, Power-law, Logarithmic, and Rényi entropies—have been explored primarily in cosmological contexts. However, their validity on galactic and sub-cosmological scales (galaxies, galaxy clusters, and globular clusters) remains largely untested. The authors aim to determine which, if any, of these generalized entropies can consistently explain gravitational dynamics across this entire structural hierarchy without invoking dark matter.

Methodology
The authors employ the entropic force framework to derive modified gravitational force laws from different entropy formulations. They categorize the entropies into two types:

1. Type-II Entropies: Additive corrections to the standard area law (e.g., Rényi, Kaniadakis, Logarithmic, Barrow with restricted parameters).
2. Type-I Entropies: Power-law generalizations of the area law (e.g., Tsallis, Barrow with extended parameters).

The methodology involves:

• Theoretical Derivation: Deriving the modified Newtonian force laws and mass-velocity relations for each entropy type.
• Galactic Scale Testing: Evaluating the ability of these force laws to reproduce flat galactic rotation curves and the Tully-Fisher relation.
• Galaxy Cluster Testing: Applying the derived force laws to 40 X-ray galaxy clusters. Using the King $\beta$-model for intra-cluster gas density and hydrostatic equilibrium equations, the authors reconstruct the mass profiles $M(r)$ and fit the nonextensive parameter $\delta$ (or equivalent parameters) to observational X-ray data.
• Globular Cluster Testing: Extending the Tsallis framework to 33 Milky Way globular clusters using high-precision kinematic data (Baumgardt & Hilker, 2018). They solve the spherical Jeans equation with a Hernquist density profile to fit line-of-sight velocity dispersion profiles and determine the best-fit $\delta$ values.
• Statistical Analysis: Comparing the goodness-of-fit ($\chi^2_{red}$) and the behavior of the nonextensive parameter $\delta$ across different scales and system properties (mass, radius, temperature).

Key Contributions and Results

1. Failure of Type-II Entropies: The paper demonstrates that Type-II entropies (Rényi, Kaniadakis, Logarithmic) and standard Barrow entropy (with $\Delta \in [0,1]$) fail to reproduce flat galactic rotation curves. When applied to galaxy clusters, these models yield mass profiles that are either unphysical or inconsistent with X-ray observations. Specifically, the authors show that the correction terms required to fit rotation curves (scaling as $R^4$) are not naturally produced by these entropies without vanishing correction terms.

2. Uniqueness of Tsallis Entropy: Tsallis entropy is identified as the only formulation that successfully passes all observational tests:

   • Galaxies: It successfully reproduces flat rotation curves and the Tully-Fisher relation. The nonextensive parameter is found to be $\delta \approx 0.5$ at galactic scales ($\sim 10$ kpc).
   • Galaxy Clusters: Applying the Tsallis-modified force law ($F \propto R^{-2\delta}$) to 40 galaxy clusters yields a consistent fit with a mean parameter $\langle \delta \rangle = 0.9770 \pm 0.004$. The predicted mass profiles align with X-ray observations without requiring dark matter.
   • Globular Clusters: Analysis of 33 globular clusters reveals $\langle \delta \rangle = 1.0046 \pm 0.0025$, which is statistically consistent with Newtonian gravity ($\delta = 1$). This explains why globular clusters generally do not exhibit the need for dark matter, while accommodating slight deviations in specific systems (e.g., M15).

3. Scale Dependence vs. System Independence: A critical finding is that while the value of $\delta$ changes with the astrophysical scale (from $\sim 0.5$ in galaxies to $\sim 1.0$ in globular clusters), it exhibits no correlation with macroscopic system properties (mass, radius, temperature, or density) within a given scale. This suggests $\delta$ is not a function of instantaneous state variables but encodes the dynamical history and relaxation state of the system.

4. Theoretical Interpretation: The authors interpret the variation in $\delta$ through the lens of nonextensive statistical mechanics. They argue that $\delta$ measures the degree of "violent relaxation" and the presence of "hidden constraints" (Casimir invariants).

   • In relaxed, collisional systems (globular clusters), the system approaches a Gaussian state ($\delta \to 1$).
   • In collisionless, unrelaxed systems (galaxies and clusters), the system remains in a non-Gaussian state ($\delta < 1$).
   • This framework resolves the Carroll-Remmen criticism regarding entropic gravity by providing a well-defined thermodynamic entropy (via nonextensive statistics) that supports entanglement entropy, unlike standard extensive entropy.

5. Recovery of MOND: The paper notes that the Tsallis framework, via the "zero-energy bits" mechanism, naturally recovers MOND phenomenology in the appropriate limit, offering a theoretical origin for MOND that other entropic approaches lack.

Significance and Claims
The paper claims that the entropic gravity paradigm, when confronted with multi-scale observational data, uniquely selects Tsallis entropy as the correct statistical foundation for emergent gravity. The authors argue that:

• Gravity is consistently described by nonextensive statistical mechanics.
• Tsallis entropy is the unique generalized entropy capable of explaining galactic rotation curves, galaxy cluster masses, and globular cluster dynamics without dark matter.
• The framework predicts the existence of dynamically relaxed galaxy clusters with $\delta = 1$, which would be observationally dark matter-free (dynamical mass equals baryonic mass). The discovery of such a cluster would serve as a decisive test to distinguish Tsallis gravity from both $\Lambda$CDM and MOND.
• The success of Tsallis entropy across scales implies a deep connection between macroscopic gravitational phenomena and the microscopic quantum structure of spacetime, specifically regarding entanglement and non-extensivity.

The authors conclude that for the entropic gravity program to be consistent with observations from globular clusters to galaxy clusters, it must inevitably be built upon Tsallis entropy.