The Galactic Halo Rotation by Weyl Incorporated Gravity

Abstract: A modification of the Einstein-Hilbert Lagrangian by introducing a coupling between the Weyl tensor and the stress-energy tensor was proposed to explain flat galactic rotation curves without the exotic (non-baryonic) dark matter (DM) [1]. The proposed coupling constant was previously determined by fitting the rotational velocities of the Milky Way and M31 modeled with constant density, yielding the same coupling constant for both [2,3]. In this work, we have modified the formalism for a variable density by modeling the galactic systems with realistic, spherically symmetric and radially varying density profiles for the baryonic matter and this analysis is applied to seven edge-on spiral galaxies of the local cluster [4-10] and the Milky Way.

• The paper introduces a novel Weyl-incorporated gravity model that fits galactic rotation curves using a single universal coupling parameter.
• It employs radially-varying baryonic density profiles from NFW, Moore, and Burkert models to achieve velocity fits within 2-3% of observations.
• The results support a covariant alternative to dark matter, linking galactic dynamics with potential quantum gravity improvements.

Weyl Incorporated Gravity and Galactic Halo Rotation Curves

Introduction and Motivation

The persistent discrepancy between observed galactic rotation curves and predictions derived from luminous baryonic matter distributions under General Relativity (GR) has traditionally been addressed by invoking a dominant, non-baryonic dark matter (DM) component. Despite extensive searches, no non-baryonic DM candidates consistent with the Standard Model of particle physics (SMpp) have been conclusively observed, and the baryon census from the CMB and structure formation leaves significant uncertainties, including the "missing baryon problem." This context motivates critical examination of gravitational dynamics at galactic and extragalactic scales.

The framework analyzed in "The Galactic Halo Rotation by Weyl Incorporated Gravity" (2604.01643) offers a minimalist covariant modification to the Einstein-Hilbert action. Rather than introducing extensive new degrees of freedom, the approach incorporates a single direct non-linear coupling of the stress-energy tensor $T_{\mu\nu}$ with the Weyl tensor $C_{\alpha\mu\beta\nu}$, parameterized by a coupling constant $\lambda$. The primary objective is to account for halo rotation curves using only baryonic mass (including baryonic dark matter in virial clouds), thereby obviating the necessity for exotic non-baryonic DM.

Theoretical Framework

The proposed Lagrangian is given by

$$\mathcal{L} = \sqrt{-g} \left( R - 2\Lambda - \kappa T + \lambda\, C_{\alpha\mu\beta\nu} T^{\alpha\beta} T^{\mu\nu} \right),$$

where $\lambda$ governs the new interaction term embodying direct matter-gravity couplings. This yields amended field equations:

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu} R + g_{\mu\nu}\Lambda = \kappa T_{\mu\nu} + \lambda I_{\mu\nu},$$

with $I_{\mu\nu}$ encapsulating the Weyl-matter interaction functional derivatives. The model is termed Modified Relativistic Dynamics (MORD), and the underlying philosophy is that the interaction, inspired by the QED source-field vertex, offers a natural path toward a renormalizable quantum theory of gravity, while potentially addressing DM phenomenology.

Methodology and Analysis

The authors systematically improve upon their prior analysis (which utilized constant-density halos) by introducing radially varying, spherically symmetric baryonic DM density profiles, specifically the Navarro-Frenk-White (NFW), Moore, and Burkert forms, for the Milky Way and seven other spiral galaxies (M31, M33, M81, M82, NGC5128, NGC4594, and M90). The baryonic DM fraction is modeled by structures observed in halo virial clouds, consistent with both kinematic and CMB constraints.

The metric is assumed spherically symmetric, and the resulting nonlinear ODEs for the gravitational potential $\mu(r)$ are solved numerically, subject to regularity at the core. The rotational velocity profile is expressed as

$v^2(r) = \frac{r \nu'(r)}{2},$

with $\nu'(r)$ determined implicitly via the modified field equations and input density profile. A shooting method is employed: for each galaxy and density model, $C_{\alpha\mu\beta\nu}$0 is iteratively tuned so that the predicted rotational velocity at the outermost observed galactocentric radius (100 kpc) matches empirical data.

Key Numerical Results

A striking result is that a single parameter value $C_{\alpha\mu\beta\nu}$1 provides an excellent fit for the outer flat halo rotation velocities across all eight galaxies and the three tested density profiles. Specific highlights include:

• Milky Way: Modeled rotation velocities at $C_{\alpha\mu\beta\nu}$2 kpc are 153–159 km/s (observed $C_{\alpha\mu\beta\nu}$3 km/s), enclosed baryonic DM halo mass $C_{\alpha\mu\beta\nu}$4.
• M31: Modeled velocities 230–232 km/s, observed $C_{\alpha\mu\beta\nu}$5 km/s, halo mass $C_{\alpha\mu\beta\nu}$6.
• M33: Modeled velocities 121–122 km/s, observed $C_{\alpha\mu\beta\nu}$7 km/s, halo mass $C_{\alpha\mu\beta\nu}$8.
• Agreement persists for M81, M82, NGC5128, NGC4594, and M90, with variations in core radius, density, and total mass all adequately reproduced by the universal $C_{\alpha\mu\beta\nu}$9 value within quoted uncertainties.

The authors emphasize that this result stands in sharp contrast to conventional particle DM scenarios, which typically require extensive multiparameter fitting, and to phenomenological modified gravity approaches (e.g., MOND), which are either incompatible with covariant frameworks or are empirically excluded at high significance by recent Gaia wide-binary star results.

Theoretical and Practical Implications

The numerical stability and cross-galaxy universality of the fitted $\lambda$0 value underscore the viability of the Weyl-matter coupling. This finding restricts the solution space to a single parameter regime of modified gravity, potentially eliminating the need for exotic non-baryonic DM contributions at galactic scales.

Practically, this approach enables robust rotation curve modeling with standard baryonic physics and spherically symmetric baryonic DM, significantly reducing the theoretical degeneracy. The method could, in principle, be extended to more complex geometries (e.g., axisymmetric or triaxial halos, disks, or the slow-rotation regime analogous to the Kerr metric), with the expectation that the universal value of $\lambda$1 persists or acquires only minor corrections.

On the fundamental side, the proposed Lagrangian structure links phenomenological galactic dynamics and quantum gravity renormalizability. The direct source-field interaction posited here could play a critical role in future attempts to reconcile gravitational theory with quantum field theory, although explicit renormalization group studies remain to be performed.

Future Directions

Further work is required to generalize the present formalism to more realistic, non-spherical halo and disk structures, and to incorporate vertical (non-circular) velocity components which may be relevant in detailed local or extra-planar halo dynamics. A systematic survey of galaxy clusters, where field strengths and potential boundary effects are more pronounced, could test the limits of MORD and explore any possible deviations in the inferred value of $\lambda$2.

Additionally, the formal quantum aspects of the Weyl-matter coupling deserve detailed exploration. If successful, this framework could facilitate a unified treatment of cosmological large-scale structure formation and quantum corrections to the metric, potentially resolving the quantum gravity problem while accounting for DM phenomenology with minimal parameter extension.

Conclusion

The study rigorously demonstrates that a covariant, minimal modification of the Einstein-Hilbert action—specifically, a direct source-field interaction term between the Weyl tensor and the matter stress-energy tensor—enables accurate and universal fits to galactic halo rotation curves using only baryonic mass profiles. The necessity of exotic non-baryonic DM is circumvented. The inferred coupling constant $\lambda$3 is both numerically stable and astrophysically viable across a diverse ensemble of galaxies and halo models. This approach motivates further theoretical and observational study, both as a practical model for galactic kinematics and as a possible foundation for quantum gravity.