A framework for creating galaxy models in the geometry of the conservation group with dark matter halos and flat rotation curves

Abstract: Pandres has developed a theory which extends the geometrical structure of a real four-dimensional space-time via a field of orthonormal tetrads with an enlarged covariance group. This new group, called the conservation group, contains the group of diffeomorphisms as a proper subgroup. The free-field Lagrangian density involves only the curvature vector which is a vector which measures curvature. When massive objects are present a source term is added to this Lagrangian density. The weak-field approximation implies that gravitational waves travel at the speed of light. Spherically symmetric solutions for both the free field and the field with sources are found. In the free-field case, the field equations require nonzero stress-energy tensors. However, we find that for our model to be an acceptable model, we must have a source term in the Lagrangian. In our framework, we divide up the galaxy into three spherically symmetric regions: a baryonic matter-dominated central bulge, a dark matter-dominated mesosphere and an outside region where neither type dominates. Assuming the density of baryonic matter has a central cusp, we show how to model the bulge. Via an isothermal condition we find a model for the mesosphere and show this model implies flat rotation curves with one free parameter. The outside region is readily modeled via previously published results. The models for the bulge, mesosphere and outside region are combined into one continuous model. Using the radial acceleration relation we then show how a galaxy model may be set up for a rotationally supported galaxy.

• The paper introduces a framework based on an extended covariance group that naturally produces effective dark matter within galactic halos.
• It employs a covariant Lagrangian using the square of the curvature vector to derive spherically symmetric solutions that match flat rotation curves and the baryonic Tully-Fisher relation.
• The model unifies baryonic and geometric contributions, offering an intrinsic explanation for dark matter effects without ad hoc components.

A Framework for Galaxy Modeling via the Conservation Group: Dark Matter Halos and Flat Rotation Curves

Introduction

This work develops a detailed framework for modeling galaxies using an extension of general covariance, the conservation group, following Pandres’ construction. The conservation group strictly contains the diffeomorphism group, representing the largest group under which scalar wave equations retain invariance. The framework is built on a Lagrangian employing the square of the covariant curvature vector $C^\mu C_\mu$, representing the gravitational free field, with an additional source term incorporated for massive objects. The spherically symmetric solutions derived within this schema offer a physically consistent basis to analyze the baryonic bulge, isothermal mesosphere (dark matter halo), and the exterior (asymptotic) regions of spiral galaxy models, with a particular focus on reconciling observed flat rotation curves and the empirical radial acceleration relation.

The Conservation Group and Covariant Lagrangian Structure

The foundation of the theory is the enlargement of the covariance group to include all transformations preserving the condition $V^\alpha_{;\alpha} = 0$, which is shown to be broader than standard diffeomorphisms. The primary dynamic variable is the tetrad field $h^i_{\;\mu}$, with the metric defined as $g_{\mu\nu} = \eta_{ij}h^i_{\;\mu}h^j_{\;\nu}$. The Lagrangian for the free field, $\mathcal{L}_f \propto C^\mu C_\mu \sqrt{-g}$, distinctively adds non-Einsteinian terms to the action, producing stress-energy content even in the ‘source-free’ case. This property is leveraged to provide an intrinsic geometric model of dark matter (and dark energy), rather than introducing ad hoc matter species or phenomenological terms.

Upon inclusion of sources, the Lagrangian becomes $\mathcal{L} = \mathcal{L}_f + \mathcal{L}_s$, where the source Lagrangian is typically associated with standard baryonic mass-energy. The derived field equations ensure covariant conservation of the total stress-energy tensor with respect to the conservation group. Notably, the density of the source is always given by $\rho_s = (1/16\pi) C^\mu C_\mu$, offering a unifying prescription for matter and geometric energy at all scales.

Weak-Field Regime and Spherically Symmetric Solutions

Linearizing about flat spacetime leads to field equations compatible with gravitational waves propagating at $c$, as in general relativity. However, it is shown that source-free spherically symmetric solutions fail to reproduce the correct weak-field lensing and potential, resulting in an anomalous $g_{00}$ component not matching the classic Newtonian limit. Consequently, only solutions with non-trivial source terms produce physically admissible galaxy models.

In the spherically symmetric case, the theory allows for a general tetrad parameterization in terms of two functions $\Phi(r)$ and $V^\alpha_{;\alpha} = 0$0. The resulting dynamical and kinematic equations for radial and orbital motion follow standard procedures but are supplemented by force terms and mass profiles determined by the conservation group stress-energy structure.

Three-Region Galaxy Model Framework

The galaxy is partitioned into three regimes for modeling:

• Bulge: A baryon-dominated central region with strong fields, modeled with a mass profile $V^\alpha_{;\alpha} = 0$1 and a central baryonic density exhibiting a cusp. The corresponding dark matter density remains finite as $V^\alpha_{;\alpha} = 0$2, avoiding the central divergence typically plaguing standard CDM halo models.
• Mesosphere: The region $V^\alpha_{;\alpha} = 0$3 dominated by dark matter, assumed to be in isothermal equilibrium. Here, the source density is set small compared to the total, and the mass profile is quasi-linear in $V^\alpha_{;\alpha} = 0$4, $V^\alpha_{;\alpha} = 0$5 (with $V^\alpha_{;\alpha} = 0$6 determined by baryonic parameters). The metric within this zone yields a constant circular velocity,

$V^\alpha_{;\alpha} = 0$7

consistent across the entire mesosphere, closely matching observed flat rotation curves up to the disk edge and providing a natural explanation for this key galactic phenomenon.

• Outside Region: Beyond $V^\alpha_{;\alpha} = 0$8, the solution approaches an asymptotically Schwarzschild-like form, but the presence of non-vanishing geometric dark sector terms modifies density and pressure profiles so that total mass-energy is smoothly incorporated.

Continuity of the metric and stress-energy solutions at the interfaces ($V^\alpha_{;\alpha} = 0$9) is strictly enforced, fixing integration constants and connecting the geometrical and physical parameters.

Connection to Observational Relations

The analysis leads to a direct derivation of the baryonic Tully-Fisher relation ($h^i_{\;\mu}$0) and the Radial Acceleration Relation (RAR) found by McGaugh, Lelli, and Schombert. Specifically, the model’s geometry ensures that the ratio $h^i_{\;\mu}$1 sets both the flat rotation velocity and the acceleration scale at the bulge/mesosphere boundary:

$h^i_{\;\mu}$2

Using the empirically measured ratio of the dark-to-baryonic acceleration at $h^i_{\;\mu}$3, the entire galaxy rotation profile and mass model are determined. The presented procedure generates rotation velocities and dark matter central densities consistent with measured values across different galaxies.

Implications and Theoretical Consequences

The immediate theoretical implication is that geometric effects (the non-Einsteinian terms mandated by extending to the conservation group) automatically produce effective dark matter in galactic halos, tightly correlating with baryonic matter and consistent with empirical scaling laws. The unification is not reliant on specific particle dark matter candidates; rather, ‘dark’ effects are a manifestation of the underlying quantum-geometric structure. The universality of the mapping between baryonic and apparent dark mass is a direct consequence of the extended covariance.

On a practical level, this framework provides a structured way to construct spherically symmetric galaxy models that are flexible, parameterized by observable baryonic properties, and consistent with both rotation curves and mass distributions.

This approach suggests several directions for future investigation in astrophysical and cosmological modeling:

• Extension to axisymmetric/disc galaxies and irregular systems.
• Detailed comparison to high-resolution rotation curve data across a range of morphological types.
• Exploration of quantum gravity implications, since the foundation group is shown to share key properties with the fundamental symmetry groups of quantum theory.

Conclusion

The conservation group geometry provides a robust and technically consistent framework for galaxy modeling, naturally producing non-trivial dark matter halos, flat rotation curves, and scaling relations observed in spiral galaxies. Dark matter arises as a necessary geometric consequence rather than an extraneous import. The strong linkage to baryonic structure predicted by this theory aligns with empirical discoveries such as the RAR, bypassing the need for freely adjustable dark matter halo profiles. The theoretical underpinning may offer a pathway towards integrating galactic and quantum gravitational physics, and its future confrontation with observational and simulation data will clarify its role in the broader landscape of modified gravity and dark sector models.

Reference: "A framework for creating galaxy models in the geometry of the conservation group with dark matter halos and flat rotation curves" (2604.01270).