Galaxy Rotation Curves and Dark Energy: A Gravitational Entropy Perspective ------The GEE Framework: Temperature Conversion Drives Hubble Parameter

Abstract

This paper proposes a phenomenological framework (the GEE framework) based on gravitational dissipative structures and the holographic principle, in which the observational effects conventionally attributed to "dark matter" and "dark energy" in the standard cosmological model are reinterpreted as effective projections of gravitational degrees of freedom during coarse-graining.

The central physical picture of this paper is the isomorphism of Prigogine dissipative structures at two scales: at the thermodynamic scale, the most ubiquitous Prigogine dissipative structure is the cyclone; at the gravitational scale, the most ubiquitous manifestation is the galactic rotation, structurally isomorphic to the cyclone. Terrestrial typhoons, Jupiter's Great Red Spot, Martian global dust storms, and Venusian polar vortices are all quasi-steady ordered structures formed by open systems maintained by a continuous energy flow, far from equilibrium, and self-organized through nonlinear positive feedback. Galaxies, galaxy clusters, and accretion disks are quasi-steady rotational structures formed by self-gravitating systems during virialization. In the Prigogine framework, both share the same abstract structure: open system, far from equilibrium, nonlinear positive feedback, continuous entropy production, and self-organized ordered structure. The physical identity of the GEE framework is precisely a Prigogine dissipative-structure theory in a long-range-force system.

Yet structural isomorphism is not physical identity. From cyclone to galactic rotation there are three hard fractures: first, opposite signs of heat capacity -- the cyclone is a positive-heat-capacity system, the galactic rotation is a negative-heat-capacity system; second, different energy currencies -- the cyclone corresponds to the thermal energy of the canonical ensemble, the galactic rotation to the gravitational potential energy of the microcanonical ensemble; third, a time-scale gap of about 20 orders of magnitude -- cyclone lifetimes are days to centuries, galactic-rotation lifetimes are billions of years. Of the three fractures, only the second can be softened by a temperature-equivalent reduction. This paper defines

Here  is the local gravitational temperature in the microcanonical-ensemble sense; when the self-gravitating system reaches local thermal equilibrium it is approximately equal to the microcanonical statistical temperature ; in virial approximation it equals the dynamical temperature

 is the ratio of two temperature scales, used to compare local gravitational potential depth with thermal kinetic energy density; it does not represent a phase transition between two thermodynamic temperatures.  has negative heat capacity and does not possess the stable thermodynamic limit of the canonical ensemble;  is the canonical-ensemble thermodynamic temperature.

The first and third remain hard fractures: negative heat capacity solves the mechanical stability problem of the galactic rotation, but not the energy-source problem; the galactic rotation still requires external energy flows ,  to maintain the far-from-equilibrium state, so that entropy production  persists. The cyclone and the galactic rotation are structurally isomorphic in mathematical form but opposite in the sign of heat capacity: after the cyclone's energy flow is cut,  drops,  rises, and the structure collapses; after the galactic rotation's energy flow is cut,  rises,  rises, but the structure is maintained by negative-heat-capacity self-regulation. They are two ends of the dissipative-structure spectrum, not the same mechanism; this paper calls this "structural isomorphism, opposite mechanisms."

As a Prigogine dissipative structure, the galactic rotation also requires a continuous external energy flow, and the input energy required to maintain the dissipative structure must be at least several times the dissipation rate; in the late epoch the main energy source is the cosmic expansion energy . Negative heat capacity solves only the mechanical stability problem, not the energy-source problem.

Convention: Throughout this paper we adopt natural units

unless otherwise stated. GEE assumes zero particle mass for the dissipative-structure gravitational gradient:

The framework introduces a dimensionless effective gravitational entropy order parameter  and a dissipative-structure order parameter , and fuses gravitational relaxation with the holographic principle: gravitational relaxation provides the dynamical background for the formation of dissipative structures, while the holographic principle provides the energy-density formulas for the dissipative-structure gravitational gradient and for the effective energy density driving cosmic accelerated expansion. The two meet through the gravitational entropy .  serves as a specified proportionality relation, stipulating that the energy released by gravitational relaxation is proportional to the system mass, but it does not directly enter the fusion equation.

Hubble parameter evolution: Planck's 67.4 is a model-dependent inference of today's , not the actual expansion rate at recombination. Therefore this paper does not claim that GEE has solved the Hubble tension. In GEE,  is a shared fit parameter;  is obtained by solving the full background equations together with the GEE components; the temperature-conversion order parameter  is only a phenomenological function controlling delayed interaction; it is not directly identified with .

Core statement: GEE is currently a phenomenological dissolution framework. It provides a parameter table, a fitting pipeline, and schemes for MCMC and Bayes-factor calculations. Before the full background solution and MCMC are completed, it cannot claim to have explained flat rotation curves, to have solved the Hubble tension, or to have obtained support from three independent observational datasets.

Prigogine virial equation: The standard virial theorem assumes a closed, dissipationless system, giving

In the GEE framework, the comparison between the local gravitational temperature  and the local thermodynamic temperature  introduces a Prigogine dissipative term , and the virial equation is generalized to

where

The self-consistent equation gives , . This is an algebraic domain lower bound, not an independent phase-transition criterion. Formation of a dissipative structure requires separately  and .

 is the ratio of two temperature scales, used to judge the relative size of local gravitational potential depth and thermal kinetic energy density.  is the local gravitational temperature in the microcanonical sense and does not have canonical-ensemble heat capacity;  is the canonical-ensemble thermodynamic temperature.

When , ; when , , and the dissipative term provides additional effective gravity. The Prigogine virial equation predicts that the standard virial theorem may overestimate mass; whether three observational datasets support it must be rechecked.

Holographic dissolution of the dissipative-structure gravitational gradient: GEE does not use particle components. The observational effects conventionally attributed to "dark matter" are reinterpreted in GEE as the effective projection of the galactic dissipative-structure gravitational gradient under a holographic cutoff. We adopt a holographic model with the Granda-Oliveros (Ricci) cutoff. This cutoff is introduced as a phenomenological assumption; its first-principles derivation from the first law of thermodynamics on the horizon is left to Sec. 15.1. In a universe containing only baryons and radiation, the background interaction is

In the background continuity equations,  flows from the expansion energy density to the dissipative-structure gravitational gradient:

The effective power-law index is defined as a derived quantity:

During the dissipative-structure-dominated era, if  is approximately constant, its average is denoted . From the continuity equation one obtains

hence

Since , , algebraically

If early  is growing, i.e. , then

Thus there is no epoch with .  is a derived quantity, and

The present density ratio

only constrains normalization and cannot by itself determine . The dimension of the interaction term is

Phenomenological construction of the k-essence Lagrangian: The k-essence Lagrangian is constructed phenomenologically from the holographic energy density and the k-essence definition ( is a phenomenological assumption):

During the dissipative-structure gravitational-gradient-dominated era, taking , the Lagrangian is approximately

where  has dimensions of energy density. The exponent  is determined by the effective sound speed:

A typical value is , corresponding to an effective sound speed

 corresponds to extreme fine-tuning; its naturalness is left to Sec. 15.1. The relation between  and  has been computed from the k-essence equation of motion given the form of :

During the dissipative-structure gravitational-gradient-dominated era, , . Without interaction

For large , this exponent is approximately , i.e. the non-interacting case gives , which does not match the required slope ; if  is introduced, the background evolution must be re-solved.  is a consequence of the phenomenological assumption; its applicability requires numerical verification.

Dissolution of the effective energy density driving cosmic accelerated expansion: GEE does not use material components. The observational effects conventionally attributed to "dark energy" are reinterpreted in GEE as the effective energy density  driving cosmic accelerated expansion. It is generalized to a two-layer structure consisting of a holographic component and an early peak (EDE) component, coupled to the dissipative-structure gravitational gradient through an interaction. The early peak fraction is dominated by Jeans-mass halos at . Cross-validation using both the Press-Schechter and Sheth-Tormen mass functions yields

The observed value is . The ST upper bound touches the  boundary and is labeled marginal. The integral result depends on the free amplitude , the power spectrum, the collapse threshold, and numerical details; when  is not independently fixed, the result cannot be regarded as an independent prediction.

An interaction between the expansion energy density and the dissipative-structure gravitational gradient is introduced:

Here  controls the delayed onset of the interaction. Early suppression comes from the smallness of , not from . As the universe expands, the density combination grows,  becomes effective at low redshift, flips the sign of , and brings it into agreement with the DESI DR2 (DESI+CMB+SNe) preference.  gives , compatible with DESI DR2's  at the  boundary;  gives , closer to the DESI DR2 central value. The model is compatible with current data in this parameter range and requires further testing with future DESI DR3/Euclid data.

Equation-of-state sign: From

one obtains

If one defines an effective equation of state after absorbing the interaction, then

Unified dissolution equation

where

 is the energy-momentum tensor of the holographic projection of the dissipative-structure gravitational gradient,  is the effective energy-momentum tensor of the expansion energy density, and  is the interaction energy-momentum tensor between the expansion energy density and the dissipative-structure gravitational gradient.

Key numerical values (nine-parameter version)

Derived quantities:

Current statistical status: In the Planck+SH0ES combined data, an approximate estimate gives GEE's  difference from the standard cosmological model as

This estimate does not include parameter-count penalties. GEE introduces 9 new phenomenological parameters; together with the 5 shared standard parameters, the fit vector has 14 dimensions, i.e. a net increase of 8 relative to the 6-parameter standard model. For , the correct substitutions are

Both still prefer the standard cosmological model. A precise Bayes factor requires a full MCMC analysis, with the calculation scheme given in Sec. 15.4.6. The expectation is

i.e., the standard cosmological model is decisively preferred, unless the likelihood improvement of GEE far exceeds the current approximate estimate. The currently reportable conclusion is: in the combined data, GEE and the standard cosmological model are statistically indistinguishable; with Planck data alone, the standard cosmological model is preferred; but this conclusion requires full MCMC confirmation.

Statement layering: In this paper, all instances of "derivation" are divided into two categories: mathematical derivation and phenomenological construction. Any formula for which a first-principles derivation is not provided shall, in accordance with the convention in Sec. 3.0.0, be regarded as a phenomenological assumption or phenomenological construction, and its rigorous derivation is deferred to Sec. 15.1.