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, reinterpreting the observational effects conventionally called "dark matter" and "dark energy" in the standard cosmological model as effective projections of gravitational degrees of freedom during coarse-graining.

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 , fusing gravitational relaxation with the holographic principle: gravitational relaxation provides the dynamical background for the formation of dissipative structures, while the holographic principle provides the effective energy-density formulas for the dissipative-structure gravitational gradient and for 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: In the GEE framework there are two temperatures---the thermodynamic temperature  and the gravitational entropy temperature . We define the temperature ratio

The core statement is . Temperature conversion drives the Hubble parameter to evolve monotonically from

to

with the instantaneous present value

The "Hubble tension" of the standard model does not exist in GEE, because  is not a single constant.  is the instantaneous value of  at the present epoch, not an artificial compromise prior.

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

In the GEE framework, the conversion between the gravitational entropy temperature  and the thermodynamic temperature  introduces a dissipative term , and the virial theorem is modified to

where

This yields a corrected rotation curve and a corrected dynamical mass. The dissipative virial theorem predicts that the standard virial theorem overestimates mass by about  times. Three independent observational datasets (Wu 1994; Wu et al. 1997; Zhang et al. 2026) support this prediction, with consistent sign and self-consistent magnitude. This is the first observational evidence for the testability of the GEE framework.

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 (Assumption 19); its first-principles derivation from the first law of thermodynamics on the horizon is left to Sec. 11.1. In a universe containing only baryons and radiation, we assume the effective energy density of the dissipative-structure gravitational gradient evolves as a power law

during the dissipative-structure-dominated era, where  is an independent phenomenological parameter. The GO cutoff naturally gives  in the matter-dominated era; the observational requirement

corresponds to

In the nine-parameter version, no independent  is introduced; instead, the difference between the GO cutoff and the observational requirement is absorbed into , and one derives

Through a coarse-graining map and the Poisson equation, an effective gravitational potential correction is derived from local Ricci curvature fluctuations:

where  is a dimensionless phenomenological coupling.

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-dominated era, taking , the Lagrangian is approximately

where  has dimensions of energy density. The exponent  is determined by observational constraints on the effective sound speed of the dissipative-structure gravitational gradient:

A typical value is , corresponding to an effective sound speed

which satisfies the observational constraint .  corresponds to extreme fine-tuning (); its naturalness is left to Sec. 11.1. The relation between  and  has been rigorously computed from the k-essence equation of motion given the form of :

During the dissipative-structure-dominated era, , , and , approximately mimicking matter behavior. The applicability conditions for this approximation are  and , which are satisfied during the dissipative-structure-dominated era.  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

compatible with the observed value  (the ST upper bound touches the  boundary and is labeled marginal). The early peak decays after the peak according to a parameterized form (Sec. 7.2), without violating recombination constraints. An interaction between the expansion energy density and the dissipative-structure gravitational gradient is introduced:

where  controls the delayed onset of the interaction. In the early universe, the gravitational temperature is suppressed by the thermodynamic temperature, , and the interaction is not yet effective; as the universe expands, , the interaction 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.

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 a difference between the GEE  and the standard cosmological model  of

This estimate does not include parameter-count penalties; GEE has 9 more phenomenological parameters than the standard cosmological model, and AIC/BIC are expected to penalize GEE. A precise Bayes factor requires a full MCMC analysis, with the calculation scheme given in Sec. 11.4.6. The expectation is

i.e., the standard cosmological model is strongly to 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.