A Strict Gravitational-Radius Cosmology with Linear Expansion

AbstractWe present a self-consistent alternative cosmological framework based on a strict gravitational-radius postulate ( R_g = GM/c^2 ) without the conventional factor of two. The theory is formulated covariantly through a modified quasi-local mass definition and a vacuum sector designed to prevent the hidden re-emergence of the Schwarzschild factor in geometric matching conditions. The resulting cosmology admits an exact linear expansion law ( a(t) \propto t ), corresponding to an effective equation of state ( w = -1/3 ), without invoking dark energy or a cosmological constant. We analyze the background dynamics, linear perturbations, and structure growth, demonstrating a natural freeze-out of growth at late times. Particular attention is given to the inevitability of the factor 2 in weak-field kinematics and its decoupling from the geometric gravitational radius via screening mechanisms. Distinct observational signatures are discussed, including luminosity distances, growth suppression, and the sign of the integrated Sachs–Wolfe effect. The model is presented as a falsifiable and phenomenologically competitive alternative to the standard (\Lambda)CDM cosmology.1. IntroductionThe standard cosmological model, (\Lambda)CDM, provides an excellent phenomenological description of a wide range of observations, yet it relies on the introduction of dark components whose physical nature remains unknown. This motivates the exploration of alternative frameworks in which cosmic acceleration and large-scale dynamics emerge from modified gravitational principles rather than new energy components.A particularly intriguing direction concerns the role of the gravitational radius. In general relativity, the Schwarzschild radius ( r_s = 2GM/c^2 ) arises from specific matching conditions between geometry and mass definitions. However, this factor of two is not imposed by dimensional analysis and may instead reflect a particular choice of quasi-local mass and vacuum structure.In this work, we investigate a cosmological model based on a strict gravitational-radius postulate,[R_g \equiv \frac{GM}{c^2},]taken as fundamental at the level of geometric matching. We show that, when implemented consistently, this postulate leads to a cosmology with linear expansion, well-defined perturbation dynamics, and testable observational consequences.2. Axiomatic Setup of Strict (R_g) GravityThe construction of the model rests on three axioms:Geometric gravitational radiusThe gravitational radius associated with a quasi-local mass (M) is defined as[R_g = \frac{GM}{c^2},]without an additional numerical factor.Modified quasi-local massThe physical mass entering the gravitational radius is identified with a Misner–Sharp–type quasi-local mass, modified to avoid implicit duplication of the Schwarzschild factor.Vacuum sector with screeningThe vacuum sector is allowed to differ from that of general relativity, such that weak-field kinematics remain standard while geometric matching conditions are modified.These axioms are sufficient to construct a covariant theory in which standard local tests of gravity are preserved, while global and cosmological properties differ from those of (\Lambda)CDM.3. Modified Field EquationsThe gravitational field equations are written in the form[G_{\mu\nu} = 4\pi G , T_{\mu\nu}^{\text{eff}},]where the effective energy–momentum tensor includes both matter and vacuum contributions. The reduced coupling constant reflects the strict gravitational-radius normalization.Energy–momentum conservation,[\nabla^\mu T_{\mu\nu}^{\text{eff}} = 0,]is maintained by construction, ensuring covariance and consistency.