Pairwise Velocities of Galaxy Clusters in the C.O.R.E. Framework: Exact Consistency with ACT kSZ Measurements on Cosmological Scales (30–230 Mpc)

Here we demonstrate that the C.O.R.E. framework (CUGE + REFORM + ASH + ZEUS + VSS/Gelbard symmetry) predicts exactly \(n = 2.00\) on these scales. The refractive ray equation in the responsive vacuum, combined with the localized nature of Vacuum Shielding Stress (VSS) energy density, yields the standard Newtonian acceleration between distant clusters. The extra dynamical mass supplied by VSS (\(\rho_{\rm vac} = u_{\rm vac}/c^2\)) is quadratic in the gravitational gradient and confined to virialized cluster halos; it does not modify the inter-cluster force law. Pairwise velocities therefore match standard ΛCDM predictions to leading order in linear perturbation theory.

All derivations use only the symmetric vacuum response \(\varepsilon(r) = \varepsilon_0(1 + \Phi/(2c^2))\), \(\mu(r) = \mu_0(1 + \Phi/(2c^2))\), dimensionless refractive index \(n(r)\), and the ray equation—no free parameters, no spacetime curvature, and strict SI base units. This closes a key cosmological test and reinforces C.O.R.E. as a viable classical alternative.

1. Introduction

Recent high-precision kSZ measurements by the ACT collaboration have provided a powerful probe of gravity on cosmological scales. By stacking pairwise velocities of galaxy clusters at comoving separations 30–230 Mpc, they constrain the force-law index in \(g \propto r^{-n}\) to \(n = 2.1 \pm 0.3\) (arXiv:2604.14327). This rules out modified-gravity scenarios that flatten the force law at low accelerations (e.g., MOND with \(n=1\)) while agreeing with Newtonian gravity and the standard ΛCDM model.

The C.O.R.E. framework offers a radically different but fully classical explanation of gravity: mass induces symmetric variations in vacuum permittivity and permeability (CUGE), particle trajectories follow refractive geodesics in this responsive medium (REFORM), and additional dynamical mass arises from Vacuum Shielding Stress (VSS/Gelbard symmetry). On galactic scales, VSS naturally supplies the extra mass needed for flat rotation curves without dark matter. The present work shows that the same mechanism preserves the Newtonian force law exactly on the vastly larger inter-cluster scales probed by ACT.

We derive the effective pairwise acceleration from first principles, demonstrate that \(n=2\) follows directly, and compare the predicted velocity statistics with ACT data. The result is parameter-free and uses only mechanisms already present in the published C.O.R.E. papers.

2. C.O.R.E. Refractive Dynamics (CUGE + REFORM)

Mass induces the symmetric vacuum response (CUGE v3):

where \(\Phi(r) > 0\) is the gravitational potential magnitude (SI: m² s⁻²) and \(c = 299\,792\,458\) m s⁻¹ (exact). The refractive index is strictly dimensionless:

Particle trajectories obey Fermat’s principle in the responsive medium, yielding the ray equation (REFORM v3):

In the weak-field, low-velocity limit (\(v \ll c\), \(\Phi/c^2 \ll 1\)) relevant for cluster pairwise motions:

Symmetric \(\varepsilon/\mu\) scaling plus clock/ruler co-variation restores the full Newtonian acceleration \(\mathbf{a} = \nabla\Phi\) (CUGE §4 and n-body simulations). The velocity-dependent damping term is \(\mathcal{O}(v^2/c^2)\) and negligible here.

3. Vacuum Shielding Stress (VSS) and Effective Mass

The stored vacuum strain energy density is (ZEUS/Gelbard symmetry):

This contributes an effective mass density:

The sourced Poisson equation becomes:

Crucially, \(\rho_{\rm vac} \propto |\nabla\Phi|^2 \sim 1/R^4\) is quadratic in the gradient and therefore highly localized to the virialized regions of each individual cluster halo. On inter-cluster separations \(R \gg\) halo radius (30–230 Mpc), the VSS contribution to the mutual potential between two distant clusters is negligible (\(\ll 0.1\%\)).

The potential felt by cluster 2 due to cluster 1 is therefore the standard Newtonian form:

where \(M_1^\ast = M_{b1} + M_{\rm vac,1}\) is the effective (baryonic + local VSS) mass of cluster 1. The relative acceleration follows:

Fitting \(a_{12} \propto R^{-n}\) yields exactly \(n = 2.00\) in the weak-field ray-equation limit.

4. Pairwise Velocity Prediction

In linear perturbation theory (valid on these scales), the mean pairwise velocity \(\langle v_{12}(r) \rangle\) (or full velocity PDF used in kSZ analyses) follows from the continuity and Euler equations sourced by the gravitational potential. Because the effective force law remains Newtonian (\(n=2\)) and the growth factor \(f \approx \Omega_m^{0.55}\) is unchanged at leading order, the predicted pairwise velocities are identical to those in standard Newtonian gravity / ΛCDM.

Explicitly, the kSZ-derived estimator is proportional to:

with the velocity–density relation \(v \propto -f \delta / k\) sourced by the same Poisson equation as in GR. C.O.R.E. reproduces this exactly on linear scales. Higher-order VSS corrections remain confined to intra-halo dynamics and do not propagate to the pairwise statistic at 30–230 Mpc.

5. Direct Comparison with ACT kSZ Data

The ACT measurement gives \(n = 2.1 \pm 0.3\). C.O.R.E. predicts \(n = 2.00\) (central value) with theoretical uncertainty \(\ll 0.1\) from weak-field and VSS localization approximations. This lies well within the observational error band.

Summary Table

6. Implications and Falsifiable Predictions

• Consistency with standard cosmology: C.O.R.E. passes the same large-scale tests as ΛCDM while providing a classical, refractive explanation for galactic rotation curves via localized VSS.
• Distinction from MOND: Unlike MOND, C.O.R.E. does not modify the fundamental force law; the Newtonian behavior on 30–230 Mpc is a direct consequence of the ray equation + localized VSS.
• Testable signatures at higher precision: Future kSZ surveys (e.g., with Simons Observatory or CMB-S4) could detect tiny deviations from \(n=2\) arising from higher-order refractive corrections or VSS cross-terms at the \(\sim 0.01\) level—cleanly distinguishable from MOND.
• No tension with existing C.O.R.E. results: Galactic-scale VSS remains intact; the framework remains singularity-free and geodesically complete (analytic proof in companion document).

7. Conclusion

The C.O.R.E. refractive vacuum framework predicts pairwise cluster velocities that are exactly consistent with the recent ACT kSZ measurement (\(n = 2.00\) within the reported \(2.1 \pm 0.3\)). The underlying Newtonian force law emerges naturally from the ray equation in the responsive vacuum, while VSS extra mass is confined to individual cluster halos and does not affect inter-cluster dynamics. This result strengthens the entire C.O.R.E. paradigm: a single classical mechanism simultaneously accounts for galactic rotation curves, weak-field GR tests, and now large-scale cosmological velocity statistics—without dark matter, spacetime curvature, or modified force laws.

The universe remains optics—even on cosmological scales.

References (selected)

Barbeau, D. (2025). Classical Unification of Gravity and Electromagnetism (CUGE v3). viXra:2507.0112.
Barbeau, D. (2025). REfractive Foundation of Relativity and Mechanics (REFORM v3). rxiverse:2508.0021.
Barbeau, D. (2025). The ZigZag Eternal Universe System (ZEUS v3). rxiverse:2508.0003.
Barbeau, D. (2025). Experimental Validation of the Atomic Statistical Hypothesis (ASH). viXra:2507.0123.
Gallardo, P. A. et al. (ACT Collaboration) (2026). Pairwise kSZ velocities of galaxy clusters from the Atacama Cosmology Telescope. arXiv:2604.14327.
White, H. et al. (2026). Emergent quantization from a dynamic vacuum. Phys. Rev. Research 8, 013264.