The Cosmological Origin of the MOND Acceleration Scale An AQUAL-Consistent Identification

This paper presents a formal mathematical derivation addressing one of the most persistent numerical coincidences in astrophysics: the near equivalence between the MOND acceleration constant a0a_0a0 and the cosmological acceleration scale cH0cH_0cH0. By stopping at Chapter 6 and concentrating specifically on the AQUAL formalism, the work moves beyond a purely empirical observation and develops a coherent theoretical identification. What follows is an overview of the paper’s logical structure and key results. 1. The Core Identification: a0≈cH02πa_0 \approx \frac{cH_0}{2\pi}a0≈2πcH0 The paper begins by reinterpreting the MOND acceleration scale a0a_0a0, traditionally treated as a free parameter calibrated to fit galaxy rotation curves, as a derived quantity. By linking it directly to the Hubble constant H0H_0H0 and the speed of light ccc, the theory removes the need to regard a0a_0a0 as an independent constant of nature. The central identification is: a0≡cH02π.a_0 \equiv \frac{cH_0}{2\pi}.a0≡2πcH0. Using current measurements of the Hubble constant, this expression yields approximately 1.05×10−10 m/s21.05 \times 10^{-10}\,\text{m/s}^21.05×10−10m/s2, which lies squarely within the observational range required to reproduce galactic rotation curves without invoking dark matter. The coincidence thus becomes a prediction rather than a parameter choice. 2. Embedding Within the AQUAL Formalism To ensure theoretical consistency, the identification is embedded within the AQUAL (A Quadratic Lagrangian) framework introduced by Jacob Bekenstein and Mordehai Milgrom. AQUAL provides a nonlinear modification of the Poisson equation derived from a Lagrangian density, preserving conservation of energy, momentum, and angular momentum—properties that simpler ad hoc modifications of Newtonian gravity often violate. The scalar-field Lagrangian density is written as: L=−a028πGF(∣∇ϕ∣2a02)−ρϕ.\mathcal{L} = -\frac{a_{0}^{2}}{8\pi G}\mathcal{F}\left(\frac{|\nabla\phi|^{2}}{a_{0}^{2}}\right) - \rho\phi.L=−8πGa02F(a02∣∇ϕ∣2)−ρϕ. In the deep-MOND limit, selecting the interpolating function F(y)=23y3/2\mathcal{F}(y) = \frac{2}{3}y^{3/2}F(y)=32y3/2 recovers the exact acceleration scaling observed in the outer regions of galaxies. This demonstrates that the cosmological identification of a0a_0a0 remains dynamically consistent within a Lagrangian-based gravitational theory. 3. Recovery of the Baryonic Tully–Fisher Relation A major result of the paper is the derivation of the Baryonic Tully–Fisher Relation (BTFR) as a direct consequence of the cosmological identification. In standard MOND formulations, the normalization of the BTFR is determined empirically. Here, it is fixed by the global expansion rate of the universe. The resulting relation is: v4=GM(cH02π).v^4 = GM\left(\frac{cH_0}{2\pi}\right).v4=GM(2πcH0). Thus, the observed scaling between baryonic mass and asymptotic rotation velocity emerges naturally from cosmological boundary conditions rather than from parameter fitting. 4. Thermodynamic Interpretation: The Unruh Note The paper includes a brief but conceptually important thermodynamic observation connected to the Unruh effect. The acceleration scale a0a_0a0 corresponds to a specific vacuum temperature scale. This suggests that the transition from Newtonian to MOND-like dynamics may occur when local gravitational accelerations fall below the thermal background associated with the cosmological horizon. In this interpretation, the MOND threshold is linked to vacuum thermodynamics rather than arbitrary modification. 5. Conclusion: From Constant to Boundary Parameter The paper concludes that a0a_0a0 should no longer be regarded as a mysterious numerical coincidence or an independent fundamental constant. Instead, it functions as a cosmological boundary parameter set by the expansion of the universe itself. Under this reinterpretation, the apparent “missing mass” problem in galaxies may reflect a geometric consequence of embedding local gravitational systems within an expanding spacetime, rather than the presence of unseen matter.