k_a and the a² Invariant: A Unified Acceleration Scale from Galaxies to Atomic Clocks

Modern gravity phenomenology exhibits at least three apparently unrelated small acceleration scales: the Milgrom scale a₀ organizing galaxy rotation curves, the cosmic acceleration scale a_Λ ~ cH₀, and the sensitivity of precision clock tests to tiny violations of local position invariance. Conventional frameworks—ΛCDM with cold dark matter on the one hand, and modified-gravity models on the other—typically treat these scales as independent or emergent features of very different sectors: dark halos, dark energy, and laboratory metrology.

Here I show that a broad class of scalar refractive-index theories of gravity admits a single, universal "acceleration-squared" invariant a² ≡ a·a, linked to the gradient energy of a scalar refractive field ψ via a dimensionless self-coupling k_a. In the weak-field, quasi-static limit the field equation can be written as ∇·a + (k_a/c²)a² = −4πGρ, with a = −c²∇ψ the physical free-fall acceleration and ρ the mass-energy density.

I then show how this structure naturally generates a single preferred acceleration-squared scale a_★² ∝ (c²/k_a)Gρ that simultaneously: (i) reproduces MOND-like scaling g ≃ √(a_★ g_N) in galaxies when the k_a a² term dominates the bare Poisson term; (ii) yields a cosmic background value a_★² ~ c²H₀² in an FRW universe with density of order the critical density; and (iii) enters directly into species-dependent gravitational redshift anomalies for atomic clocks, via scalar couplings K_A encoding the internal structure of each atomic transition.

Remarkably, the phenomenological parameters (k_a, d_e) governing this structure appear to satisfy simple numerical relations involving the fine-structure constant α ≈ 1/137. Specifically: a₀ = 2√α cH₀ (within current H₀ uncertainties), k_α = α²/(2π) (a characteristic scale compatible with cross-species clock data; the PTB Yb⁺ E3/E2 comparison constrains any universal pure-α coupling to |k_α| < 10⁻⁸, indicating composition-dependent structure), and k_a = 3/(8α). These relations contain no free parameters beyond α and H₀, and suggest a vertex-counting structure familiar from quantum electrodynamics.

This paper develops the a² formalism, derives the α-relations, and identifies falsifiable predictions for near-term experiments. The construction connects to the broader Density Field Dynamics (DFD) framework developed in companion work.