Structure Formation in Modular Entropic Gravity: The MEG-Native Picture

MEG's screening is environmental, the one-loop-derived screening law ℓ_env(ρ_b) = ℓ₀[1 + (ρ_b/ρ₀)^{1/3}]^{-1/4} activates only where the local baryon density exceeds the universal screening density ρ₀ ≈ 10⁻²¹ kg/m³. On linear cosmological scales, the mean baryon density is far below this threshold (ρ̄_b/ρ₀ ≲ 10⁻² throughout the structure formation epoch z ≲ 30), and the enhanced gravitational coupling μ_prim ≈ 5.4 persists completely unscreened. The screening threshold is crossed at z ≈ 133; above this redshift, the environmental response is partially screened, but this affects only the environmental component, not the primordial vacuum memory from inflation.

The persistent baryon-coupled Poisson enhancement (1+μ_prim)ρ_b δ_b is derived, not assumed. It follows from three established results: (i) the oscillating entropy modes have ρ̄_χ ∝ a⁻³ with ⟨w⟩ ≈ 0 (P45), (ii) the adiabatic condition δ_χ = δ_b (both components seeded by the same inflationary curvature perturbation, connected to P46's modular variance integral), and (iii) negligible environmental screening on linear scales. Under this adiabatic persistent-kernel condition, baryons grow with effective matter density Ω_m^eff = Ω_b(1+μ_prim) ≈ 0.32, matching the ΛCDM growth factor D(a) ∝ a during matter domination. The gravitational potential persists at ~80% of its recombination value, with the ~20% loss attributable to standard Λ-induced decay.

The Helmholtz Jeans scale identified in P45 at k_μ ≈ 0.08 Mpc⁻¹ at recombination evolves as k_μ(a) = a/ℓ₀, reaching k_μ ≳ 8 Mpc⁻¹ by z = 10. All structure-formation modes (k ≲ 1 Mpc⁻¹) are deeply sub-Helmholtz throughout the growth epoch. There is no Jeans suppression during structure formation.

The Lyman-α forest test is passed: in the forest window (k ≈ 0.1–10 h Mpc⁻¹ at z ~ 2–5), MEG predicts the same shape as ΛCDM for both the 3D matter spectrum and the 1D flux spectrum, with a nearly scale-independent enhancement (~10% in matter power, ~15–20% in flux power) degenerate with standard astrophysical nuisance parameters.

The physical picture: baryons fall into their own enhanced gravitational field. The vacuum entropy field modifies the gravitational coupling, not the source. Screening activates only where local density is high — inside virialised structures — which is precisely the regime where MEG's rotation curve predictions operate. The cosmological (unscreened, μ ≈ 5.4) and galactic (screened, μ ≈ 0.06) regimes are unified by a single density-dependent screening law. No dark matter particles. No temporal transitions. No two-fluid clustering problem.

Part of the MEG programme.