Quantum Fundamental Speed Theory: Mathematical Derivations of Dark Matter and Dark Energy from the Speed Field
We present a quantum formulation of the Fundamental Speed Theory (QFST) built from the classical FST Lagrangian, where the dimensionless speed field νμ plays a dual role: it modifies gravity classically in galaxies, while its quantum excitations behave as ultra-light dark matter on cosmological scales.On galactic scales, the classical FST equation fits rotation-curve data without dark matter (171 SPARC galaxies, χ²ν = 0.170). Linearization yields a screening scale μ = 1/λscreen, implying a particle mass m = ħ/(c λscreen) = 3.88 × 10−24 eV, with no free parameters. Quantum corrections to the galactic equation are suppressed by (ℓPl/λscreen)² ≈ 10−103, ensuring classical-quantum separation.Dark energy arises from the logarithmic potential V(ν̃) = −V0 ln(1 + ν̃²/ν₀²). We obtain a positive vacuum energy density ρΛ = [c⁴/(16πGL₀²)] · U(ν̃now) and determine ν̃now = 8.21 × 10−12 by matching the observed ρΛ. At the background level, QFST reproduces the standard expansion history as tested by Cosmic Chronometer H(z) data. Full discrimination from ΛCDM requires a Boltzmann-code implementation at the perturbation level.In a spatially flat FLRW spacetime, we derive the reduced kinetic scalar including Hν terms, construct the stress-energy tensor T(ν)μν, and obtain closed-form expressions for ρν and pν. The equation of state w(z) = pν/ρν is determined by the coupled background system (Friedmann equations plus the ν(t) equation of motion). Detailed derivations and dimensional checks are provided in the appendices.
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