The Harmonic Phase / Timing Field Framework

Ultralight scalar dark matter models have been proposed as a resolution to the small-scale discrepancies of the ΛCDM cosmological paradigm. In this work we develop the Harmonic Phase / Timing Field framework, identifying relativistic proper time (τ) with the phase accumulation (θ) of a universal scalar field. Starting from the Einstein–Hilbert action coupled to a Klein–Gordon scalar field, we perform a covariant Madelung decomposition separating density and phase degrees of freedom. The resulting relativistic Hamilton–Jacobi equation leads to a phase–time identity linking proper time to scalar phase evolution at the Compton frequency. In the weak-field limit the equations reduce to the Schrödinger–Poisson system governing ultralight scalar dark matter halos. This hydrodynamic description naturally produces soliton cores and interference granules in galactic halos. We derive scaling relations connecting scalar mass, oscillation frequency, and the quantum Jeans scale, and discuss observational constraints from Lyman-α forest measurements, dwarf galaxy cores, and pulsar timing arrays. Finally, we show that fluctuations of the scalar timing field produce fractional frequency shifts potentially detectable by next-generation optical lattice clocks.