Gravitational-Wave Propagation Test of Scalar Temporal Flow

The Scalar Temporal Field Ontology (STFO) models time as a scalar field τ (xμ) whose spacetime gradients govern inertial behavior, curvature, and the emergence of geometric structure. In this framework, matter and radiation propagate on an effective disformal metric geffμν = gμν + B(τ )∇μτ ∇ν τ , while the scalar sector contributes a stress–energy tensor T (τ )μν to the Einstein equations. This paper develops a gravitational-wave (GW) propagation test of scalar temporal flow, using multi-messenger constraints on the GW speed and population-level tests of modified dispersion. Linear tensor perturbations on a cosmological background acquire a modified propagation equation with an effective GW speed cT , friction term, and possible dispersion induced by the background values of A(τ ), B(τ ), and ∇μτ . Using the joint observation of GW170817 and GRB 170817A, which constrains |cT /c − 1| ≲ O(10−15) at z ≃ 0.01, together with LIGO–Virgo–KAGRA (LVK) bounds on GW dispersion, we translate existing data into constraints on combinations of B(τ ) and the scalar-gradient amplitude X ≡ −gμν ∇μτ ∇ν τ . We then formulate a concrete analysis pipeline in which: (i) multi-messenger events constrain the local value of cT and hence B(τ )X; (ii) the compact-binary population is used to bound a scalar-induced dispersion parameter; and (iii) the GW luminosity distance dGWL is compared against electromagnetic distances dEM L to test for scalar-induced damping. The resulting constraints confine STFO to a regime in which cosmological GW propagation is extremely close to general relativity, while preserving the possibility of nonlinear, tau-saturated phenomena in compact and stellar environments. This work provides a falsifiable bridge between the STFO framework and current GW observations, and defines a clear roadmap for sharpening bounds with forthcoming LVK and third generation detectors.