From Rotation Curves to Cosmic Time: Probing High-Redshift Expansion through Nested Spiral Dynamicss

We present a novel method for estimating the cosmic expansion rate $H_z$ at high redshift ($z\approx 2\text{？}\text{-}\text{？}10$ ) using the internal dynamics and morphology of nearby galaxies—without relying on direct redshift-distance measurements. The approach models rotation curves of systems showing a clear structural transition between a central bar (interpreted as a fossil spiral) and an outer spiral disk. Within a dual-Lagrangian framework, these nested structures are assigned separate dynamical regimes, enabling $H_z$ to be extracted from the mass $M$ and critical radius $r_c$ of the proto-bar region. The critical radius is defined locally by the condition $v_H\left(r_c\right)=v_{\text{esc}}\left(r_c\right)$ , which yields a local critical density ${\rho }_c\propto H_z^2$ . When this purely local relation is mapped onto cosmic time using the ΛCDM matter-era scaling $H\left(t\right)\propto t^{？1}$ , it reproduces the universal ${\rho }_c\propto t^{？2}$ law. Applying the method to seventeen