📝 SummarXiv

Abstract

We present a geometric embedding method that exploits the exact kinematic relation $\dot{z} = H_0(1 + z) - H(z)$ to transform redshift misalignment between Cosmic Chronometer (CC) and Sandage-Loeb (SL) datasets into fundamental constraints in observable space. The approach recognizes that $H_0$ encodes the orientation of the FLRW observational plane defined by $(z, H(z), \dot{z})$ coordinates, enabling direct algebraic determination without parametric assumptions or interpolation schemes. Validation using available CC measurements and forecasted redshift drift data from FAST, CHIME, SKA, and ELT demonstrates 1.9\% precision for optimal data combinations, yielding $H_0 = 66.26 \pm 1.26$ km s$^{-1}$ Mpc$^{-1}$ while maintaining complete cosmological model independence. While no actual SL measurements currently exist, requiring us to rely on simulations for validation, our geometric constraints show superior resilience against sparse redshift coverage compared to Gaussian Process (GP) methods, which exhibit systematic biases and large uncertainties when datasets lack substantial overlap. This kinematic framework establishes geometric embedding as a robust tool for precision cosmological measurements, offering a fundamentally different approach to $H_0$ determination through pure observational analysis based on FLRW kinematic principles. The full potential of this method awaits implementation with real SL measurements from next-generation facilities.