FLRW Kinematic-Induced Measurement of the Hubble Constant from Cosmic Chronometer and Redshift Drift Observations

TL;DR

This work tackles the Hubble constant tension with a model-independent approach that leverages the FLRW kinematic relation $\dot{z} = H_0(1+z) - H(z)$ to geometrically embed Cosmic Chronometer and Sandage-Loeb data into a common observable plane. By treating $H_0$ as the plane's orientation in $(z, H(z), \dot{z})$ space and deriving $H_0$ algebraically without interpolation or dark-energy priors, the method yields precise, cosmology-independent estimates. Validation with current CC data and forecasted SL measurements from FAST, CHIME, SKA, and ELT demonstrates a 1.9–2.1% precision in $H_0$ (e.g., $H_0 = 66.26 \pm 1.26$ km s$^{-1}$ Mpc$^{-1}$), while showing superior resilience to sparse redshift coverage compared with Gaussian Process reconstructions. The approach remains FLRW-based and fully data-driven, highlighting a robust pathway for precision cosmology that can cross-check the standard model once real SL data are available from future facilities.

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.

Figures (6)

• Figure 1: Comparison of fiducial model fitting to simulated SL redshift drift data under different noise realizations across redshift range $z = 0 - 5$. The three columns represent noiseless theoretical data (left), random noise realization (middle), and optimized noise seed selection (right). Each panel displays the redshift drift signal $\dot{z}(z)$ with random MCMC samples (shaded regions), best-fit fiducial models (solid lines), true fiducial values (dashed lines), and simulated data points with error bars. This comparison demonstrates the impact of noise seed optimization on fiducial cosmological parameter recovery accuracy.
• Figure 2: Comparison of fiducial model fitting for CC and simulated SL Data. The top row shows the random MCMC samples (shaded regions), best-fit fiducial models (solid lines), true fiducial values (dashed lines), and simulated data points with error bars. The bottom three rows display residuals between observations and best-fit fiducial models for different parameter constraints: all parameters free (second row), $H_0$ marginalized over other fixed parameters (third row), and $\Omega_m$ marginalized over other fixed parameters (bottom row), with 1$\sigma$ parameter estimates displayed in the upper-left corners.
• Figure 3: These panels show the 2D joint distributions of the Hubble constant ($H_0$) and matter density parameter ($\Omega_m$), combining CC data with various simulated SL datasets under the assumption of a Flat $\Lambda$CDM model: Radio SL (left panel), ELT SL (middle panel), and Full SL (right panel). Blue, red, and green contours represent constraints from SL, CC, and their joint analysis, respectively.
• Figure 4: GP reconstructions of CC (left) and SL (right) data utilizing RBF (blue) and Matérn 5/2 (orange) kernels. The shaded regions represent the 68% (darker) and 95% (lighter) confidence intervals, illustrating the uncertainty in the GP fits.
• Figure 5: Visualization of the geometric embedding method in $(z, H(z), \dot{z})$ space with projections in the $H(z)$-$\dot{z}$ (top left), $z$-$\dot{z}$ (top right), $z$-$H(z)$ (bottom left) planes and a 3D view (bottom right). Black error bars represent original measurements. Green and blue error bars show geometric embedding results for $\dot{z}$ and $H(z)$ respectively. Orange error bars display GP reconstruction. The figure demonstrates how our method transforms redshift misalignment into a geometric constraint without requiring interpolation.