A linearized approach to radial-velocity extraction – II. shot-noise-limited precision via spectral factorization

ABSTRACT We generalize the short-time Fourier transform (STFT) formalism for radial velocity extraction to cases where the underlying spectral components are unknown. The method factorizes a spectroscopic time series into principal spectra and time-dependent kernels, enabling simultaneous recovery of both. In Fourier space, each inverse-wavelength slice is decomposed by singular value decomposition, and radial velocity shifts are inferred from phase differences between epochs. In the high-SNR regime, this provides a linearized and statistically tractable estimate of differential velocities. The method is validated on synthetic and SOAP simulations and applied to EXPRES observations of HD 34411 and $\tau$ Ceti, recovering coherent signals and reaching the instrumental precision limit of ${\sim }30$ cm s-1. Apart from p-mode modulation, the residuals show no significant long-term correlations and allow the detection of signals with semi-amplitudes down to ${\sim }50$ cm s-1 with $\lesssim 10$ cm s-1 uncertainty. The framework thus enables extreme-precision radial velocity measurements in the presence of spectral variability, representing a step toward detecting and characterizing Earth-like planets around solar-type stars.