A Linearized Approach to Radial-Velocity Extraction. II: Shot-Noise-Limited Precision via Spectral Factorization

TL;DR

This paper introduces a spectral factorization method based on a generalized short-time Fourier transform to achieve shot-noise-limited precision in radial velocity measurements, crucial for detecting Earth-like exoplanets.

Contribution

It extends the STFT formalism for unknown spectral components, enabling simultaneous recovery of spectra and velocity shifts with high precision in spectroscopic data.

Findings

Validated on synthetic and real data, reaching ~30 cm/s precision.

Detected signals with semi-amplitudes down to ~50 cm/s.

Residuals show no significant long-term correlations.

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 ~30 cm/s. Apart from p-mode modulation, the residuals show no significant long-term correlations and allow the detection of signals with semi-amplitudes down to ~50 cm/s with $\lesssim10$ cm/s 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.