Red noise-based false alarm thresholds for astrophysical periodograms via Whittle's approximation to the likelihood

Abstract

Astronomers who search for periodic signals using Lomb-Scargle periodograms rely on false alarm level (FAL) estimates to identify statistically significant peaks. Although FALs are often calculated from white noise models, many astronomical time series suffer from red noise. Prewhitening is a statistical technique in which a continuum model is subtracted from log power spectrum estimate, after which the observer can proceed with a white-noise treatment. Here we present a prewhitening-based method of calculating frequency-dependent FALs. We fit power laws and autoregressive models of order 1 to each Lomb-Scargle periodogram by minimizing the Whittle approximation to the negative log-likelihood (NLL), then calculate FALs based on the best-fit model power spectrum. Our technique is a novel extension of the Whittle NLL to datasets with uneven time sampling. We demonstrate FAL calculations using observations of $α$~Cen~B, GJ~581, HD 192310, synthetic data from the radial velocity (RV) Fitting Challenge, and {\it Kepler} observations of a differential rotator. The {\it Kepler} data analysis shows that only true rotation signals are detected by red-noise FALs, while white-noise FALs suggest all spurious peaks in the low-frequency range are significant. A high-frequency sinusoid injected into $α$~Cen~B $\log R^{\prime}_{HK}$ observations exceeds the 1\% red-noise FAL despite having only 8.9\% of the power of the dominant rotation signal. In a periodogram of HD 192310 RVs, peaks associated with differential rotation and planets are detected against the 5\% red-noise FAL without iterative model fitting or subtraction. Software for calculating red noise-based FALs is available on GitHub.