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

The search for Earth-like planets orbiting Sun-like stars through radial velocities (RV) relies on algorithms that can filter out the Doppler planet signal from stellar signals and noise. The goal of this project is to improve the method of calculating the detection significance of the planet signals, quantified by False Alarm Levels (FALs), for the Lomb-Scargle (LS) periodogram. These FALs are typically calculated based on the assumption that the data contains uncorrelated white noise, and thus are independent of frequencies. However, the periodograms of RV and other astrophysical datasets in many cases of interest show a negative slope in log(P(f)), where P is the estimated power and f is the frequency, which indicates the presence of red noise. Therefore, the FALs should be calculated based on a red noise model. We propose two red noise models that can be used to calculate frequency-dependent FALs: i) AR(1), an autoregressive model of order 1, a persistence model, which represents a correlation between an observation and its immediate past, ii) power law, which is motivated by stellar processes like convective turbulence that can be the physical sources of red noise. We fitted both red noise models and a white-noise model to the LS periodograms of multiple RV and activity-indicator time series and found that both red-noise models are significantly better fits than the white noise model for datasets whose power spectrum had a red-noise background. We demonstrate the FAL calculations using archival observations of α Cen B and Barnard's star. A high-frequency oscillation injected into α Cen B log R′HK observations correctly exceeds the 1% red-noise FAL despite having only 8.9% of the power of the dominant rotation signal. In a periodogram of Barnard's star RVs, peaks associated with planets b, c, d, and e are detected against the 5% red-noise FAL without iterative model fitting and subtraction. Software for calculating red noise-based FALs for Lomb-Scargle periodograms is available on GitHub.