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

TL;DR

This work tackles false alarm level estimation for Lomb-Scargle periodograms in red-noise time series with uneven sampling by extending Whittle's likelihood to uneven cadences. It fits two physically motivated noise continua (AR(1) and power-law PSD) via Whittle NLL minimization, selects the best model through Monte Carlo evaluation, and then derives frequency-dependent FALs from time-domain realizations. The approach enables robust prewhitening of the power spectrum and reliable detection of true periodic signals, demonstrated across α Cen B, an RV Fitting Challenge dataset, GJ 581, KIC 6102338, and HD 192310, with software available for public use. This method improves over white-noise FALs by accounting for red-noise backgrounds and is applicable to any unevenly sampled time series in astronomy and beyond.

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.

Figures (16)

• Figure 1: GLS periodogram $\hat{S}^{LS}(f_k)$ (green) of activity indicator $\log R^{\prime}_{HK}$ of $\alpha$ Cen B dumusque with a red noise background. The vertical navy dotted line is the rotation signal, the horizontal black dash-dot line is a white noise fit to the spectrum, and the purple solid line and red dotted line are the best-fit versions of two red noise models, AR(1) and power law (see section § \ref{['sec:models']} for a detailed description of the red noise models used in this work).
• Figure 2: Top --- GLS periodogram $\hat{S}^{LS}(f_k)$ (blue) of a simulated AR(1) time series (Eq. \ref{['eq:ar1_ts']} with $\phi=0.75$ and $\sigma=1$). The dashed red line shows the analytic power spectrum $S(f)$ (Eq. \ref{['ar1ps']}) of the AR(1), $\phi = 0.75$, $\sigma = 1$ process. Bottom --- AR(1) power spectrum for different values of $\phi$.
• Figure 3: Top --- A 2D color plot of the Whittle NLL function $-\mathcal{L}(\theta)$ against AR(1) model parameters for the GLS periodogram of the dumusque$\alpha$ Cen B $\log R^{\prime}_{HK}$ time series. The x-axis and y-axis of the colored plot represent $\phi$ and $\sigma_w$ respectively. Bottom --- $-\mathcal{L}(\theta)$ for power law models of the same time series. Here the x-axis and y-axis represent normalization $a$ and exponent $p$ respectively. The color bars are normalized between 0 and 1 for visualization purposes. The blue curves show the marginalization of $-\mathcal{L}(\theta)$ with respect to a single parameter. The minimum values of $-\mathcal{L}(\hat{\theta})$ are marked with red stars.
• Figure 4: The observed H$\alpha$ time series of GJ 581 GJ581-Robertson and a new realization created by adding white noise to each observation (red).
• Figure 5: The Whittle NLL distributions created by fitting all three noise models to GLS periodograms of 10000 realizations of each time series studied in this paper. $-\mathcal{L}(\theta)$ distributions from the AR(1) model are colored orange, $-\mathcal{L}(\theta)$ distributions from the power-law model are in blue, and $-\mathcal{L}(\theta)$ distributions from the white-noise fits are in green. From top to bottom, the time series are $\alpha$ Cen B $\log R^{\prime}_{HK}$dumusque, simulated RVs from System 11 of the RV Fitting Challenge RVchallengedata, GJ 581 H$\alpha$GJ581-Robertson, Kepler photometry of KIC 6102338 borucki10 and HD 192310 RVs Laliotisetal2023AJ....165..176L.