Avoiding the cascade of spurious frequencies in the analysis of stellar light curves

In the frequency analyses of stellar light curves the objective is to extract true pulsation frequencies down to the limit of the noise. It is well known, though, that this in practice is never reached. One of the most puzzling cases hampering the frequency analysis and preventing to reach this level of precision is the so-called cascade of spurious frequencies. This happens especially when the density of modes per frequency bin is high. Contrary to what is intuitive, a high signal-to-noise ratio can also be a factor influencing the appearance of spurious frequencies since sidelobes produced by the spectral window can be above the mean noise level. This has been shown by Balona 2014 [1] and other authors. Mitigation strategies have been suggested but no general solution has been found so far to this phenomenon. Here we show with simulations and data from observations of Kepler satellite, that a simple solution is found by improving the fitting algorithm. The clear benefit of investing in a better fitting algorithm is finding the true pulsation frequencies, which is the first and mandatory step to perform asteroseismology. Thus, these results can be of significant interest for any future studies and, especially, high amplitude and high quality data of multimode pulsators. Abstract Avoiding the cascade of spurious frequencies in the analysis of stellar light curves Javier Pascual-Granado1 ([email protected]) (1) Instituto de Astrofísica de Andalucía, Glorieta de la Astronomía s/n 18008 Granada (Spain) Issues The Solution Improvement References Acknowledgements Conclusions [1] Balona, L. A. 2014, MNRAS, 439, 4, 3453-3460 [2] Hocke, Annales Geophysicae 16, 356-358 (1998) [3] Barceló-Forteza, S., Pascual-Granado, J., et al. 2024, MNRAS, 535, 3, 2189-2209 [4] Pamos Ortega, D. et al. 2022, MNRAS, 513, 1, 374–388 Schematic periodogram of known frequency components (in red) and extracted components (in green). Adapted from [1] Zooming one peak in the power spectrum of a high-amplitude pulsator. Blue is the original periodogram and red is the residuals after extracting the main peak. The Problem This work has been developed during a research stay at ISTA as part of the preparatory activities for the development of a research line aimed to detect and characterize magnetic fields in δ Scuti stars. JPG wants to thank the kind support received from Lisa Bugnet, Lucas Barrault and the whole Asterics team during his stay. The authorsDacknowledgeDfinancialDsupportDfrom the Severo Ochoa grant CEX2021-001131-S funded by MICIU/AEI/ 10.13039/501100011033 and funding support from Spanish public funds for research under project PID2023-149439NB-C42 of MICIU/AEI. ●The median level in the whole spectrum is 1.27. Therefore, the red sidelobes are significant and will be detected in subsequent iterations as signal. ●The median level in the interval shown in the plot is 54.02 after extracting the main peak. ●Which noise level is better for SNR estimation? ●The 1st one might overestimate SNR, the 2nd one might underestimate the SNR. ●The origin of these fictitious components: They come about because the extracted frequency of the unresolved peak differs from the true frequency. Even though, we are using 5.4 years of data and we have very high frequency resolution, the pre-whitened data still contain significant signal because the amplitude of the unresolved peaks is high. ●Provided that the S/N level is sufficiently high, successive frequency extraction will lead to an ever increasing number of frequencies at low amplitudes which do not actually exist. Optimizing the fitting we can minimize the bias in SNR estimation introduced by correlated residuals. ●Since in the periodogram only amplitudes are represented, it is usually assumed that phase estimation do not affect the prewhitening procedure. ●Our result shows that phases are wrongly ignored when assessing the quality and robustness of the prewhitening process and our conclusion is that current methods of prewhitening could benefit a lot by improving the estimation of phases. ●In fact, phase information is already contained in the complex Fourier transform so, if we use this information as initial guess for phases the least squares fitting can improves considerably and reduce the variance of the residuals. ●The cascade of spurious frequencies can be avoided with optimal fitting including phase estimation. ●However, as shown in [3], this only cannot explain all the excess of frequencies found. ●We have used Multimodes2 for this research, which is an extension of a Python code to extract the most significant frequencies of a set of lightcurve of the original Multimodes code authored by D. Pamos Ortega [4] . See QR code: Example of the dense spectrum found in TIC230016910 by analysing ~5.4 years of TESS data. Red bars are the extracted frequencies. Even contraining the analysis to SNR>5.7 still more than 1400 frequencies are extracted before the prewhitening cascade stops. The situation is even worse using Period04 or SigSpec where 3960 frequencies have been found with a significance above 10. TIC230016910. Lomb-Scargle periodogram of the residuals after extraction of 1400 frequencies. A non-white spectral distribution of the residuals might be originated by the cascade bug during prewhitening process. Let’s analyze the numerical origin of the cascade bug… We will start assuming, for simplicity, a linear least-squares, that is, frequencies are fixed, and compare the influence of amplitude and phase errors. Let’s denote the two sinusoids as follows: y1(t) = A1 sin(ωt+ 1)ϕ y2(t) = (A1 + ∆A1) sin(ωt+ 1 + ∆ 1)ϕ ϕ It can be easily proved by trigonometric relations and combining like terms that the resulting wave of the interference is: y(t) = A1 sin(ωt+ 1)(1− cos(∆ 1))− (A1 + ∆A1) cos(ωt+ 1) sin(∆ 1)ϕ ϕ ϕ ϕ We can observe that the resulting waveform will still oscillate at the frequency ω, but its amplitude and phase will be influenced by the changes ∆A1 and ∆ 1. ϕ If ∆A1 and ∆ 1 are small, we can use approximations to simplifyϕ the analysis further. If we ignore quadratic terms we see the most significant term is: −A1 sin(∆ 1) cos(ωt+ 1)ϕ ϕ That is, we can conclude that phase is a relevant parameter for assessing the quality of a least squares fitting and, therefore, dealing with cascade bugs. Compare this example, with the upper plot. The median level in the interval shown in the plot is 2.87 after extracting the main peak, which is ~20 times lower. In 1998, Hocke [2] extended the LombScargle periodogram to perform phase estimation by using similar equations. Here we use eq.12 from [2]: and eq. 2: which is the original expression from Scargle 1982 seminal paper. High-quality data from space observations requires to adapt data analysis techniques in order to exploit the level of precision found. This will allow to detect and characterize new physics emerging as features and high-order perturbative terms both in individual and pattern of frequencies. Future work: The next steps should focus on validating, benchmarking, and applying the improved algorithm broadly, with an emphasis on demonstrating its impact on asteroseismology. If proven robust, the work could set a new standard in frequency analysis of stellar pulsations for the application to the next space mission PLATO2.0