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REVIEW 3 major objections 5 minor 34 references

Gaussian Process modelling of granulation and oscillations in red-giant stars

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Gaussian processes can pull red-giant oscillation parameters straight out of light curves.

desk verdict Solid, honest methods paper: the celerite-to-asteroseismic parameter mapping is genuinely new and nu_max recovery is robust, but the reported uncertainties need a calibration check before the precision claim is trusted. read the letter →

arxiv 1908.10662 v1 pith:JXSU7MLH submitted 2019-08-28 astro-ph.SR astro-ph.EP

Filipe Pereira , Tiago L. Campante , Margarida S. Cunha , João P. Faria , Nuno C. Santos , Susana C. C. Barros , Olivier Demangeon , James S. Kuszlewicz
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Enrico Corsaro
This is my paper · ORCID
classification astro-ph.SRastro-ph.EP
keywords asteroseismologyredgiantsGaussianprocessesgranulationnu_maxceleritekernelsstellarnoisetransitmodelling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that a Gaussian-process regression with a physically motivated kernel can model the granulation and solar-like oscillation signals of red-giant stars directly in the time domain, and can recover the frequency of maximum oscillation amplitude, $\nu_\mathrm{max}$, accurately. This matters because TESS will deliver on the order of $10^5$ red-giant light curves in which these stellar signals have amplitudes and timescales comparable to planetary transits, so a fast time-domain model with physical meaning can serve both asteroseismology and exoplanet transit fitting. The paper validates the claim on TESS-like artificial light curves and on 19 Kepler low-luminosity red giants, comparing recovered parameters with power-spectrum fits. It also reports a preliminary result that fitting the GP stellar model simultaneously with a transit model improves the precision and accuracy of the planet-to-star radius ratio, $R_p/R_\star$.

What carries the argument

The central object is the celerite SHOTerm kernel, a covariance function whose power spectral density is that of a stochastically driven, damped harmonic oscillator. At quality factor $Q=1/\sqrt{2}$ its PSD reduces to the Harvey-like form with an exponent of 4, the function usually used to model red-giant granulation in the frequency domain; for $Q>1$ the same kernel approximates the Gaussian-like shape of the oscillation bump. The model sums a granulation kernel (or two, in Model 2), an oscillation-bump kernel with $1.2<Q<18$, and a white-noise term. The kernel's hyperparameters $S_0$, $Q$, and $\omega_0$ are then mapped one-to-one onto the asteroseismic parameters $a_\mathrm{gran}$, $b_\mathrm{gran}$, $P_g$, and $\nu_\mathrm{max}$ via the normalization relations derived in Appendix A, so the GP fit returns physical stellar parameters rather than merely a flexible noise model.

What would settle it

Fit the same two-component kernel to high-cadence, high-signal-to-noise Kepler light curves of red giants whose granulation background has been independently measured, and compare the recovered $a_\mathrm{gran}$, $b_\mathrm{gran}$, and $\nu_\mathrm{max}$ against flexible power-spectrum fits; systematic discrepancies beyond the few-percent biases reported here, or failure to recover an injected Harvey component with a different slope (for example exponent 2 instead of 4), would invalidate the claimed parameter mapping.

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Extended reading notes

Core claim

The paper's central claim is that a Gaussian-process model built from damped-harmonic-oscillator kernels can represent the photometric signal of a red-giant star, granulation plus the solar-like oscillation bump, directly in the time domain, and that the fitted kernel parameters can be translated into the same physical quantities normally extracted by power-spectrum fitting. The translation is carried out through the mapping $a_\mathrm{gran}=\sqrt{\sqrt{2}\, S_{0,\mathrm{gran}}\,\omega_{0,\mathrm{gran}}}$, $b_\mathrm{gran}=\omega_{0,\mathrm{gran}}/(2\pi)$, $P_g=4S_{0,\mathrm{bump}}Q^2$, and $\nu_\mathrm{max}=\omega_{0,\mathrm{bump}}/(2\pi)$. On TESS-like simulated light curves the model recovers $\nu_\mathrm{max}$ with about 1% bias and 3-4% scatter, and on 19 Kepler low-luminosity red giants the time-domain and power-spectrum determinations of $\nu_\mathrm{max}$ agree to within about 1-2%. The paper also reports a preliminary combined fit of the GP stellar model with a transit model that reduces the bias in $R_p/R_\star$ from 1.71% to -0.06% and the scatter from 6.49% to 5.80%. The authors note that a second granulation component's characteristic frequency is not constrained in the simulated data and that white noise is underestimated, which motivates their recommendation of the simpler one-granulation model for typical TESS time series.

Load-bearing premise

The argument depends on the assumption that the chosen kernel's power spectral density, a damped harmonic oscillator with quality factor fixed at $1/\sqrt{2}$ for granulation, matches the real granulation signal of red giants closely enough that the fitted kernel hyperparameters correspond one-to-one with the asteroseismic amplitudes, frequencies, and $\nu_\mathrm{max}$.

Editorial extensions

If this is right

  • For 27.4-day TESS-like light curves of red giants, the simpler one-granulation model recovers $\nu_\mathrm{max}$ with roughly 1% bias and 3-4% scatter on simulated data, meaning asteroseismic characterization can be done in the time domain without building a full frequency-domain model.
  • On real Kepler low-luminosity red giants, the time-domain GP and power-spectrum fits agree in $\nu_\mathrm{max}$ to within about 1-2%, indicating that the kernel choice does not introduce large systematic offsets in the main quantity of interest.
  • Modelling the stellar GP simultaneously with a transit improves the estimated $R_p/R_\star$ from a 1.71% offset with 6.49% scatter to a -0.06% offset with 5.80% scatter in simulated injections, so evolved-host transit parameters benefit directly from treating stellar signals as correlated noise.
  • Because the celerite kernels allow linear-scaling likelihood evaluation, the method can be applied to long TESS light curves, and the implementation is publicly available.
  • The second granulation component's frequency is left unconstrained by TESS-like data, and white-noise levels are systematically underestimated in the two-component model, so the paper recommends Model 1 for typical TESS red-giant light curves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if the same kernel mapping holds at longer baselines and higher signal-to-noise, GP-based time-domain analysis could take over the role of periodogram-based asteroseismic fitting for large red-giant samples, avoiding binning choices and providing a single model that also handles transits; this is a plausible extension, not a paper claim.
  • Editorial inference: the cleanest test of the physical mapping is to let the granulation quality factor float in the fit; if the posterior converges near $Q=1/\sqrt{2}$ across many Kepler giants, the one-to-one calibration is confirmed, whereas systematic deviation would mean the reported $a_\mathrm{gran}$ and $b_\mathrm{gran}$ carry model-dependent bias. Such a test is not reported in the paper.
  • Editorial inference: because the oscillation bump is treated as a single broad component, the recovered $\nu_\mathrm{max}$ describes the envelope rather than individual mode frequencies; combining this time-domain model with a mode-by-mode search could produce a pipeline that first extracts stellar parameters and then measures frequency spacings from the same framework.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a Gaussian-process (GP) framework, built on the celerite package, to model granulation and oscillation signals in red-giant light curves directly in the time domain. Two models are considered: Model 1 uses one granulation kernel, an oscillation-bump kernel, and white noise; Model 2 adds a second granulation component. The method is applied to 20 TESS-like artificial low-luminosity red-giant stars and to 19 Kepler LLRGB stars, and the recovered parameters are compared with the injected values and with DIAMONDS power-spectrum fits. The paper reports a 1.14% bias and 3.34% scatter in nu_max on the simulated data, relative biases of 1.99% and 0.92% against DIAMONDS for Models 1 and 2, and a preliminary improvement in Rp/Rstar when the GP stellar model is fit jointly with a transit model. The authors are transparent about limitations: the second granulation frequency is unconstrained for every simulated star, white noise is underestimated in several configurations, and the GP and DIAMONDS uncertainties for nu_max sit below the paper's own stated lower bound from Eq. (13).

Significance. If the main result survives revision, the paper makes a useful contribution: it provides a publicly available and physically motivated celerite-kernel model for red-giant stellar signals, and the nu_max accuracy is supported by simulation and by agreement with an independent power-spectrum method. I give credit to the public implementation, the reproducible workflow, and the unusually clear reporting of unconstrained parameters. The method is of direct practical relevance for TESS light curves and for joint transit-and-stellar-noise modelling. However, the stated precision of the nu_max measurement is not yet established, because the GP credible intervals are smaller than the stated lower bound from Eq. (13) and no coverage test is presented. The transit result is explicitly preliminary but is advertised in the abstract, which raises the bar for documenting that experiment. With these caveats, the paper's significance is moderate rather than high: the core ideas are sound, but the central precision claim needs either empirical calibration or reformulation.

major comments (3)
  1. [Sec. 4.3, Eq. (13)] The paper's central claim that nu_max can be measured both accurately and precisely is not fully supported. The authors call Eq. (13) a lower limit for the uncertainty in nu_max, yet report <sigma_GP> = 3.43 microHz and <sigma_DIAMONDS> = 1.91 microHz against <sigma_nu_max> = 7.15 microHz. The subsequent appeal to DIAMONDS' successful literature applications cannot validate the GP HPD intervals, since it provides no direct test of whether the 68.3% GP intervals contain the true nu_max in the simulations or in injected signals. I request either a coverage test on the TESS-like simulations with the empirical coverage probability reported, or a reformulation of the abstract and conclusions that claims accuracy only and presents the GP uncertainties as model-based estimates that may be underestimated. This matters for downstream uses of the quoted uncertainties, such as scaling relations or ensemble asteroseismology.
  2. [Sec. 3.2, Figs. 2-3] The simulation validation is partially internal, because the artificial light curves are generated using the same Kallinger et al. (2014) Harvey-like granulation functional form that defines the GP kernel through Eqs. (9)-(12). The paper's own results show the limits of this mapping: the second granulation frequency b_gran,2 is unconstrained for every simulated star, b_gran,1 is recovered with a 8.05% bias in Model 1, and white noise is systematically underestimated in Model 2. Consequently, the claim of recovering physically meaningful parameters is convincingly established only for nu_max; it is not established for the individual granulation parameters. I request either a simulation test using a different background shape (e.g., a different exponent or an additional background component) or a revised conclusion that limits the physical-parameter claim to nu_max and the oscillation envelope.
  3. [Sec. 5, Fig. 10] The transit experiment is presented as a partial validation of the abstract's claim about improving Rp/Rstar, but the manuscript does not state the number of injected transits, the orbital periods or depths, the limb-darkening model, or the SDE threshold used for detection. Without these details the comparison in Fig. 10 cannot be reproduced or evaluated. If the transit claim is retained in the abstract, the experimental setup should be documented; alternatively, the sentence should be removed or moved to the context of future work.
minor comments (5)
  1. [Sec. 4.2, Fig. 7] The text reports a relative white-noise bias of -48.71% with 16.89% scatter for Model 2, while the inset in Fig. 7 shows 1 sigma = 19.69%; this discrepancy should be reconciled.
  2. [Eq. (13)] The printed form of Eq. (13) is typographically ambiguous: the expression '(1 + 4 (H_BR/sigma_g)^{2/3})' could be read as a sum rather than as the expected ratio formula from Kallinger et al. (2010). Please verify the formula and clarify whether it is a strict lower bound or an empirical expectation, since the text relies on that distinction.
  3. [Fig. 5 and Fig. 9] These figures show the PSD of the GP output and its components, but they do not overlay the input or DIAMONDS-comparable PSD model components; adding the true or reference background and oscillation envelope would help the reader judge how well the time-domain fit reproduces the frequency-domain structure.
  4. [Sec. 2.4] The authors correctly note that the Gelman-Rubin diagnostics are not strictly valid for correlated MCMC chains; however, the threshold of 1.1 and the practice of discarding up to 50% of each chain should be described with a reference or a brief justification, because these choices can affect the reported HPD intervals.
  5. [Fig. 10] The x-axis label 'Star' is uninformative; the figure would be clearer if the axis were labelled 'Star index (ordered by SDE)' and the SDE values were tabulated.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: granulation recovery on synthetic data is self-consistent by construction, but the core nu_max claim rests on external Kepler/DIAMONDS comparison.

  1. other [Section 2.3 (Eqs. 8-9) and Section 3.1]
    "To model the granulation power spectral density, a scaled version (to predict TESS granulation amplitudes) of model F of Kallinger et al. (2014) was adopted, which contains two granulation (or Harvey-like) components. ... Equation (9) has the same functional form as the equation commonly used to model the granulation in a power spectrum analysis (e.g., Kallinger et al. 2014). Since Eq. (9) corresponds to the PSD of the kernel in Eq. (8), this kernel can be used to capture the same granulation signal in the time domain."

    The synthetic granulation time series used for validation are generated from the same Harvey-like functional form that the celerite SHOTerm kernel at Q = 1/sqrt(2) reproduces exactly. Therefore, the reported recovery of the granulation parameters agran and bgran from the simulated light curves is partly a self-consistency check of the kernel's assumed PSD shape rather than an independent test of the physical model. This does not compromise the paper's central claim about nu_max, because the oscillation signal in the simulations was injected via individual modes rather than with the GP bump kernel, and because the Kepler results are benchmarked against DIAMONDS, an independent power-spectrum fitting method.

full rationale

The paper's main derivation is a standard Bayesian GP parameter estimation: the kernel hyperparameters are mapped to asteroseismic parameters through the explicit, derived correspondences in Eqs. (9)-(12) and Appendix A, and no target quantity is fitted and then renamed as a prediction. The Kepler comparison against DIAMONDS is an external benchmark that does not depend on the GP assumptions, and the reported nu_max agreement (0.92-1.99% relative bias) supports the central claim independently of the synthetic simulations. The only partial circularity is that the TESS-like artificial granulation signal is generated with the same Harvey-like model family that the kernel assumes, so the granulation recovery on synthetic data is partly by construction; the paper itself acknowledges the resulting difficulties by finding that the second granulation frequency is unconstrained. This is a minor validation-design issue rather than a load-bearing circular derivation, and the self-citations in the paper (e.g., Campante et al. 2016, 2018) are used for noise models and scaling relations, not as an unverified uniqueness argument.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The fitted quantities are the GP kernel hyperparameters, whose physical interpretation rests on the assumed PSD shapes of celerite SHOTerm kernels and on standard red-giant scaling relations. No new physical entities are introduced.

free parameters (8)
  • a_gran,1 (mesogranulation amplitude) = per star, not tabulated; median over 5-10 subsets
    Fitted GP hyperparameter for the first Harvey-like granulation kernel; recovered values are compared to injected and PSD-fit values.
  • b_gran,1 (mesogranulation characteristic frequency) = per star, not tabulated
    Fitted GP hyperparameter; recovered with moderate bias in simulations and close agreement on Kepler data.
  • P_g (oscillation bump height) = per star, not tabulated
    Mapped from the fitted S0,bump and Q_bump via Eq. (12); there is no bona fide input in simulations because the data are generated from individual modes.
  • Q_bump = per star, within [1.2, 18]
    Quality factor of the oscillation-bump SHOTerm; the prior bounds are defined empirically for this work in Sec. 2.3.
  • nu_max = per star; central result
    Frequency of maximum oscillation amplitude, mapped as omega_0/(2 pi); recovered with roughly 1% bias and a few percent scatter.
  • white noise = per star, not tabulated
    Constant added in quadrature to the diagonal of the covariance matrix; biased low in both models in the TESS-like tests.
  • a_gran,2 (granulation amplitude, Model 2) = per star, not tabulated
    Second granulation component added in Model 2; recovered with large scatter in simulations.
  • b_gran,2 (granulation frequency, Model 2) = unconstrained for all simulated stars
    Second granulation characteristic frequency could not be constrained in the TESS-like artificial data, and shows large bias on Kepler data.
assumptions (6)
  • domain assumption Granulation power is described by a Harvey-like profile with exponent 4, as in Kallinger et al. (2014).
    Used in Eq. (9) and Eq. (10) to identify the GP kernel with the standard granulation background, and used to generate the simulated light curves in Sec. 3.1.
  • domain assumption The oscillation power excess can be approximated by a celerite SHOTerm with Q > 1, whose PSD approximates the Gaussian-like oscillation bump.
    Introduced in Sec. 2.3 and Fig. 1; this model mismatch with the Gaussian bump used in power-spectrum fitting explains part of the parameter biases.
  • domain assumption The TESS photometric noise model from Sullivan et al. (2015) and Campante et al. (2016), plus a 20 ppm hr^(1/2) systematic term, describes real TESS noise.
    Used in Sec. 3.1 to generate the photometric uncertainty of the artificial TESS-like light curves.
  • domain assumption Scaling relations for red-giant oscillations provide realistic input values for the simulated stars.
    Used in Sec. 3.1 to choose nu_max and granulation amplitudes for the artificial data; the same physical scalings are also used to interpret recovered parameters.
  • domain assumption Non-overlapping 27.4-day subsets of a longer light curve are independent realizations of the stellar signal.
    Assumed in Secs. 3.1 and 4.1 when combining results from multiple TESS-like or Kepler subsets; long-term variations are discarded because they are not detectable in 27.4 days.
  • domain assumption The celerite O(n) approximation, which restricts kernels to sums of exponentials or SHOTerms, is sufficient for red-giant granulation and oscillation signals.
    Adopted in Sec. 2.2 from Foreman-Mackey et al. (2017); this restricts the covariance model to the functional family used throughout the paper.

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Pith. "Pith review of Gaussian Process modelling of granulation and oscillations in red-giant stars." pith.science (2026). https://pith.science/paper/JXSU7MLH

@misc{pith2026190810662,
  author       = {Pith},
  title        = {Pith review of: Gaussian Process modelling of granulation and oscillations in red-giant stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JXSU7MLH}},
  note         = {Machine review of arXiv:1908.10662}
}
abstract

The analysis of photometric time series in the context of transiting planet surveys suffers from the presence of stellar signals, often dubbed "stellar noise". These signals, caused by stellar oscillations and granulation, can usually be disregarded for main-sequence stars, as the stellar contributions average out when phase-folding the light curve. For evolved stars, however, the amplitudes of such signals are larger and the timescales similar to the transit duration of short-period planets, requiring that they be modeled alongside the transit. With the promise of TESS delivering on the order of $\sim\!10^5$ light curves for stars along the red-giant branch, there is a need for a method capable of describing the "stellar noise" while simultaneously modelling an exoplanet's transit. In this work, a Gaussian Process regression framework is used to model stellar light curves and the method validated by applying it to TESS-like artificial data. Furthermore, the method is used to characterize the stellar oscillations and granulation of a sample of well-studied \textit{Kepler} low-luminosity red-giant branch stars. The parameters determined are compared to equivalent ones obtained by modelling the power spectrum of the light curve. Results show that the method presented is capable of describing the stellar signals in the time domain and can also return an accurate and precise measurement of $\nu_\text{max}$, i.e., the frequency of maximum oscillation amplitude. Preliminary results show that using the method in transit modelling improves the precision and accuracy of the ratio between the planetary and stellar radius, $R_p/R_\star$. The method's implementation is publicly available.

Figures

Figures reproduced from arXiv: 1908.10662 by the authors.

Figure 1
Figure 1. Solid (blue) curve depicts the power spectral density of a granulation profile (Eq. 9). Dashed (orange) and dotted (red) curves are the power spectral densities of the functions used to capture the signal from the oscillation bump in a GP and power spectrum analyses, respectively. Study of stellar signals in the literature is commonly performed in the frequency domain by means of power￾spectrum fitting. To be able t… view at source ↗
Figure 2
Figure 2. Comparison of the parameters in the fit of Model 1 to the TESS-like artificial data with the input used to generate those data. Data points represent the relative deviation with respect to the input value, with error bars corresponding to the uncertainties returned by the GP regression method. Black solid and dashed lines represent the median and standard deviation of the data points, respectively, with their numeri… view at source ↗
Figure 3
Figure 3. Comparison of the parameters in the fit of Model 2 to the TESS-like artificial data with the input used to generate those data. Data points represent the relative deviation with respect to the input value, with error bars corresponding to the uncertainties returned by the GP regression method. Parameters that have been flagged as not constrained (see end of Section 2.4) have dotted edges. Black solid and dashed line… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Predictive model (Model 1) output by the GP regres￾sion (mean and 1σ interval) when applied to one of the artificial TESS-like time series. The plot is zoomed in on the first ∼ 3 days of simulated data to improve visualization. 10 0 10 1 10 2 Frequency ( Hz) 10 1 10 0 …
Figure 5
Figure 5. Figure 5: Power spectral density of the same (full) light curve depicted in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Comparison of the parameters in the fit of Model 1 to the Kepler time series data both by means of a GP regression and power-spectrum fitting. Data points represent the relative deviation with respect to the value determined using the PSD-fitting procedure, with error …
Figure 7
Figure 7. Figure 7: Comparison of the parameters in the fit of Model 2 to the Kepler time series data both by means of a GP regression and power-spectrum fitting. Data points represent the relative deviation with respect to the value determined using the PSD-fitting procedure, with error …
Figure 8
Figure 8. Figure 8: Predictive model (Model 2) output by the GP regres￾sion (mean and 1σ interval) when applied to one of the Kepler LLRGB stars in the sample. The plot is zoomed in on the first ∼ 3 days of observations to improve visualization. 10 0 10 1 10 2 Frequency ( Hz) 10 1 10 0 10…
Figure 9
Figure 9. Figure 9: Power spectral density of the same (full) light curve depicted in [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.