Gaia Data Release 3 Pulsations in main sequence OBAF-type stars

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Introduction
Intermediate-and high-mass (IHM) main sequence stars (M ≥ 1.3 M , spectral types O, B, A, and F) have convective cores whose physical conditions cannot be extrapolated from dwarf stars like our Sun.The transport processes caused by internal convection, rotation, and magnetism in IHM dwarfs are far less understood than those in Sun-like stars, yet these processes have a large impact on their evolution and their age determination.A large fraction of the IHM stars are oscillators located in the β Cep, slowly pulsating B (SPB), δ Sct, or γ Dor instability strips in the Hertzsprung-Russell (HR) diagram.β Cep, SPB, and δ Sct stars all exhibit self-excited oscillation modes caused by the con-version of thermal energy into mechanical energy by the socalled κ-mechanism (Pamyatnykh 1999).γ Dor oscillations are driven by a combination of the κ-mechanism on one hand, and radiative flux modulation caused by convection at the bottom of the convective envelope on the other hand.The first mechanism plays a more important role for the hotter γ Dor stars, while the second one is predominant for the cooler γ Dor stars (Guzik et al. 2000;Dupret et al. 2005;Xiong et al. 2016).For all four types of these pulsators, asteroseismology is a particularly promising tool to investigate their internal physics (e.g.Aerts 2021).
Space telescopes like CoRoT (Auvergne et al. 2009), Kepler (Koch et al. 2010), and TESS (Ricker et al. 2016) were dedicated to the gathering of long uninterrupted high-cadence photometry with µmag precision.This type of photometric light curve assembled from space brought unprecedented capabilities to test and calibrate the theory of stellar structure for stars of various masses and evolutionary stages (see Hekker & Christensen-Dalsgaard 2017;García & Ballot 2019;Kurtz 2022, for extensive observational reviews).Thanks to these high-quality data, the high-dimensional problem of asteroseismic modelling of IHM stars becomes achievable (Aerts et al. 2018).Asteroseismic modelling requires identification of the pulsation modes in terms of their spherical wave numbers ( , m) and radial order n.Compared to the simple case of Sun-like stars, the Coriolis force as well as the non-linear interplay between rotation, convection, and magnetism adds a high level of complexity to the modelling of β Cep, SPB, γ Dor, and δ Sct stars.This is why detailed asteroseismic modelling has so far only been carried out for a few tens of such IHM dwarfs, and with inhomogeneous approaches and levels of detail (Degroote et al. 2010;Kurtz et al. 2014;Saio et al. 2015;Murphy et al. 2016;Schmid & Aerts 2016;Deng et al. 2018;Szewczuk & Daszyńska-Daszkiewicz 2018;Mombarg et al. 2019;Wu et al. 2020;Mombarg et al. 2020Mombarg et al. , 2021;;Pedersen et al. 2021;Sekaran et al. 2021).
In this paper, we present a performance verification of Gaia DR3 in terms of the mission's capacity to detect radial and nonradial oscillations of oscillating IHM stars on the main sequence.The Gaia space mission was primarily designed as an astrometric mission to reach µas precision.To reach this goal, the CCD detections of the sources do not need to have a very high signalto-noise ratio (S/N) per field of view transit, nor does the sampling need to be dense.
As a result, Gaia photometric time series of an individual star were not meant to be particularly attractive for asteroseismology.The median number of measurements per star is only around 50 in Gaia DR3 and their time sampling is sparse.The photometric precision in the G passband is of the order of a mmag.This compares excellently with ground-based surveys, but is not (and never was intended to be) able to compete with the photometric precision of dedicated space missions like Kepler or TESS (cf.Appendix A).However, the main advantage of the Gaia mission for asteroseismology is the sheer number of sources for which it gathers data.Unlike the Kepler or TESS space missions focusing on a relatively small part of the sky or only on brighter stars, respectively, Gaia is an all-sky survey covering stars spanning more than 12 orders of magnitude in brightness.In addition, its high angular resolution allows us to also probe the denser parts of the sky.Therefore, Gaia enables the detection of oscillations in a far larger number of variable stars.Gaia already proved to be an excellent instrument to detect radially oscillating highamplitude variables (e.g.Clementini et al. 2019;Mowlavi et al. 2018;Ripepi et al. 2022;Gaia Collaboration et al. 2019), but its capacity has not yet been established for the low-amplitude main sequence IHM stars.While the µmag-precision space mis-sions so far resulted in asteroseismic modelling of thousands of red giants (Hon et al. 2018;Yu et al. 2018Yu et al. , 2020Yu et al. , 2021;;Stello et al. 2022) and of hundreds of Sun-like dwarfs with masses M ≤ 1.3 M (García & Ballot 2019), the samples of asteroseismically modelled IHM dwarfs are much smaller and do not yet fully cover the parameter space in terms of rotation, binarity, and metallicity.It is in this area that Gaia's time series is helpful and will allow for the discovery of thousands of oscillating IHM candidates, with the aim being to achieve unbiased and complete samples.These will then become suitable for ensemble asteroseismology in the future, once dedicated high-cadence high-precision monitoring for their members has become available.
In terms of impact, Gaia DR3 (and later data releases) may lead to the discovery of tens of thousands of new IHM pulsating dwarfs, delivering the frequencies and amplitudes of their dominant (non)radial oscillation mode(s), aside from a high-precision parallax.Here, we assess Gaia's performance on this front, offering as well an observational census of the instability strips along the upper main sequence for samples of new pulsating IHM dwarfs with M ≥ 1.3 M .The borders of those instability strips depend on the physics of internal rotation, gravitational settling, radiative levitation, shear mixing, and so on, as well as on the bulk metallicity and possible binarity at birth.Given the complex (often non-linear) interplay of these astrophysical phenomena during the evolution and the initial conditions at birth, samples of thousands of (non-)radial pulsators are required to interpret the borders of the instability strips in terms of the excitation physics (Townsend 2005;Szewczuk & Daszyńska-Daszkiewicz 2017) and transport processes (Aerts et al. 2019).
This paper is one among several publications led by the Gaia Data Processing Analysis Consortium (DPAC) accompanying the Gaia DR3.We present the Gaia photometric dataset of variable IHM dwarfs in Sect. 2 and develop our strategies to detect IHM main sequence pulsators in Sect.3, distinguishing non-radial gravity-mode (g mode) pulsators from (non-)radial pressure-mode (p mode) pulsators.We compare the samples of IHM dwarf pulsators with those in the literature in Sect. 4 and move on to Gaia's performance in Sect.5, highlighting some dedicated applications.We conclude in Sect.6 with an explicit encouragement and invitation to the worldwide astrophysics community to exhaustively exploit the Gaia time-series data in its full glory.

The dataset
Within the Gaia Data Processing and Analysis Consortium (DPAC), the CU7 coordination unit responsible for analysing variable stars processed a grand total of 1 840 651 642 sources, of which 25% of the most variable sources were selected for further investigation.This variability was tested in the time domain rather than the Fourier domain for computational reasons, as described in Eyer et al. (2022).These variable sources were then further characterised in the Fourier domain and classified by multiple classifiers each with their own setup Rimoldini et al. (2022a).We also refer to the documentation provided by Rimoldini et al. (2022b) for further processing details.
For this paper, we started with those 450 605 variable sources from the Gaia DR3 table vari_classifier_result with at least 40 photometric measurements in the Gaia G-band, and which the CU7 classification team classified as either β Cep, δ Sct, SX Phe, SPB, or γ Dor star.No constraints were put on the apparent G magnitude.To this sample we added the 54 476 IHM main sequence pulsator candidates from the Gaia DR3 ta-ble vari_ms_oscillator in Gaia DR3, which used a different set of classification criteria.We filtered out any duplicates common to both samples to end up with a sample of 460 519 candidate main sequence variables with at least 40 Gaia G data points.Both classifications mentioned here are based on a list of attributes that includes the main frequency ν 1 found in the generalised Lomb-Scargle periodogram (Zechmeister & Kürster 2009).This main frequency optimises the fit of the following model to the data: where G n is the observed Gaia G magnitude at time t n , C is a constant term, and A and ϕ are the amplitude and phase of the sine wave, respectively.The former classification did not automatically exclude those variables for which ν 1 is not statistically significant.Hence, for this sample the classification may be partly based on detected variability that has no astrophysical origin.For this reason we imposed a first threshold on the false alarm probability (FAP; Baluev 2008) of ν 1 , which we limit to be no greater than 0.01.
Examination of the time series revealed that a stricter FAP threshold is required for some cases.The histogram of the first frequency ν 1 for our sample of stars is shown in Fig. 1 in dark blue.This histogram shows that, although the quality of the photometric data reduction is steadily and significantly improving with every data release, the photometric data are not yet completely free of instrumental artefacts.We see high peaks around harmonics of 4 d −1 , which is the spin frequency of the spacecraft (cf.Gaia Collaboration et al. 2016).The strong wings around these peaks suggest that sources in a fairly broad frequency range are affected.For these stars, the Fourier analysis therefore extracted the signature of an instrumental variation rather than an astrophysical variation.Holl et al. ( 2022) and Distefano et al. (2022) also observed instrumental periodic variations in the photometric data of Gaia DR3, be it while focusing on a different frequency range (ν < 0.5 d −1 ) from that relevant for this article.Although these authors were not able to clean the affected time series from instrumental variations, they were able to efficiently filter out the impacted sources.Their filtering method proved not as efficient for our sources, casting doubt on whether the instrumental frequencies observed at very low frequencies are of the same origin as those we observe at high frequencies.There are multiple possible reasons for this; in addition to genuine instrumental variations, some of the peaks observed in Fig. 1 might be caused by frequency aliasing of very low frequencies into high frequencies.For the final release (Gaia DR5), we expect the instrumental variations to be mostly calibrated out.
To mitigate these variations for the benefit of the current paper, we experimented with many different observables to filter out the affected sources.The only effective one proved to be a stricter FAP threshold, which we limit to be no greater than 10 −3 for sources with ν 1 ∈ [4, 25] d −1 .This strongly reduces the wings of the high peaks seen in Fig. 1.It does not completely eliminate the peaks themselves, and so we also removed all sources with a frequency ν 1 near 4, 8, 12, 16, 20, and 24 d −1 using a proximity threshold of 0.05 d −1 .Lastly, we also discarded those sources with ν 1 < 0.7 d −1 to avoid dominant long-term variability of unknown origin, because most of the dominant g modes found in Gaia DR3 data of the best characterised 63 bona fide Kepler γ Dor and SPB stars have frequencies above this value (cf.Appendix A).The histogram of ν 1 of the resulting sample of 108 663 sources is shown in light blue in Fig. 1.
Looking further at the main frequency ν 1 , we find that the CU7 classifiers assign some of the stars to the β Cep, SPB, γ Dor,  or δ Sct classes even if ν 1 is far from the typical frequency ranges associated with the dominant modes of such stars in this particular type of variables, even when taking into account frequency shifts due to rapid rotation.An underlying reason is that the classifiers did not assign sufficient weight to the main frequency of the variability.In a next step we therefore further filtered the main frequency ν Figure 2 shows the sky distribution of our sample in galactic coordinates, and Fig. 3 shows the projection on the XY galactocentric plane of the subset of 71 490 sources with a relative parallax uncertainty 0 < σ / < 0.25.The galactocentric coordinates (X gal , Y gal ) were computed using the median photogeometric distances computed from Gaia EDR3 data by Bailer-Jones et al. (2021).
The gaps in the spatial distribution are caused by Gaia's scanning law and the lower constraint on the number of time points (N ≥ 40) for our sources.Finally, Fig. 4 shows exam- ples of phase diagrams of high-S/N light curves for each of the OBAF-type pulsators.

The IHM instability strips as observed by Gaia
In a global sense, pulsation theory successfully predicts the occurrence of modes in OBAF-type main sequence stars in the HR diagram (see e.g.Dupret et al. 2005;Townsend 2005;Miglio et al. 2007;Daszyńska-Daszkiewicz et al. 2013;Xiong et al. 2016;Szewczuk & Daszyńska-Daszkiewicz 2017;Burssens et al. 2020, and references therein).However, discrepancies between mode excitation predictions and observations occur, both at the level of the observed samples as a whole and for the various detected modes in individual pulsators.As an example of the former, Murphy et al. (2019) found that the observational instability strip of about 2000 δ Sct stars detected in Kepler data covered a broader temperature regime than predicted by state-of-theart pulsation theory based on time-dependent convection theory (Dupret et al. 2005).For individual hotter stars, two examples of IHM pulsators with detected modes predicted not to be excited were discussed in great detail by Daszyńska-Daszkiewicz et al. (2017) and Moravveji et al. (2015) for a β Cep and SPB star, respectively.These two B-type pulsators are representative of numerous additional cases, pointing to overly low opacities for iron peak elements used as input physics for the computation of stellar models.Moreover, even in state-of-the-art stellar models, very approximative treatments of internal rotation and core convection are used, leading to unreliable predictions about their non-radial oscillations (Aerts 2021).On the other hand, from the observational side, the empirical location of IHM stars in the HR-or Kiel diagrams is often plagued by large systematic uncertainties on the observed stellar distance, interstellar reddening, and bolometric corrections (Pedersen et al. 2020), as well as on the star's gravity, helium abundance, metallicity (z), and, to a lesser extent, effective temperature (Burssens et al. 2020;Gebruers et al. 2021).
For the present Gaia sample, we face the additional challenge that the classification was based on one frequency only, which makes the presence of contaminants in our sample likely.In particular, rotationally modulated stars are prevalent at spectral types A and B (e.g.Bp and Ap stars, Kurtz 2000;Briquet et al. 2007;Balona et al. 2011b;Bowman et al. 2018).Bp and Ap stars tend to be slow to moderate rotators showing surface spots, leading to variability in the same frequency domain as gmode pulsations.On top of that, although the window function of Gaia time series is significantly better than those of typical ground-based surveys, the highest peak in the normalised spectral window of the Gaia DR3 time series can still reach 60% or more for certain regions in the sky.We cannot therefore exclude that a fraction of the low-frequency oscillators are actually δ Sct or β Cep stars for which an alias frequency was found instead of the true frequency.Likewise, high-frequency oscillators might have been classified as SPB or γ Dor star because of frequency aliasing (cf.Appendix A).
For our sample of OBAF-type IHM stars, we extract the effective temperature T eff from the field teff_gspphot in the table astrophysical_parameters_supp in Gaia DR3.The luminosities were computed using the absolute G magnitude mg_gspphot field in the same table, and a bolometric correction BC (depending on T eff ) from Pedersen et al. (2020) for stars hotter than 10000 K and from Eker et al. (2020) for the cooler stars.In both cases, we used the metallicity-independent BC prescription, as we do not have a reliable estimate of [M/H] for most of our stars.We propagated the errors on T eff and M G to obtain the errors on the bolometric corrections and the luminosities.The typical T eff uncertainty of the stars in our sample as reported in table astrophysical_parameters_supp is about 150 K, leading to a typical uncertainty of 0.08 on log L. However, as Andrae (2022) discusses in depth, the uncertainties on T eff and M G in the Gaia DR3 release are likely underestimated, in some cases by a factor of three.Given the experience of ground-based spectroscopy, we can expect a minimum uncertainty of about 400 K on T eff , especially for the hotter B-type pulsators.
As our sample includes stars from magnitudes G =6 to G =18, and the uncertainty on the gspphot stellar parameters is often too large to make them useful for placement in the HR diagram, we limit ourselves to the nearby IHM pulsators with a relative Gaia DR3 parallax uncertainty 0 < σ / < 0.03. Figure 5 shows a HR diagram with the nearby Gaia sources classified as γ Dor or SPB as described in Section 2. In the same figure, the SPB-star instability strip from Burssens et al. (2020) Dupret et al. (2005) for a value of the mixing length, α MLT = 2.
The highest concentration of g-mode pulsators is, as expected, inside the γ Dor instability strip.The slight offset towards lower luminosities is easily explained as due to systematic uncertainties in the reddening or bolometric correction BC.The low-frequency variables blueward of the γ Dor instability strip fall inside or above the δ Sct instability strip (cf.Fig. 6).Aside from the possibility that they might be hot γ Dor stars (Li et al. 2020) or γ Dor stars with an uncertain location in the HR diagram, they could also represent hybrid p-and g-mode γ Dor/δ Sct pulsators (Grigahcène et al. 2010;Bowman et al. 2016), aliased δ Sct stars, Bp/Ap stars (Balona et al. 2019), fastrotating SPB stars (Townsend 2005;Szewczuk & Daszyńska-Daszkiewicz 2017), or rotationally modulated g-mode pulsators following the recent pulsation excitation theory for fast rotating early-type stars (Ouazzani et al. 2020;Lee & Saio 2020;Lee 2021).This latter theory predicts g-mode pulsators to occur in  rapid rotators between the SPB and γ Dor instability strips, that is, at effective temperatures representative of spectral types A0-A3.This concerns fast rotating spotted stars with gravito-inertial envelope modes coupled to inertial core modes as observed in Kepler observations (Saio et al. 2018).
Occurrences of rapidly rotating g-mode pulsators between the SPB and δ Sct instability strips are well established from modern space photometry.Indeed, after initial discoveries from CoRoT data of SPB stars (Degroote et al. 2009(Degroote et al. , 2011)), numerous detections were achieved from ground-based cluster studies as well (Saesen et al. 2010(Saesen et al. , 2013;;Mowlavi et al. 2013;Moździerski et al. 2014;Mowlavi et al. 2016;Saio et al. 2017;Moździerski et al. 2019).These detections are interpreted as fast-rotating SPB stars from instability computations based on the so-called traditional approximation of rotation applied to g modes (Salmon et al. 2014), following the interpretation by Aerts & Kolenberg (2005) based on spectroscopic time-series data.Many fastrotating SPB and Be pulsators were also found near the red edge or below the SPB instability strip from Kepler and TESS data (Balona et al. 2011a;Pedersen et al. 2019;Burssens et al. 2020;Sharma et al. 2022).Our Gaia results in Fig. 5 reveal plenty of g-mode pulsators between the SPB and δ Sct instability strips.Within this group of g-mode pulsators, we observe an underdensity between the hot A stars and the blue edge of the δ Sct instability strip.At this point, we cannot exclude that this is an artefact rather than having an astrophysical origin.
The distribution of stars within the SPB instability strip in Fig. 5 is concentrated in or near the cooler bottom half, in close agreement with mode density excitation predictions (cf.Fig. 1 in Pápics et al. 2017).These g-mode pulsators are in the same position as those observed in high-cadence space photometry of slow-and fast-rotating B and Be stars (Neiner et al. 2009;Huat et al. 2009;Diago et al. 2009;Neiner et al. 2012;Baade et al. 2016;White et al. 2017;Baade et al. 2018a,b;Pedersen et al. 2019;Burssens et al. 2020;Pedersen et al. 2021;Szewczuk et al. 2021;Sharma et al. 2022).It is noteworthy that the Gaia CU7 classification scheme also has a class of γ Cas variables representing classical Be stars (Rivinius et al. 2013).It has long since been known from line-profile variability that many Be stars are rapidly rotating non-radial p-and/or g-mode pulsators (Rivinius et al. 2003).High-cadence space photometry indeed reveals most of the γ Cas variables to have non-radial pulsations aside from rotational modulation (Balona et al. 2011a;Balona 2016), including the prototype γ Cas itself (Labadie- Bartz et al. 2021;Smith & Henry 2021).We did not include the γ Cas variables in our study, because this class definition within CU7 does not rely on them being non-radial pulsators.We therefore anticipate that many more rapidly rotating non-radial B-type pulsators occur in Gaia DR3, but were assigned to the γ Cas class.
In Fig. 6 we show the HR diagram dedicated to the stars that were classified as either β Cep or δ Sct stars.As before, we imposed a relative parallax of better than 3% for the δ Sct stars.For the β Cep stars, which are much more rare, we relaxed this constraint to 0 < σ / < 0.2.The vast majority of δ Sct stars are well concentrated in the expected instability strip of p modes, demonstrating both the good performance of the variability classifier as well as the precision of the astrophysical parameter determination by the CU8 processing pipeline, at least for the nearby stars.Higher up the main sequence, the sample of nearby very luminous β Cep stars is small.Most of them have an observed location in the HR diagram outside the expected instability strip, which is difficult to explain in terms of metallicity.The most likely explanation is a systematic bias in the astrophysical parameters of hot stars as derived from Gaia DR3 gspphot data.Indeed, the observed shifts to cooler temperatures can easily occur due to poorly estimated reddening.However, the effects of fast rotation may also play a role, as it is not or only incompletely treated in published instability strips for p modes, while its impact on the properties of the modes is large (Daszyńska-Daszkiewicz et al. 2002).Many fast rotators with p modes have indeed been found in high-cadence space photometry of earlytype B and Be stars (Balona et al. 2011a;Burssens et al. 2020;Balona & Ozuyar 2020).
The classifier also picked up a sample of stars with high frequencies ν 1 right below the SPB instability strip, with T eff around 10000 K or slightly cooler.These variables are in line with CoRoT and TESS discoveries of what appears to be pmode pulsators in that position of the HR diagram (Degroote et al. 2009;Balona & Ozuyar 2020).The variability of such pulsators is not yet well understood.We note that the frequencies of gravito-inertial modes in moderate to fast rotators can easily get shifted into the regime of p modes (e.g.Aerts & Kolenberg 2005;Salmon et al. 2014).Such frequency shifts may be particularly large for g modes with |m| > 1, which complicates the classification of the variability based on the frequencies from photometry alone.This is illustrated nicely by the two cases of the slowly rotating variable star Maia (Struve 1955) and the magnetic moderately rotating SPB pulsator HD 43317 (Pápics et al. 2012).While Maia was originally classified as a pulsator by Struve (1955) and even led Struve to introduce a seemingly new class of pulsators (the so-called Maia variables), White et al. (2017) showed it to be a rotationally modulated star rather than a pulsator.On the other hand, the high-frequency mode of HD 43317 at 4.3 d −1 was originally misinterpreted as a p mode, while it is a rotationally shifted quadrupole g mode (Buysschaert et al. 2018).These two examples reveal that misclassifications of B-type variables based on photometric data alone are easily made and that correct interpretation of the frequencies of such stars in the regime of a few to several d −1 is only possible from adding independent data -such as high-precision spectroscopy-to photometric light curves.Given the recent Kepler-guided understanding of fast rotating BAF-type pulsators with mode frequencies between pure g and pure p modes, such variables were not yet included as a separate variability class in the CU7 classifiers.Therefore, their strongly shifted non-radial mode frequencies due to fast rotation in the inertial frame of the observer may imply that these variables get spuriously classified as p-mode pulsators below the SPB instability strip in Fig. 6.
An even larger overdensity of high-frequency variables also pops up at cooler temperatures well below the δ Sct instability strip.A closer look at this part of the HR diagram is shown in Fig. 7.This concerns mainly stars with dominant frequency below 7 d −1 , obeying a period-luminosity relation (albeit a noisy one) as can be seen from the colour gradient in the figure.We postulate that this may be a cooler-temperature, lower mass relation analogous to the period-luminosity (PL) relations found for rapidly rotating B-type pulsators with g modes occurring below the SPB instability strip.Such PL relations were initially deduced by Saio et al. (2017) based on the B-type cluster pulsators found by Mowlavi et al. (2013Mowlavi et al. ( , 2016)).Sharma et al. (2022) found similar PL relations from TESS photometry of rapidly rotating g-mode pulsators occurring between the SPB and δ Sct strips (cf.Fig. 5).Sharma et al. (2022) interpreted these PL relations in terms of dipole (l = |m| = 1) and quadrupole (l = |m| = 2) g modes.It may well be that the PL relation revealed in Fig. 7 can be explained similarly, that is, as the result of a mixture of dipole or quadrupole gravito-inertial, Rossby, or Yanai modes in very fast rotators of spectral type F, as such types of modes have been detected in Kepler photometry of cool γ Dor stars (Van Reeth et al. 2016, 2018;Saio et al. 2018).2020), respectively.This sample has the advantage that the classification was carried out manually based on dense Kepler time series, and is therefore very reliable.The drawback is that the sample covers only the fairly small magnitude range observed by Kepler and is very localised in the sky.This sample therefore does not cover the large magnitude range of our Gaia DR3 sample, nor does it cover all the window functions caused by the sky-coordinate-dependent Gaia sampling.
Focusing first on the δ Sct stars, we find that 746 of them (35%) are identified as an OBAF-type pulsator by the CU7 classifiers.The majority were therefore either not found to be variable in the time domain or ended up as another type of variable.This illustrates the challenges that come with identifying these small-amplitude multi-periodic variables with the fairly small number of Gaia measurements.Of these 746 pulsators, 73 (10%) have a FAP of smaller than our threshold of 10 −3 (cf.Section 2).Our measures to mitigate the presence of instrumental effects have a significant impact on the selection of bona fide pulsators as well.The vast majority (82%) of the δ Sct stars that did pass the FAP criterion were correctly identified as δ Sct stars by the CU7 classifiers.
The results for the γ Dor stars are similar.We find that 233 of them (39%) passed the CU7 classifier as an OBAF-type pulsator, and 26 of these stars have a FAP value smaller than our threshold of 10 −2 for g-mode pulsators.For most of these stars, their small amplitudes (often below 5 mmag, cf.Appendix A) make the peaks in the Fourier spectrum barely statistically significant and distinguishable from peaks purely caused by noise fluctuations.
The second literature compilation with which we compare our sample is that of Gavras et al. (2022), who compiled a large database of variable stars using existing catalogues published in the literature (before the Gaia DR3 release), including those of Stankov & Handler (2005) 2020) among others.The compilation ensured an all-sky coverage and a large range of G magnitude (and thus S/N levels).It consists of catalogues compiled using a manual variability classification, as well as catalogues based on a probabilistic machine learning approach.We refer to Gavras et al. (2022) for more details.Although these latter authors adopted a stringent classification score threshold, we should expect a small fraction of these literature classifications to be inaccurate.
For 12 042 stars in our sample, the cross-match catalogue of Gavras et al. (2022) includes a primary frequency found in the literature.The comparison of these literature frequencies with the ones derived by CU7 using Gaia DR3 photometric time series in the G passband is shown in Figure 8. Overall there is an excellent match between the Gaia DR3 and the literature frequencies.A small sample of sources forms a parallel line above and below the main diagonal.These are sources for which ground-based observations found a one-day alias of the true frequency, while the oscillation frequency found by Gaia is not affected by Earth's day-night rhythm.For a smaller sample, the main frequency derived from Gaia DR3 data is twice the frequency found in the literature.We find that most of them are stars catalogued as a binary star in the literature, but erroneously classified as a g-mode pulsator by the CU7 classifiers.
The literature cross-match compilation of Gavras et al. ).The miss rate (i.e. the false negative rate FN/(FN+TP)) is therefore fairly good for δ Sct stars (32%) but high (≥ 57%) for the other OBAF-type pulsators.This is not surprising, given that the stars in our 88K sample have a median number of time points equal to 56.Nevertheless, the OBAFtype pulsators of the literature that do end up in our 88K sample are mostly correctly classified.We remind the reader that the CU7 classification pipeline used XGBoost and Random Forest classifiers to classify 12 million variable stars into 25 variability classes.More details can be found in Rimoldini et al. (2022a).
Starting from our 88K sample of Gaia DR3 OBAF-type pulsators, it turns out that 66 β Cep stars, 119 SPB stars, 793 γ Dor stars, and 11064 δ Sct (+ SX Phe) stars also appear in the literature compilation of Gavras et al. (2022), although not necessarily classified as the same type.For the β Cep stars, the main contaminants are the γ Cas variables (20%) and the δ Sct variables (6%).For SPB stars, the main contaminants are α 2 Canum Venaticorum (ACV) variables (24%), rotationally modulated stars (21%), and γ Cas variables (9%).The γ Dor stars are confused most often with RS Canum Venaticorum variables (33%), rotationally modulated stars (23%) and, as expected, with binary stars (12%).Finally, the δ Sct (+ SX Phe) stars are sometimes confused with binaries (6%), RR Lyr stars (2%), or RS Canum Venaticorum variables.Assuming again that the literature class label is always reliable, we can conclude that the false discovery rate (i.e.FP/(FP+TP)) is fairly good to excellent for the highfrequency variables, as low as 12% for the δ Sct stars, but is high for the low-frequency variables (≥ 65%).However, we note that we used our entire 88K sample for this assessment, including the faint stars.As can be seen from the previous section, the comparison gives much better results for the nearby and brighter stars.
To investigate why the classification of g-mode pulsators is so difficult with Gaia DR3 data in more detail, we analyse in Appendix B the Gaia DR3 time series in the Gaia G-band for 63 well-known bona fide g-mode pulsators listed in Aerts et al. (2021).For these stars, we know the true oscillation frequency from densely sampled Kepler photometric time series.Our analysis confirms the comparison with the literature above.For about 15% of these stars, the main frequency identified from Gaia photometry coincides with the main frequency found from Kepler photometry.For two of these stars, the second frequency also coincides.The main culprit for mismatches is either an aliased frequency making the g-mode pulsator look like a δ Sct star, or an instrumental frequency as described in Section 2.

The period-luminosity relation of δ Sct stars
δ Sct stars pulsating in the radial fundamental (F) and/or in the first overtone (1O) mode show a period-luminosity relation, albeit with a larger scatter than for Cepheids (see e.g.McNamara et al. (2007), McNamara (2011), Ziaali et al. (2019), Jayasinghe et al. (2020), Poro et al. (2021)).For these modes, the pulsation constant Q ≡ P ¯ / ¯ is more or less constant over the instability strip, and the dominant stellar property that correlates with the stellar mean density is the luminosity L, hence the PL relation (Breger 2000).
To establish the relation as seen using the Gaia DR3 dataset, we first computed the reddening-insensitive period-Wesenheit (PW) relation.The absolute Wesenheit index W was computed using where M G is the absolute magnitude in the G-band taken from the Gaia DR3 astrophysical_parameters table (gspphot).Inspection of the PW relation obtained with this sample revealed some contamination from the well-defined group of lowluminosity stars with a frequency f < 6.6 d −1 (P>0.15days) described in the last paragraph of Section 3.Because of their compactness, it proved easy to select and remove them with the criteria f < 6.6 d −1 and W < 1.0 mag.In this way, we discarded 233 objects.Figure 9 shows the PW relation for nearby δ Sct stars with σ / < 0.05 and T eff ∈ [6400, 8700] K colour coded with the empirical Q value (in days).The latter was computed using the relation derived by Breger (1990): As before, the quantities L/L , g and T eff were taken from the Gaia DR3 astrophysical_parameters table (gspphot).
Comparing the empirical pulsation constant Q with the typical values given by Breger (1979), we see that the Gaia DR3 data not only show the ridge of the fundamental mode (dark blue), but also the ridges of the first and the second overtone.
In the remainder of this section, we focus on the most populous ridge in Fig. 9 which is that of the fundamental mode pulsators, for which we follow a more precise procedure.The highamplitude δ Sct (HADS) variables are known to pulsate in the F mode, the 1O mode, or both (McNamara et al. 2007).Figure 10 shows a histogram of the amplitude A G,1 of the main frequency ν 1 for our sample of δ Sct stars, in which the HADS population is clearly visible.To avoid too much contamination of the nonradially pulsating lower amplitude population, we only retained δ Sct stars with A G,1 ≥ 50 mmag.As before, we constrained the effective temperature of our sample stars to the range of the δ Sct instability strip, i.e. 6400 K ≤ T eff ≤ 8700 K.This left us with 8894 stars.In addition we also filtered on the quality of the astrometric parallax as this is a key ingredient in the derivation of the PW relation.To do so, we followed the approach of Rybizki et al. (2022), who used a neural network based on 17 proper Gaia catalogue entries to discriminate objects with poor astrometry by  means of a single parameter which they called 'astrometric fidelity'.We retained only objects with their fidelity_v2 > 0.5.This last selection defined our final sample of 8760 δ Sct stars useful for the PW relation determination.We plotted the δ Sct stars in this sample with a relative parallax uncertainty of better than 20% in Fig. 11.The points are colour coded with the frequency ν 1 so that the frequency gradient due to the periodluminosity relation is clearly visible.
To calculate the PW relation, we adopted the so-called photometric parallax (see Riess et al. 2021, and references therein), which was recently used to derive the period-Wesenheitmetallicity relation for the classical Cepheids in the Gaia bands by Ripepi et al. (2022).This approach allows us to use the parallax linearly, retaining the Gaussian property of its uncertainty.Moreover, as we do not make any selection in parallax (even negative parallaxes can be used) we are not introducing biases in the PW relation.The photometric parallax (in mas) is defined as follows: where W is the absolute Wesenheit magnitude in the G passband, and w is the corresponding apparent Wesenheit magnitude: The absolute Wesenheit index W can be written as where P is the period and P 0 a 'pivoting' period (in days) chosen approximately as the mean of the period distribution (i.e.log 10 P 0 = −1.1).This serves to reduce the correlation between the zero point α and the slope β of the PW relation.
Denoting with EDR3 the parallaxes of the pulsators corrected for the zero-point offset (ZPO) (see Lindegren et al. 2021), we seek to minimise the following quantity: where In our case, σ EDR3 is composed of the standard error of the parallax as reported in the Gaia EDR3 catalogue, here conservatively increased by 10%, and the uncertainty on the individual ZPO corrections, that is 13 µas (Lindegren et al. 2021).To calculate the uncertainty on the photometric parallax we followed the procedure detailed by Ripepi et al. (2022): σ phot = 0.46 • σ µ • phot , where µ = w − W is the distance modulus and σ 2 µ = σ 2 w + σ 2 W its variance.We calculated σ w from error propagation assuming a constant error of 0.01 mag in each of the three Gaia bands (G, G BP , G RP ), where σ W is the intrinsic dispersion of the PW relation.Similarly to the case of classical Cepheids, we assumed a value of 0.1 mag for this quantity.
We minimised Eq. ( 7) using the Python minimisation routine optimize.minimize(included in the Scipy package Virtanen et al. 2020).To provide robust uncertainties on the coefficients α and β of the PW relation, we used a bootstrap procedure in which we repeated the fit to the data of Eq. ( 7) 1000 times, each time with a randomised bootstrap sample, and obtained a different α and β value for each repetition.Our best estimate for these two parameters is then the median of the resulting distributions, while for the uncertainty we used 1.4826 • MAD (median absolute dispersion).
In a first attempt, we fitted the PW relation over the whole period range of 0.04-0.20 days, but it became clear that a single linear fit is not able to represent the empirical PW relation as the data show a break or a non-linearity (see Fig. 12).We therefore applied two approaches, first fitting the PW with a piecewise linear function, and then with a quadratic PW relation of the form For the piecewise linear model, we find that the breaking point occurs approximately at log P = −1.05± 0.05.The best piecewise linear fit to the data is shown in the top panel of Fig. 12 and the corresponding fitted parameter values are listed in the first two rows of Table 1.The quadratic PW relation fit to the data is shown in the bottom panel of Fig. 12, while the corresponding fit parameters are listed in the bottom row of Table 1.
As for the piecewise linear fit, the slopes obtained for the two intervals of the dominant pulsation period are different with a significance of about 10σ, indicating that the PW relation of the δ Sct stars with a pulsation period P shorter than ∼0.09 days has a much steeper slope than the one valid for the longer-period pulsators.Also, the quadratic term has a high significance, indicating that the non-linearity or the two-regime nature of the δ Sct PW relation is a genuine physical feature.1) and the bottom panel shows the quadratic fit to the data (see first two lines of Table 1).
It is instructive to compare the δ Sct PW relation with that of classical Cepheids as both classes of pulsators pulsate due to the well-known κ mechanism.Therefore, the instability strip of the δ Sct stars can be considered the low-luminosity extension of the classical Cepheid instability strip.To make the comparison, we considered the classical Cepheids' PW relations obtained in the Gaia bands by Poggio et al. (2021), which were derived in the same way as for the δ Sct stars in this paper.Figure 13 shows the same δ Sct stars and PW relations as in the top panel of Fig. 12 with, in addition, the location of the F-and 1O-mode classical Cepheids and the relative PW relations from Poggio et al. (2021).For clarity, we only show objects with relative errors on the Gaia EDR3 parallax of better than 20%.The distribution of the classical Cepheids pulsating in the first overtone extends up to that of the δ Sct F-mode pulsators. 1It is clear that the PWs of both variability classes predict similar luminosities for modes in the period range ∼0.20-0.25 days, where the δ Sct and classical Cepheid period distributions overlap.The slopes of the Fand 1O-mode classical Cepheids PW relations as calculated by Poggio et al. (2021) are −3.317±0.028and 3.624±0.017,respectively.Comparing these values with those listed in the first two rows of Table 1, we find that the slopes for the relations found for the classical Cepheids are intermediate between those calculated for the low-and high-period δ Sct samples.If we were to ignore the observed curvature in the δ Sct PW data and fit a single straight line, this would produce an intermediate slope similar to that of the classical Cepheids.12) and those of F-(red dots) and the first overtone 1O-mode (blue dots) classical Cepheids.For the latter pulsators, the data and the PWs were taken from Poggio et al. (2021).For clarity, only objects with relative error on the photometric parallax better than 20% are plotted.

Pulsation amplitude attenuation due to rotation
There are at least two ways in which stellar rotation can affect the observed photometric oscillation amplitudes. 2 The first is through a change in the photometric visibility, that is, how the pulsation manifests itself on the stellar surface, and the second is through a change in the intrinsic oscillation amplitude.Theoretical predictions of photometric visibilities for modes in rotating stars have been analysed in depth in the literature (see e.g.Daszyńska-Daszkiewicz et al. (2007) and Reese et al. (2013Reese et al. ( , 2021) ) for p modes and Henneco et al. (2021); Dhouib et al. (2021a,b) for g modes).For non-radial oscillations in fast rotators, the surface pattern of the modes can no longer be described by a single spherical harmonic Y m .Depending on the type of mode, its frequency, and the angular rotation speed, the pulsation patterns can have a complex structure making them appreciable only at certain latitudes, which affects the geometrical disc-integration factor.
The effect of rotation on the intrinsic amplitude of κ-driven pulsators is poorly understood.The vast majority of available pulsation codes rely on linear pulsation theory, which does not allow the mode amplitudes to be derived.Analysing the effect of rotation on the pulsation amplitudes in rapidly rotating IHM pulsators requires non-linear pulsation theory, which takes into account the effects of mode coupling in the presence of the Coriolis and centrifugal forces.While nonlinear mode coupling is detected in many IHM pulsators (Bowman et al. 2016;Van Beeck et al. 2021), modelling of their observed mode amplitudes has not yet been achieved.
Observations suggest that there is a link between the rotation of a star and its photometric pulsation amplitudes, but the relationship between the two is not well understood (e.g.Aerts et al. 2014, in the case of B-type pulsators).This is partly due to the limited sample sizes of IHM pulsators with the appropriate information.Given the sample sizes of new IHM pulsators presented here, and the precision of v sin i delivered by Gaia for these stars, we focus here on the observed relationship between rotation and amplitude of the dominant pulsation mode for the δ Sct stars.The extreme case of the δ Sct star Altair rotating at an equatorial speed of ∼ 310 km s −1 (Bouchaud et al. 2020) re-Table 1. Results for the PW fitting for δ Sct variables.The fitted equations are W = α + β(log 10 P + log 10 P 0 ) or W = α + β(log 10 P + log 10 P 0 )+γ(log 10 P + log 10 P 0 ) 2 , where we took log 10 P 0 = 1.1.The Notes column lists the period interval over which the different PW relations were calculated.The high-amplitude δ Sct stars tend to rotate slower (typically v sin i ≤ 30 km/s) than normal δ Sct stars, which tend to have v sin i ≥ 150 km/s (see e.g.Breger 2000;Antoci et al. 2019).This suggests that different angular momentum transport mechanisms are at work in these two groups of δ Sct stars, or different efficiencies of the same mechanism.The origin of this transport is poorly understood (Aerts et al. 2019).To investigate the relation between the oscillation amplitude and the stellar rotation, we cross-matched the stars classified as δ Sct stars in our sample described in section 2 with those in the Gaia-DR3 archive for which a v sin i value is available from Gaia spectroscopy (vsini_esphs) with a formal precision better than 50%.This returned 3515 sources, which we partitioned into ten bins in v sin i, each with a width of 15 km s −1 .For each bin, we computed the median value of the photometric amplitude A G,1 .As expected, the bins are not equally populated.All bins contain at least 220 stars, except for the bin with the most rapid rotators for which the constraint on the v sin i precision resulted in only 29 stars.The result is shown in Fig. 14.The dark blue line denotes the median amplitude for each bin, and the light blue band shows the 25%-75% quantile interval of photometric amplitude A G,1 in each bin.A clear and steady decline in median amplitude can be seen with increasing v sin i at a rate of roughly 46 µmag per km s −1 .We note that the sample of 3525 sources contains both HADS and normal δ Sct stars.The vast majority of HADS occur in the lowest v sin i bin, where they are responsible for the strong decrease in amplitude in that bin.The corresponding curve for normal δ Sct stars is far less pronounced in the lowv sin i bins, yet is clearly present in all v sin i bins.We cover δ Sct stars with slow to moderate rotation rates of v sin i < 150 km s −1 .This is well below the v sin i = 270 km s −1 of the fast-rotating δ Sct star Altair, whose dominant mode has an amplitude of only about 0.6 mmag in the white-light broad-band filter of the MOST space telescope (Le Dizès et al. 2021).

Summary and Conclusions
In this paper, we present our investigation of the Gaia DR3 time series of stars classified as β Cep, SPB, γ Dor, or δ Sct by the CU7 classification pipeline.These stars are non-radial pulsators of intermediate and high mass and have proven to be excellent targets for asteroseismology (Aerts 2021;Kurtz 2022).They are often multi-periodic and have much lower amplitudes than RR Lyr stars or long-period variables.This makes them more challenging to detect and discriminate from other types of variables, in particular with the modest size and sparseness of a typical Gband time series in Gaia DR3.For this reason, we focused on time series with a least 40 data points.
Roughly half a million stars with at least 40 data points were classified in one of the four classes by the variability processing pipeline of CU7.Closer inspection of the results of the automated Fourier analysis of this pipeline showed that a variation of instrumental origin was detected for the majority of these stars.This points to imperfect calibrations of the photometric time series at mmag level.We expect these instrumental variations to be calibrated out in the Gaia DR5 release.We succeeded in filtering out most of the stars that show instrumental variations, at the cost of significantly reducing the sample size of IHM pulsators.Nevertheless, the approximately 106K pulsators of this kind constitute an unseen sample in terms of size, magnitude range, and sky coverage.Our comparison with a subsample of the remaining stars in Section 4 shows that the main frequency ν 1 agrees very well with that found in the literature.
To assess the quality of our sample of IHM pulsators, we compared the position of the stars in the HR diagram with the theoretical instability strips.This requires the empirical T eff and luminosity of each star, which we took from the Gaia DR3 gspphot data as derived by DPAC-CU8.We focused on the nearby stars with the most precise location in the HR diagram as observed by Gaia.For these stars, the empirical location of SPB, δ Sct, and γ Dor stars matched the theoretical instability strips quite well.We did find a fraction of ∼20% pulsators to occur outside the strips, in between the SPB and δ Sct strip on the one hand, and below the γ Dor strip on the other.These pulsators have a dominant frequency expected for fast rotators pulsating in gravito-inertial modes (Salmon et al. 2014;Saio et al. 2017;Aerts et al. 2019).For some of the β Cep stars, we noticed discrepant positions with respect to their strip.This may be caused by a systematic bias of the T eff and/or the luminosity L as derived from gspphot data for the hottest stars.However, it may also suggest that the instability strips of OB-type stars are incompletely covered by the current physical descriptions of their interiors.The HR diagrams also revealed the presence of rotationally modulated stars.This is as expected since both ground-based data (e.g.Briquet et al. 2007) and space photometry (Degroote et al. 2011;Balona 2016;Bowman et al. 2018;Balona et al. 2019, among others) already showed that rotationally modulated and pulsating stars co-exist inside the instability strips, and that the two populations of stars largely overlap.
Our analysis of the period-luminosity relation of δ Sct stars provides a good demonstration of the uniqueness of Gaia data, as Gaia DR3 is able to bring both ingredients to investigate this relation.To our knowledge, the empirical PW relation in Fig. 12 is the most extensive reported in the literature for δ Sct stars so far.We find evidence that the relation has two different regimes depending on the period of the dominant oscillation mode.
Finally, we are able to confirm that stellar rotation has a direct impact on the observed oscillation amplitude of δ Sct stars, in the sense that increasing rotation decreases the amplitude.The v sin i data necessary to establish this result were derived from Gaia DR3 spectroscopic esphs data.We were able to quantify the order of magnitude of the gradient, arriving at roughly 46 µmag per km s −1 .
Despite the not yet perfect photometric calibration of the Gaia DR3 data, we were able to demonstrate the great potential of combined and homogeneously sampled photometric, spectroscopic, and astrometric Gaia data to investigate OBAF-type main sequence pulsators, and we look forward to the more extensive Gaia DR4 and DR5 data for which we expect the time series to be less prone to high-frequency instrumental variations.oscillations at mmag level, albeit it for a very small fraction (3%) of pulsating dwarfs pulsators.
We also checked for all 63 stars if it is meaningful to use the fraction of the variance in the data explained by a harmonic fit with the dominant frequency as a selection criterion to distinguish intrinsic g-mode frequencies from instrumental frequencies.We found this not to be feasible, as this merit function attains similar values for instrumental and intrinsic frequencies.The corresponding phase diagrams also do not allow to distin-guish between intrinsic and aliased/instrumental frequencies in a meaningful way.This is illustrated in   rion to distinguish between instrumental and true frequencies in Gaia's DR3 time series.

Fig. 1 .
Fig. 1.A histogram of the first frequency ν 1 .Dark blue: histogram of the first frequency ν 1 of the 460 519 candidate main sequence variables.The highest peaks go beyond the maximum value of the y-axis.Light blue: Same histogram for the 108 663 for which we filtered on the primary frequency ν 1 and its corresponding FAP value to avoid instrumental artefacts.Because of the 34-month time-span of the DR3 time series, the typical uncertainty on the frequency is about 10 −3 d −1 .

Fig. 2 .
Fig. 2. Sky distribution in Galactic coordinates of the 106 207 sources in the data set resulting from various steps of filtering as described in Section 2.

Fig. 3 .
Fig. 3. Distribution of the Galactic sources of the dataset shown in Fig. 2, limited to those 71 490 stars with 0 < σ / ≤ 0.25, projected on the equatorial plane of the Galactocentric reference frame.

Fig. 4 .
Fig. 4. Example phase diagrams of sources classified as main sequence OBAF-type pulsators, folded with the primary frequency ν 1 found in Gaia G-band photometry.From top to bottom the rows show three examples of stars classified as respectively β Cep, SPB, δ Sct, and γ Dor variables.

Fig. 5 .
Fig. 5. HR diagram of 10 402 nearby g-mode pulsators classified as SPB or γ Dor star with a relative parallax uncertainty of better than 3%.The red line is the SPB instability strip from Burssens et al. (2020, at hot temperatures) and the γ Dor instability strip from Dupret et al. (2005, at low temperatures), both for solar metallicity.

Fig. 6 .
Fig. 6.HR diagram of 13 974 nearby p-mode pulsators classified as β Cep or δ Sct stars.For the β Cep stars, we imposed 0 < σ / < 0.2, while we demanded 0 < σ / < 0.03 for the δ Sct stars.The blue line at hot temperatures is the β Cep star instability strip from Burssens et al. (2020) for spherical degree ∈ {0, 1, 2} and radial order n ∈ {0, • • • , 6} .The blue lines represent the blue and red edge of the δ Sct instability strip for n = 1 modes, and the fuchsia line the blue edge for n = 4 modes.All instability strips are for solar metallicity.

Fig. 7 .
Fig. 7. Zoom into the overdensity below the δ Sct instability strip at lower effective temperatures in the HR diagram shown in Fig. 6.The red line is the red edge of the δ Sct instability strip for n = 1 modes.

Fig. 8 .
Fig.8.Comparison with the main frequency ν 1 found with the CU7 pipeline using Gaia DR3 data and the main frequency f 1 reported in the literature.

Fig. 10 .
Fig. 10.Histogram of the amplitude A G,1 of the main frequency ν 1 of our sample of δ Sct stars.

Fig. 11 .
Fig. 11.Location of 3122 high-amplitude δ Sct stars with 0 < σ / < 0.2 in a colour-Wesenheit G diagram.The points are colour coded according to frequency ν 1 .Typical uncertainties of the mean G BP − G RP are between 1 and 2 mmag and between 2 and 4 mmag for the Wesenheit G index.
Fig. 12. PW fit to the data.The top panel shows the two-line fit (see first two lines of Table1) and the bottom panel shows the quadratic fit to the data (see first two lines of Table1).

Fig. 13 .
Fig. 13.Comparison between the PW relation(s) calculated in this paper for the fundamental F-mode δ Sct (as in Fig. 12) and those of F-(red dots) and the first overtone 1O-mode (blue dots) classical Cepheids.For the latter pulsators, the data and the PWs were taken from Poggio et al. (2021).For clarity, only objects with relative error on the photometric parallax better than 20% are plotted.

Fig. 14 .
Fig. 14.Photometric amplitude A 1 in the Gaia G-band as a function of the v sin i derived from spectroscopic Gaia DR3 data for a sample of 3515 δ Sct stars.The dark blue line is the median amplitude for each v sin i bin of 15 km s −1 while the light blue band shows the 25%-75% quantile interval.

Fig. B. 1 .
Fig. B.1.Histogram of the dominant frequency in the Kepler light curves of 63 asteroseismically modelled g-mode pulsators (blue) compared with the dominant frequency occurring in their Gaia DR3 G-band epoch photometry (black hatched).For reasons of visibility, the x−axis is cut at 6.5 d −1 and the y−axis at 20; 38 of the 63 g-mode pulsators have their Gaia data dominated by instrumental effects with dominant frequency above 6 d −1 rather than by their g modes.

Fig. B. 5 .
Fig. B.5.Four phase diagrams for the γ Dor pulsator KIC 7023122 whose two dominant g-mode frequencies (as listed in the legend of the x−axes) occur consistently in the periodograms of the Gaia DR3 G-band and Kepler photometry.The data are shown as black squares and the best harmonic fits for the fixed frequencies as blue lines.For visibility purposes, the phases are shown for two cycles.
Fig. B.7 for the largestamplitude γ Dor pulsator of the sample.
Figure B.7 shows two phase diagrams based on the 40 Gaia DR3 G-band data points: one based on the 'true' intrinsic dominant g-mode frequency reducing the variance by 47% and another one using the dominant frequency in the data themselves, which is either aliased or of instrumental origin.The latter frequency reduces the variance by 58%.Hence the variance reduction cannot be used as a crite-

Fig
Fig. B.7.Two phase diagrams for the largest-amplitude g-mode pulsator in the γ Dor sample.The upper graph is phase-folded with the true dominant g-mode frequency derived from Kepler data; its fit leads to a variance reduction in the Gaia DR3 G-band data of 47%.The lower graph is phase-folded with the dominant frequency in the data, which is of instrumental origin and reduces the variance with 58%.