© The Authors 2025

1. Introduction

In the Hertzsprung–Russell diagram, δ Scuti stars exist at the intersection of the classical instability strip and the zero-age main sequence (Breger 1979; Rodríguez & Breger 2001). Characterized by short periods (< 0.3 d) and small peak-to-peak amplitudes, δ Scuti stars are A0- to F5-type intermediate-mass stars (Chang et al. 2013). They represent a transitional class, with temperatures ranging from 6900 to 8900 K and masses between 1.5 M_⊙ and 2.5 M_⊙. These pulsators bridge the gap between low-mass stars with radiative cores and convective envelopes and high-mass stars that are characterized by prominent convective cores and radiative envelopes (Bowman 2017), making them crucial for studying stellar structure and evolution history. In addition, δ Scuti stars occupy a transitional role in stellar pulsation, linking large-amplitude radial pulsators like Cepheids with non-radial, multi-period oscillators (Breger 2000).

Like other well-studied pulsating variables, namely Cepheids and RR Lyrae stars, the pulsation of δ Scuti stars is mainly excited by the κ mechanism (Baker & Kippenhahn 1965; Breger 1979), where the opacity varies as a function of temperature and density in the He II partial ionization zones near 48 000 K (Chevalier 1971). Most δ Scuti stars are multimode pulsators, with both radial (l = 0) and non-radial (l = 1,2,3,...) modes, as well as fundamental (n = 1) and overtone (n = 2, 3, 4,...) modes (Uytterhoeven et al. 2011), which has been discussed in many studies (Breger 2000; Guzik et al. 2019). δ Scuti stars with amplitudes above 0.3 mag in the V band are classified as high-amplitude δ Scuti stars (HADSs; Balona 2016), while others are typical δ Scuti stars (DSCTs). HADSs and DSCTs are different in many aspects, including pulsation mode and rotational speed (Pigulski et al. 2006).

Adhering to the Leavitt law, also known as the period-luminosity (P − L) relations, pulsating stars serve as crucial tools in the determination of cosmic distances. Among these, the P − L relations of Cepheids have been the most extensively studied and utilized for decades, forming the cornerstone of our understanding of extragalactic distance scales (Leavitt & Pickering 1912; Madore & Freedman 1991). These relations have also paved the way for groundbreaking discoveries in astrophysics, including exploration of the accelerated expansion of the Universe through observations of Cepheids and Type Ia supernovae (Riess et al. 1998, 2004) and the constraint on the Hubble constant (Pierce et al. 1994; Riess et al. 2018). As Population II stars, RR Lyrae stars also exhibit well-defined P − L relations, making them particularly valuable for tracing the distance to old populations, such as globular clusters (Cacciari & Clementini 2003; Braga et al. 2015) and dwarf galaxies in the Local Group (Vivas et al. 2022; Nagarajan et al. 2022).

Although δ Scuti stars also follow the P − L relations, they are less utilized as distance indicators due to their intrinsically fainter luminosities, which limit their applicability in extragalactic studies, and their small amplitudes, ranging from 0.003 to 0.9 mag (Breger 1979), that make them more challenging to detect in time-domain surveys. As shown in Fig. 1, δ Scuti stars exhibit lower absolute magnitudes and smaller amplitudes compared to Cepheids and RR Lyrae stars. Their pulsation modes are also more complex, resulting in less tight P − L relations. Despite these challenges, δ Scuti stars represent the second most populous class of pulsating stars in the Galaxy, surpassed only by pulsating white dwarfs (Breger 1979). This abundance, combined with their broad distribution across the Galactic disk and bulge, could play a significant role in studies of Galactic structure, stellar evolution, interstellar dust, and the calibration of distances to globular clusters (McNamara 2011).

Fig. 1. Comparison of the light curves of δ Scuti stars (Soszyński et al. 2021), Cepheids (Udalski et al. 2018), and RR Lyrae stars (Soszyński et al. 2014) obtained from the OGLE Collection of Variable Stars.

Over the past decade, the number of known δ Scuti stars has grown substantially thanks to the rapid advancement of modern large-scale time-domain surveys, and their P − L relations have been increasingly studied in recent years (McNamara 1997; Barac et al. 2022; Martínez-Vázquez et al. 2022). Chen et al. (2020) discovered ∼15 000 δ Scuti stars in the Zwicky Transient Facility (ZTF) Catalog of Periodic Variable Stars, while the All-Sky Automated Survey for SuperNovae (ASAS-SN) Catalogue of Variable Stars contains ∼8400 δ Scuti stars (Jayasinghe et al. 2020). The Transiting Exoplanet Survey Satellite (TESS) found ∼15 000 δ Scuti stars with its 30-minute cadence survey (Gootkin et al. 2024). The largest collections of δ Scuti stars so far have been provided by the Optical Gravitational Lensing Experiment (OGLE), which identified over 24 000 δ Scuti stars in the Galactic bulge and disk (Soszyński et al. 2021), as well as 15 000 in the Large Magellanic Cloud (Soszyński et al. 2023).

The growing availability of high-precision parallax measurements from Gaia has revolutionized this field, enabling unprecedented accuracy in the calibration of the P − L relations and thus enhancing their utility in various astrophysical applications. Looking ahead, the Legacy Survey of Space and Time (LSST) is expected to overcome some of the current observational challenges associated with δ Scuti stars. With its unparalleled photometric precision (∼10 mmag) and sky coverage (18 000 deg²; Abell et al. 2009), LSST will facilitate the detection of faint and low-amplitude sources such as δ Scuti stars, potentially unlocking broader applications for these stars in the future.

In this paper, we study the P − L relations of δ Scuti stars in the Tsinghua University–Ma Huateng Telescopes for Survey (TMTS) catalogs of δ Scuti stars (part of the TMTS catalogs of Periodic Variable Stars; Guo et al. 2024). The multiband observation data are presented in Sect. 2, and the P − L relations of δ Scuti stars are discussed in Sect. 3. In Sect. 4 we analyze the methodology used to apply the multiband P − L relations to estimate the color excess. Section 5 studies the influence of metallicity for the P − L relations. We summarize our work in Sect. 6.

2. Data

2.1. TMTS Catalogs of δ Scuti Stars

The TMTS is a multi-tube telescope system (40 cm optical telescope × 4) located at the Xinglong Station of National Astronomical Observatories, Chinese Academy of Sciences (NAOC), with a wavelength coverage of 400–900 nm. The TMTS has a field of view up to 18 deg² (Zhang et al. 2020). Exposing for 60 s, it can reach ∼19.4 mag in the white-light band (TMTS luminous filter, 3σ limitation). Having started monitoring the Large Sky Area Multi-object Fiber Spectroscopic Telescope (LAMOST) sky areas in 2020, the TMTS has covered an area of about 7000 deg² during the first three years of the survey. Due to its cadence as short as 1 minute, this instrument demonstrates significant potential to identify short-period variables. Such a high cadence is particularly suitable for the detection of δ Scuti stars, especially those with periods < 2 h, where rapid monitoring is essential to capture their variability. In TMTS catalogs of Periodic Variable Stars (Guo et al. 2024), we employed machine learning techniques to classify 11 638 periodic variable stars into 6 main types, including δ Scuti stars, eclipsing binaries and candidates of RS Canum Venaticorum stars. The subset of δ Scuti stars in this catalog, known as the TMTS catalogs of δ Scuti stars, includes 4876 DSCTs and 628 HADSs, which provide the data analyzed in this study.

To enhance the robustness of our analysis, we applied relatively strict selection criteria to our sample. Following the methods of Lin et al. (2023) and Guo et al. (2024), we used the Lomb-Scargle periodogram (LSP; Lomb 1976; Scargle 1982) to analyze the light curves collected by the TMTS. We calculated the maximum powers in LSP (LSP_max) and retained only sources with LSP_max > 10σ threshold. Furthermore, we cross-matched our samples with Gaia Data Release 3 (DR3; Prusti et al. 2016; Vallenari et al. 2023) using a matching radius of 2.0^″ to obtain parallax measurements(ϖ) and included only sources with ϖ/σ_ϖ ≥ 10.0.

Since δ Scuti stars exhibit multimode pulsations, with different pulsation modes adhering to distinct P − L relations, the following step was to carefully classify the variable stars according to their pulsation modes. We obtained the color excess of each source from the 3D dust maps using the DUSTMAPS package (map version = “bayestar19”, mode = “mean”; Green 2018; Green et al. 2019). With the help of the CDS-Xmatch service, we performed a cross-match with the Two Micron All Sky Survey (2MASS; Cutri et al. 2003; Skrutskie et al. 2006) and AllWISE Data Release from the Wide-field Infrared Explorer (WISE; Wright et al. 2010; Cutri et al. 2021) with a matching radius of 2.0^″. 2MASS obtained full-sky photometric data in three infrared bands: J (1.235 μm), H (1.662 μm), and K_s (2.159 μm). WISE also gathered full-sky photometry across four infrared bands, and we selected the three shorter wavelength bands: W1 (3.368 μm), W2 (4.618 μm), W3 (12.082 μm).

We then derived the Wesenheit indexes (Madore 1982; Lebzelter et al. 2018), as the P − L relations in the absolute Wesenheit magnitudes are better defined (Soszyński et al. 2005):

$$\begin{array}{c}\hfill W_{\mathit{JK}}=K_S-0.686(J-K_S).\end{array}$$(1)

HADSs predominantly pulsate in the fundamental mode, while DSCTs display multimode pulsations, oscillating in both the fundamental mode and various overtone modes (McNamara 2011). Therefore, we first established a fundamental mode P − L relation using the HADSs in our catalog. Outliers in the HADS samples were examined, and sources with low signal-to-noise ratios were excluded to enhance the reliability of the relation.

For DSCTs, we calculated the perpendicular distances of each star from this fundamental mode P − L relation. The resulting distribution of these distances is shown in Fig. 2, which reveals a distinct bimodal structure, as in Ziaali et al. (2019), Jayasinghe et al. (2020), and Barac et al. (2022): a primary peak near 0, indicative of the fundamental mode; and a secondary peak around −0.26, associated with the overtone mode. Based on these findings, we classified DSCTs into three categories: fundamental mode pulsators (with distances between −0.146 and 0.4), overtone mode pulsators (with distances of less than −0.146), and rejected pulsators (with distances greater than 0.4). Examination of the rejected pulsators revealed that these sources typically possess low-quality light curves, which limits their reliability for further analysis. Nonetheless, accurately differentiating between various overtone pulsators (e.g., first-overtone, second-overtone, etc.) only through light curve analysis proves challenging. However, these distinctions are essential for studying the P − L relations, as each overtone mode relates to its own P − L relation. To improve the precision of the determination of the P − L relations, we therefore focused exclusively on fundamental mode δ Scuti stars.

Fig. 2. Distribution of the perpendicular distances to the initial fundamental mode Wesenheit WJKP − L relation for different types of pulsators. Fundamental mode pulsators are depicted in red, overtone-mode variables in blue, and candidates excluded from the analysis in gray.

We acknowledge that in the double-Gaussian distribution shown in Fig. 2, some overtone-mode pulsators may “invade” the peak of fundamental-mode pulsators. This overlap contributes to deviations from a tight P − L relation.

2.2. Optical and infrared photometry

We cross-matched the δ Scuti stars in our catalog with the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) Data Release 1 (PS1 DR1; Kaiser et al. 2002; Chambers & Magnier 2016) using a matching radius of 2.0^″ for optical photometry. Pan-STARRS collected photometry in five optical bands: g (4866 Å), r (6215 Å), i (7545 Å), z (8679 Å), and y (9633 Å). The entire cross-matching process provided each source with up to 11-band photometry, including grizy bands from PS1 DR1, JHK bands from 2MASS and W1 − W3 bands from AllWISE. Table A.1 shows a sample of our dataset. For Pan-STARRS, 2MASS, and AllWISE, each source has photometric measurements from approximately 13, 45 and 2 epochs, respectively, providing a robust temporal sampling of the light curves. The one benefit with small peak-to-peak amplitude pulsators like δ Scuti stars is that the “mean magnitude” – which can be a challenge to determine with sparse sampling of large amplitude variables – is closely measured in a single epoch. Of course, the periods are still determined from the high-precision photometry from TMTS, which continuously monitors the LAMOST sky areas throughout the night with a 1-minute cadence. This observation strategy ensures collections of uninterrupted light curves, each obtained from a single night of observations. The short cadence and uninterrupted light curves from the TMTS enable precise period determination, providing valuable constraints for the P − L relation.

To ensure the precision of our fitting, we selected only sources with photometry available in more than four bands. The final dataset comprises 1864 fundamental mode δ Scuti stars, of which 1567 are newly discovered by the TMTS, i.e., not recorded by the International Variable Star Index (VSX; Watson et al. 2006). A total of 16 781 photometric data were recorded. As illustrated in Fig. 3, approximately 99% of these variables have photometric measurements in six or more bands, with more than half of them having photometry in ≥10 bands.

Fig. 3. Number of photometric measurements of the δ Scuti stars in our dataset, as derived through the cross-matching process. More than 50% of the sources have photometric measurements across over ten bands.

Figure 4 shows the period distribution of the sources in our dataset. Compared with other catalogs, our dataset has a larger portion of variables with extremely short periods (i.e., P ≤ 2 h), which adds a longer lever arm to aid in the inference of the zero points of the P − L relations.

Fig. 4. Periods of the δ Scuti stars in our dataset. The distribution exhibits a peak around 0.1 days (2.4 hours).

3. Period–luminosity relations

In many studies like McNamara (2011) and Jayasinghe et al. (2020) of the P − L relations of δ Scuti stars, the P − L relation for each band are determined separately, with the distance modulus and color excess (which are critical in calculating the absolute magnitudes) for each variable assumed to be fixed, even though they are essentially variables with probability distributions. For a δ Scuti star with a Gaia parallax (ϖ) of 0.7 mas (the average in our dataset) and a parallax-over-error ratio of 10 – corresponding to a parallax uncertainty σ_ϖ = 0.07 mas – the resulting uncertainty in the distance modulus is ∼0.22 mag. This exceeds typical photometric uncertainties and is larger than the intrinsic scatter in most bands of the P − L relations. Consequently, uncertainties in the distance modulus have a non-negligible impact on the determination of P − L relations and are necessary to be accounted for in the analysis.

Using the Python Package PyMC (Patil et al. 2010; Abril-Pla et al. 2023), we can simultaneously fit the multiband P − L relations, extinction, and distances of the δ Scuti stars in a Bayesian context. Designed for probabilistic programming, PyMC has extensive support for prior distributions, Markov chain Monte Carlo sampling methods, and comprehensive model diagnostics. Within this framework, the distance modulus and the color excess due to dust are modeled as random variables with prior distributions. These priors are derived from Gaia DR3 and DUSTMAPS (Green et al. 2019), respectively. By incorporating multiband photometry to impose cross constraints, this approach not only tightens the P − L relations but also refines the uncertainties in the distance modulus and color excess through the posterior distributions.

Absolute magnitude is defined as

$$\begin{array}{c}\hfill M_{i,j}=m_{i,j}-{\mu }_i-A_j,\end{array}$$(2)

where i refers to the ith source and j refers to the jth band. M_i, j (m_i, j) is the absolute (apparent) magnitude of the ith source in the jth band. μ_i represents the distance modulus of the ith source, and A_j is the interstellar extinction in the jth band. The Cardelli, Clayton, and Mathis extinction law (Cardelli et al. 1989) gives the relationship between the interstellar extinction in the jth band and that in the V band:

$$\begin{array}{c}\hfill ⟨A_j/A_V⟩=a_j+b_j/R_V,\end{array}$$(3)

where the coefficients a_j and b_j are wavelength functions. Given R_V = A_V/E(B − V), A_j can be written as a function of the color excess E(B − V):

$$\begin{array}{cc}\hfill A_j& =A_V(a_j+b_j/R_V)=R_{V}E(B-V)(a_j+b_j/R_V)\hfill \\ \hfill & =E(B-V)(a_j{R}_V+b_j).\hfill \end{array}$$(4)

The relationship between period and absolute magnitude (the P − L relation) is

$$\begin{array}{c}\hfill M_{i,j}=M_{0,j}+{\alpha }_j{log}_{10}(P_i/P_0),\end{array}$$(5)

where α_j and M_0, j are the slope and intercept of the P − L relation in the jth band. P_i refers to the period of the ith source and P₀ is the mean period of all δ Scuti stars in our dataset, calculated to be 2.23 hours.

Substituting Eqs. (2) and (4) into Eq. (5) gives

$$\begin{array}{c}\hfill m_{i,j}={\mu }_i+M_{0,j}+{\alpha }_j{log}_{10}(P_i/P_0)+E{(B-V)}_i(a_j{R}_V+b_j)+{ϵ}_{i,j},\end{array}$$(6)

where ϵ_i, j is a zero-mean independently and identically distributed Gaussian random variable, with ϵ_i, j² = σ_{m, (i, j)}² + σ_{intrinsic, j}². σ_{m, (i, j)} is the observed photometric error and σ_{intrinsic, j} is the intrinsic scatter of the P − L relation in a given band.

The inclusion of intrinsic scatter is a common approach in Bayesian statistics to account for variability beyond the measurement uncertainties. Ideally, the scatter in observation data should be fully explained by the measurement errors, σ_m. However, in many cases, the observed dispersion exceeds what measurement uncertainties alone predict. To account for this discrepancy, an intrinsic scatter term (σ_int), is introduced, while this term does not include the uncertainties in distance modulus, E(B − V), or photometry. Instead, it captures model imperfection (for example, the P − L relation of δ Scuti stars is not a perfect linear relation due to complex pulsation mechanisms and stellar rotations) and un-modeled data uncertainty (including systematic errors in photometric measurements and uncertainties in median flux determination). Without invoking such an intrinsic scatter term, several issues can arise, such as underfitting – when the model fails to capture the full dispersion of the data, leading to underestimated uncertainties – and biased parameter estimates, as the fitting process can artificially minimize the residuals, potentially resulting in overfitting.

In the least-square fitting (LSF), the post-fit residual σ_LSF is computed by encompassing both measurement uncertainties and intrinsic scatter, but without distinguishing between the above two sources. In contrast, the Bayesian method explicitly incorporates intrinsic scatter as a parameter that is optimized during the fitting process, enabling a more precise decomposition of the observed scatter into measurement errors and intrinsic variability of the P − L relation.

As in Klein & Bloom (2014), we constructed a sparse matrix X to accommodate the simultaneous fitting of the 11-band P − L relations, where matrix multiplication is used to implement Eq. (6):

$$\begin{array}{c}\hfill m=X·b,\end{array}$$(7)

where X is a 16 781 × 3750 matrix, and each row corresponds to an individual photometric measurement. The 3750 columns represent the parameters of interest: 1864 distance modulus, 1864 color excess values (corresponding to 1864 sources), along with 11 α and 11 M₀ (corresponding to 11 bands), yielding a total of 1864 + 1864 + 11 + 11 = 3750. The vector m contains the 16 781 photometric measurements, while the parameter vector b represents the random variables, initially assigned with prior distributions for each δ Scuti star’s distance modulus and color excess, as well as for the α, M₀ and intrinsic scatters of the 11 bands.

The prior distributions are as follows:

$$\begin{array}{cc}& {\mu }_{i,\text{Prior}}\sim \mathcal{N}({\mu }_{\mathit{gaia}},{\sigma }_{\mathit{gaia}}^2)\hfill \\ \hfill & E{(B-V)}_{i,\text{Prior}}\sim \mathcal{N}(E{(B-V)}_{\mathrm{dustmaps}},{\sigma }_{\mathrm{dustmaps}}^2),\hfill \\ \hfill & E{(B-V)}_{i,\text{Prior}}\ge 0\hfill \\ \hfill & {\alpha }_{j,\text{Prior}}\sim \mathcal{N}(0,5^2)\hfill \\ \hfill & M_{0,j,\text{Prior}}\sim \mathcal{N}(0,3^2).\hfill \end{array}$$(8)

The DUSTMAPS library does not directly provide the standard error of the color excess for a given coordinate. To estimate this uncertainty, we employed the “sample” mode and obtained σ_dustmaps by calculating the standard error from several samples. For α and M₀, we adopted a broad normal prior distribution to avoid imposing unnecessary constraints on the estimation.

Using PyMC, we trained a model to obtain posterior estimates of all random variables. The PyMC model ran in 5000 steps, until all the variables converged. To assess convergence, we adopt the Gelman-Rubin diagnostic ($\widehat{R}$; Gelman & Rubin 1992). Convergence is indicated when $\widehat{R}<1.1$, suggesting that the chains have stabilized. The closer $\widehat{R}$ is to 1, the stronger the evidence of convergence. $\widehat{R}$ is computed as follows:

$$\begin{array}{c}\hfill \widehat{R}=\sqrt{\frac{(1-\frac{1}{n})W+\left(\frac{1}{n}\right)B}{W}},\end{array}$$(9)

where W represents the variance within the chain, B is the variance between chains, and n denotes the length of the chain. For our analysis, all variables yielded $\widehat{R}=1$, indicating robust convergence. Figure 5 shows the trace plots and posterior density distributions for the variables in our model. For all parameters, the trace plots illustrate consistent horizontal bands without notable trends, indicating good mixing and convergence. The density plots also reveal well-defined distributions with minimal multi-modality, suggesting reliable parameter estimations. Figure A.1 presents the contour plots for the joint posterior distributions of α and M₀ in the 11 bands. The contour lines approach a circular shape, indicating a weak correlation between α and M₀, which implies that the model constrains these parameters relatively independently and the result is robust.

Fig. 5. Posterior density plots indicating the range and relative likelihood (left) and trace plots showing the values of each parameter over the course of iterations (right) of μ, E(B − V), α, and M0. In the upper two panels, each color represents a distinct δ Scuti star; in the lower two panels, each color corresponds to a different photometric band. The stability of the distributions indicates that the chains have achieved satisfactory convergence.

The multiband P − L relations of δ Scuti stars are shown in Fig. 6. The zero points of the P − L relations are vertically offset to visually separate each relation, enabling a clear graphical comparison. The solid black lines represent the best-fitting P − L relations. The fitted α, M₀ and σ_intrinsic of each band are summarized in Table 1. Note that the scatter in the δ Scuti P − L relation is larger than that of Cepheids and RR Lyrae stars (i.e., ∼0.1–0.2 mag; Fouque et al. 2007; Madore et al. 2013; Dambis et al. 2014; Trahin et al. 2021). The pulsation modes of δ Scuti stars are complex due to multiple reasons, including multi-modal oscillation, excitation mechanism, stellar structure, and rapid rotation. All these factors contribute to the intrinsic scatter, making the P − L relations less tight. However, it is important to note that these effects, along with the overtone “invaders” discussed in Sect. 2.1, have been accounted for by the intrinsic scatter term introduced in our model.

Fig. 6. P − L relations of δ Scuti stars in the 11 bands, as denoted by different colors. The zero points of the P − L relations are vertically shifted to visually separate each relation. The solid black lines represent the best-fitting P − L relations.

Figure 7 compares our results with those of Laney et al. (2002), McNamara (2011), Ziaali et al. (2019), Jayasinghe et al. (2020), Poro et al. (2021), Barac et al. (2022), and Poro et al. (2024). For comparison purposes, we reformulated Eq. (5) as M_i, j = M_0, j + α_jlog₁₀(P_i), where periods are expressed in days rather than hours. As our sample consists of Galactic δ Scuti stars, and potential differences may exist between Galactic and extragalactic P − L relations, we have limited our comparison to studies on Galactic δ Scuti stars. Our results align well with previous studies. We further compared the P − L relations of δ Scuti stars with Cepheids in the J, H, and K_s bands, and with RR Lyrae stars in the W1, W2, and W3 bands. As shown in Fig. 8, δ Scuti stars are characterized by intrinsically shorter periods and lower luminosities compared to Cepheids and RR Lyrae stars. These properties inherently complicate their identification and systematic study, and underscore the need for next-generation time-domain survey like LSST with better photometric precision, as discussed in Sect. 1.

Fig. 7. Comparison of the P − L relations’ slope (α) and intercept (M0) of this paper with those from Laney et al. (2002), McNamara (2011), Ziaali et al. (2019), Jayasinghe et al. (2020), Poro et al. (2021), Barac et al. (2022), and Poro et al. (2024). Our results align well with previous investigations.

Fig. 8. Comparison of the P − L relations of δ Scuti stars and Cepheids in the J (upper left), H (upper middle), and Ks (upper right) bands, alongside the comparison of δ Scuti stars and RR Lyrae stars in the W1 (lower left), W2 (lower middle), and W3 (lower right) bands. The period range for δ Scuti stars spans 0.02–0.25 day (Chang et al. 2013), for Cepheids 3–100 days (Persson et al. 2004), and for RR Lyrae stars 0.2–1 day (Lafler & Kinman 1965).

We performed a comparative analysis between the prior and posterior distributions of the distance modulus and E(B − V) for the δ Scuti stars. As illustrated in Fig. 9, the posterior distributions align with the priors, suggesting that the model does not introduce excessive overfitting. This alignment confirms that the model predictions are data-driven while remaining consistent with prior information. However, we observe a slight systematic deviation at larger distance, where the posterior distribution appears marginally smaller than the prior, as shown in the left bottom panel of Fig. 9. To investigate this, we compared our posterior distribution with the results from Bailer-Jones et al. (2021), which combines Gaia DR3 parallax likelihoods with a prior based on a 3D Galactic model to improve stellar distance estimates. As shown in the left panel of Fig. 10, μ_Post − μ_{Bailer − Jones} exhibits a consistent pattern across the entire distance range, with similar trends at both the near and far ends, indicating that the observed bias discussed above disappears when the more sophisticated Galactic prior is incorporated. This is confirming evidence that the methodology does indeed lead to reasonable distance estimates.

Fig. 9. Comparisons of the prior and posterior distributions of the distance modulus (left) and E(B − V) (right), with residuals shown in the lower panels.

Fig. 10. Comparisons of the residuals of the distance modulus derived from Bailer-Jones et al. (2021) with our posterior distributions (left) and with Gaia DR3 (right).

Note that there is another deviation in Fig. 10: our posterior distances are slightly larger than those from Bailer-Jones et al. (2021) over the entire range. This is likely due to that the Gaia DR3 distance modulus, which serve as our prior, are systematically larger than those from Bailer-Jones et al. (2021), particularly at greater distances, as shown in the right panel of Fig. 10. Since Bayesian methods inherently follow the prior, it is expected that our posterior – based on Gaia DR3 – would be systematically larger than those from Bailer-Jones et al. (2021). It is worth noting that the “systematic deviation” seen in the left panel of Fig. 9 in turn demonstrates the effectiveness of our approach, as the distance modulus “should be” smaller under the more sophisticated Galactic prior (Bailer-Jones et al. 2021), especially at larger distance, and our model successfully adjusted the posterior toward this expectation.

Furthermore, we examine the reduction in uncertainties between the prior and posterior distributions of the distance modulus and E(B − V), as shown in Fig. 11. The comparison reveals a substantial reduction in error for both the distance modulus and E(B − V) in the dataset, with particularly pronounced improvements for measurements with higher uncertainties in the prior distribution. This is evident from the significantly lower peaks in the violin plots of the posterior distributions compared to the priors. Previous studies, such as Sesar et al. (2017), have utilized the period–luminosity–metallicity (P − L − Z) relations of pulsating stars within a Bayesian framework to constrain the parallax. This result underscores the effectiveness of our methodology in refining parameter estimates, as our framework not only preserves prior information but also incorporates observational data to provide tighter constraints on uncertainties.

Fig. 11. Violin plots depicting the prior and posterior error distribution for the distance modulus (left) and E(B − V) (right). The substantive depression in posterior distribution peaks relative to prior distributions shows a great error mitigation, particularly in measurements with larger initial uncertainties.

4. Application: Estimation of 3D dust distribution and distance

3D dust maps provide insights into Galactic structure and enable precise corrections for dust extinction and reddening (Green et al. 2014). The development of 3D dust maps has been revolutionized by advancements in large-scale, multiband photometric surveys like 2MASS and WISE, which provide extensive photometric data on stars across various wavelengths. Combined with precise stellar distance measurements, such as those derived from Gaia, these surveys enable the creation of high-resolution maps that trace the spatial distribution of dust with great precision (Leike et al. 2020; Lallement et al. 2022).

The implications of 3D dust maps extend beyond extinction correction. These maps deepen our understanding of Galactic structure and evolution, revealing features such as spiral arms shaped by the distribution of interstellar matter through reconstruction of the 3D distribution of dust in the Galaxy (Schultheis et al. 2014; Kh et al. 2018). Dust maps also provide crucial insights into the complex interplay between interstellar dust and star formation, as grain surfaces are important catalytic sites for key chemical reactions, and dust is a fundamental component of molecular clouds where stars are born (Bialy et al. 2021; Dharmawardena et al. 2022). Furthermore, they aid in the calibration of cosmological observations by refining our models of the foreground extinction that affects extragalactic studies (Zasowski et al. 2019; Chiang & Ménard 2019).

Currently available 3D dust maps, such as those presented in Green et al. (2019), Lallement et al. (2022), and Edenhofer et al. (2024), primarily rely on systematic comparisons between observed stellar colors, which are affected by interstellar extinction, and intrinsic stellar colors predicted by stellar types. By calculating color indices across multiple photometric bands, incorporating detailed stellar parameters such as effective temperature, surface gravity and metallicity, and building models through the Gaussian process – a robust framework has already been established for quantifying extinction at scale.

A notable application of our methodology is its potential to serve as an alternative or complementary approach for estimating color excess, thereby expanding existing techniques. Our method facilitates inference of E(B − V) by leveraging the multiband P − L relations of δ Scuti stars (and can be extended to other types of pulsating stars). Using the well-characterized photometric and pulsation properties of these pulsators, our framework provides an independent avenue for deriving interstellar reddening, thereby offering reliability in the estimates of dust map.

Equation (6) can be reformulated as

$$\begin{array}{c}\hfill m_{i,j}-{\mu }_i-M_{0,j}-{\alpha }_j{log}_{10}(P_i/P_0)=E{(B-V)}_i(a_j{R}_V+b_j)+{ϵ}_{i,j}.\end{array}$$(10)

Using the fitted parameters α and M₀ for all 11 bands in combination with the photometry and distance modulus of a given source as before, we can estimate the color excess of that source using PyMC. To avoid overlap between the dataset used to derive the P − L relation parameters and that used to predict the color excess, we randomly divided our dataset into a training set (70%) and a test set (30%). From the training set, we determined α and M₀ for each band using 5000 sampling iterations. For posterior estimation, instead of employing a normal prior distribution as outlined in Eq. (8), we adopted a broad uniform distribution:

$$\begin{array}{c}\hfill E{(B-V)}_{i,\mathrm{Prior}}\sim \mathrm{Uniform}(0,2).\end{array}$$(11)

This approach simulates a scenario in which no prior extinction information is available, thereby reducing the influence of preexisting assumptions. It provides a robust framework for testing the intrinsic capability of the model to independently infer E(B − V), ensuring that the results are derived solely from the underlying data and the methodological principles.

To enhance the reliability of our results, we focus exclusively on variables with photometric measurements in at least nine bands. As shown in Fig. 3, approximately 60% of the sources in our dataset fulfill this criterion. Figure 12 compares the posterior distribution of E(B − V) with the values obtained from DUSTMAPS (version = “bayestar19”; Green et al. 2019), showing a close alignment between the two. This shows that we are able to estimate E(B − V) independently without clear bias, highlighting the reliability of our framework.

Fig. 12. Estimation of E(B − V) utilizing multiband P − L relations of δ Scuti stars on held-out test sources. Blue, orange, and green squares represent sources with 9, 10, and 11 photometric measurements, respectively. The x-axis shows the E(B − V) values derived from the 3D dust map from Green et al. (2019), while the y-axis displays the corresponding estimates obtained from the application of our model, using an uninformative color excess prior.

Note that the P − L relations of δ Scuti stars are not perfectly linear due to various reasons discussed above. As a result, the estimation of E(B − V) based on these relations is subject to uncertainties arising from this imperfection. Consequently, the dust maps produced using our approach may not achieve the same level of precision as those given by Green et al. (2019). However, the intrinsic scatter is relatively small in most bands, suggesting that, while various factors related to the physical properties of δ Scuti stars contribute to the dispersion of the P − L relations, their impact remains within an acceptable range.

In addition, incorporating multiband photometry strengthens our constraints on dust extinction compared to single-band P − L relation-based methods. Since dust extinction is the only free parameter for a given δ Scuti star during the fitting process, as in Eq. (10), increasing the number of photometric bands enhances the observational constraints because of reducing degrees of freedom. Therefore, despite the existence of intrinsic scatter in the P − L relations, our extinction estimates remain reliable within meaningful limits.

Despite these limitations, our method offers a unique advantage: the simultaneous inference of both the distance modulus and dust extinction, which enables us to extend the dust map beyond the limits of Gaia parallax measurements. The 3D dust maps rely on distance measurements, with most current dust maps being primarily based on Gaia parallaxes, which remain reliable (ϖ/σ_ϖ ≥ 10.0) only within a few kiloparsecs. For sources lacking reliable Gaia parallax measurements (e.g., 10.0 ≥ ϖ/σ_ϖ ≥ 5.0, or even worse), we can constrain their distances and dust extinction through the multiband P − L relations. As an example, we simultaneously estimated the distance modulus and dust extinction for TMTS J00572464+5750191, a source with ϖ/σ_ϖ ≈ 7. Figure 13 presents the contour plot for the joint posterior distributions of the distance modulus and E(B − V) for this source, and Fig. 14 compares the prior and posterior distribution of its distance modulus, demonstrating a significantly improved constraint.

Fig. 13. Contour plots for the joint posterior distributions of the distance modulus and E(B − V) for TMTS J00572464+5750191. The expected anticorrelation between extinction and distance is evident.

Since δ Scuti stars are widely distributed throughout the Galaxy, and the detection depth of large-scale surveys such as LSST (5σ depth for r-band = 24.3 mag; Ivezić et al. 2019) and the Wide Field Survey Telescope (WFST; 5σ depth for r-band > 21 mag; Lin et al. 2024) has greatly increased, multiband photometry of distant δ Scuti stars is becoming accessible. Their P − L relations would offer an independent means to refine measurements of distance modulus and dust extinction. This capability enables us to expand the spatial coverage of 3D dust maps to more distant regions, where Gaia parallaxes become less reliable, highlighting the critical role of pulsating stars in constructing more comprehensive dust maps.

In the future, the large number of δ Scuti stars identified by LSST will facilitate the construction of 3D dust map through their P − L relations. The single-visit depth of LSST is 24.3 mag, and the typical absolute magnitude is M_0, r = 1.83 mag. As such, the extinguished maximum distance at which LSST can detect δ Scuti stars is 312 kpc, which far exceeds the stellar locus Galaxy. Even with 7.5 mag of r-band extinction toward the Galactic plane, δ Scuti stars can be detected by LSST to ∼8 kpc (though stellar crowding may ultimately be limiting). Roughly scaling from the total number of stars expected to be detected in LSST, we estimate that LSST will ultimately discover and characterize a few tens of millions of δ Scuti stars. Figure 15 compares the dust extinction derived from the map of Green et al. (2019) with δ Scuti stars in our dataset, focusing on the range between 1 and 3 kpc.

Fig. 14. Comparison between the prior distance modulus distribution (derived from Gaia DR3) and the posterior distribution, demonstrating a substantial improvement in the precision of the latter.

Fig. 15. E(B − V) values between 1 and 3 kpc estimated from the dust map of Green et al. (2019) (left) and δ Scuti stars in our dataset (right). The size of each data point represents the distance to the source, with larger points corresponding to closer sources.

Furthermore, it is worth noting that as this approach is inherently based on the P − L relations of pulsating stars, it is not restricted to δ Scuti stars; but is broadly applicable to any pulsating stars that follow the P − L relations, including Cepheids and RR Lyrae stars, whose P − L relations are tighter and better-studied. Given the widespread presence of pulsating stars across the Galaxy, this underscores the potential of our method for refining and expanding dust maps into the outer regions of the Galaxy. This capability offers great opportunity for studying the 3D structure of interstellar dust in the Milky Way, offering insights into the spatial variation of extinction and the large-scale properties of the interstellar medium.

Leveraging pulsating stars as distance and extinction tracers, our 3D dust maps enable the exploration of more distant regions, greatly broadening their applicability. For example, we might be able to explore low-surface-brightness substructures with these comprehensive dust maps, such as Milky Way satellite galaxies with tidal tails (Bournaud et al. 2004), which arise from galaxy interactions. The mapping of dust and gas distributions offers critical insights into the dynamical and chemical characteristics of these substructures, shedding light on their formation mechanisms, the influence of tidal forces and the distribution of dark matter (Hozumi & Burkert 2015). Furthermore, dust maps covering greater distances facilitates the identification of obscured star-forming regions within these systems (Knierman et al. 2013), enabling a deeper investigation of the interactions among dust, gas, and dark matter that govern their evolutionary pathways.

5. Period–luminosity–metallicity relations

The study of metallicity influence on the P − L relations of pulsating stars is important for enhancing the precision of various astrophysical measurements, as metallicity represents a potential source of uncertainty in the P − L relations (De Somma et al. 2022). Metallicity dependence can affect distance determinations and the reconstruction of reliable dust maps across different regions of the Galaxy, highlighting the need to account for these effects.

Some studies have analyzed the effect of metallicity on the P − L relations of δ Scuti stars (i.e., the P − L − Z relations). For example, Antonello & Mantegazza (1997) investigated the impact of metallicity on the P − L relations of δ Scuti stars and concluded that these relations remain consistent regardless of metallicity, in contrast to the findings of Nemec et al. (1995). Cohen & Sarajedini (2012) incorporated SX Phe stars (a subtype of δ Scuti stars with lower metal abundances) into the determination of the P − L relations. Their results demonstrated great consistency with the relations derived from metal-rich pulsators, suggesting that metallicity has a minimal impact on the P − L relations. McNamara (2011) treated the influence of metallicity as a correction term to absolute bolometric magnitudes, incorporating a linear metallicity-dependent term into the P − L relations to account for its impact.

Theoretically, metals contribute to the opacity in the outer layer of a star, affecting the radiative transfer processes within the star, which can modify the internal structure of the pulsator and nuclear fusion rates (Lenz et al. 2008; Montalbán & Miglio 2008). Metal-rich stars are generally less luminous and cooler compared to metal-poor stars at the same age and mass (Mowlavi et al. 1998), thus introducing a metallicity dependence in the P − L relation.

We investigated the influence of metallicity on the P − L relations using the metal abundance value obtained from LAMOST DR7. Among the δ Scuti stars in our dataset, 494 have [Fe/H] measurements from LAMOST. Following the approach of McNamara (2011), we incorporated the effect of metallicity as a linear term into Eq. (6), resulting in the P − L − Z relation:

$$\begin{array}{cc}\hfill m_{i,j}& ={\mu }_i+M_{0,j}+{\alpha }_j{log}_{10}(P_i/P_0)+E{(B-V)}_i(a_j{R}_v+b_j)\hfill \\ \hfill & +{\beta }_j{[Fe/H]}_i+{ϵ}_{i,j},\hfill \end{array}$$(12)

where β represents the coefficient of the metallicity term. As summarized in Table 2, metallicity appears to exert a minimal influence on the P − L relations of δ Scuti stars. All β coefficients remain close to zero within 2σ significance, and are smaller than the intrinsic scatter of their respective photometric bands, indicating that the impact of metallicity is statistically insignificant and difficult to separate from other contributing factors, such as photometric uncertainties and the inherent scatter in the P − L relations. These findings suggest that not including metallicity in the determination of P − L relations might not compromise precision, aligning with the results of many previous studies.

We conducted a comparison of the intrinsic scatter between the P − L relations and the P − L − Z relations across various photometric bands, as shown in Fig. 16. Our results indicate that incorporating metallicity seems to reduce the intrinsic scatter at longer wavelengths – as shown in Fig. 16, where the red line lies below the blue line in the H, K_s, W1, W2, and W3 bands, typically by less than 0.01 mag. At shorter wavelengths, however, the impact of metallicity is less consistent, with the intrinsic scatter showing both increases and decreases. However, the observed effects of metallicity fail to achieve a 3σ significance level in almost all bands (as illustrated by the shaded region of Fig. 16), suggesting that the trend is not statistically robust. Furthermore, all the β remain smaller than the intrinsic scatter itself, highlighting the possibility that the observed changes could be attributed to statistical fluctuations rather than to the physical impact of metallicity on the P − L relations. More studies are required to understand the impact of metallicity on δ Scuti stars, including larger sample sizes and more detailed modeling of other confounding factors, such as the influence of age, binarity, or rotational effects on pulsation properties.

Fig. 16. Intrinsic scatter in the P − L relation (blue) and P − L − Z relation (red) as a function of wavelength. Incorporating metallicity appears to reduce the intrinsic scatter at longer wavelengths.

6. Summary

With a cadence of 1 minute, the TMTS has proven to be a powerful tool for identifying short-period variable stars. Leveraging the dataset of 1864 fundamental-mode δ Scuti stars from the TMTS catalogs of Periodic Variable Stars, we developed an innovative methodology for simultaneously determining the multiband P − L relations of these pulsators. By cross-matching TMTS sources with Pan-STARRS1, 2MASS, and WISE for optical and infrared photometry, our approach has achieved highly precise P − L relations spanning multiple photometric bands. Additionally, this methodology facilitates the refinement of the distance modulus and color excess within the posterior distribution.

A notable application of this methodology lies in its capability to estimate the color excess independently. Derived without prior information of dust extinction, our E(B − V) estimates are in good agreement with results from state-of-the-art 3D dust maps. This demonstrates the utility of our approach as a complementary method for extinction estimation. This extensible method could also potentially be used to estimate dust extinction with other types of pulsating stars that follow well-studied P − L relations, such as Cepheids and RR Lyrae stars. When parallax measurements are not reliable or even unavailable, this method can be used to determine distances to δ Scuti stars. More numerous than Cepheids and RR Lyrae, δ Scuti stars could become the de facto sources for distance measurements to Galactic substructures, such as tidal tails.

We also investigated the influence of metallicity on the P − L relations of δ Scuti stars by incorporating it as a linear term in the fitting. Our findings suggest that the inclusion of metallicity can reduce intrinsic scatter at longer wavelengths, potentially refining the precision of the P − L relations in these bands. However, the observed effects do not reach 3σ statistical significance, with the metallicity coefficients remaining within the intrinsic scatter of the respective bands, making it challenging to distinguish these effects from statistical fluctuations. Further investigation is required to conclusively evaluate the physical impact of metallicity on these relations.

Data availability

The full Table A.1 is available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/699/A115

Acknowledgments

This work is supported by the National Science Foundation of China (NSFC grants 12033003 and 12288102), the Ma Huateng Foundation and New Cornerstone Science Foundation through the XPLORER PRIZE. J.L. is supported by the National Natural Science Foundation of China (NSFC; Grant Numbers 12403038), the Fundamental Research Funds for the Central Universities (Grant Numbers WK2030000089), and the Cyrus Chun Ying Tang Foundations. This work is supported by the National Natural Science Foundation of China (NSFC) grant 12373031, the Joint Research Fund in Astronomy (U2031203) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences (CAS), and the NSFC grants 12090040, 12090042. This work is also supported by the CSST project (CCST-A12). We acknowledge the support of the staffs from Xinglong Observatory of NAOC during the installation, commissioning, and operation of the TMTS system. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories of the Chinese Academy of Sciences.

References

Appendix A: Supplementary table and plot

Fig. A.1. Contour plots for the joint posterior distributions of α and M0 across the 11 photometric bands (g, r, i, z, y, J, H, Ks, W1, W2, and W3). For each band, the bottom-left panel shows the contour of α versus M0, while the marginal posterior distributions for the two variables are displayed on the top-left and bottom-right panels, respectively. The dashed vertical lines in the marginal distributions represent the 1σ credible intervals.

All Tables

All Figures

	Fig. 1. Comparison of the light curves of δ Scuti stars (Soszyński et al. 2021), Cepheids (Udalski et al. 2018), and RR Lyrae stars (Soszyński et al. 2014) obtained from the OGLE Collection of Variable Stars.
In the text

	Fig. 2. Distribution of the perpendicular distances to the initial fundamental mode Wesenheit WJKP − L relation for different types of pulsators. Fundamental mode pulsators are depicted in red, overtone-mode variables in blue, and candidates excluded from the analysis in gray.
In the text

	Fig. 3. Number of photometric measurements of the δ Scuti stars in our dataset, as derived through the cross-matching process. More than 50% of the sources have photometric measurements across over ten bands.
In the text

	Fig. 4. Periods of the δ Scuti stars in our dataset. The distribution exhibits a peak around 0.1 days (2.4 hours).
In the text

Fig. 5. Posterior density plots indicating the range and relative likelihood (left) and trace plots showing the values of each parameter over the course of iterations (right) of μ, E(B − V), α, and M0. In the upper two panels, each color represents a distinct δ Scuti star; in the lower two panels, each color corresponds to a different photometric band. The stability of the distributions indicates that the chains have achieved satisfactory convergence.

In the text

	Fig. 6. P − L relations of δ Scuti stars in the 11 bands, as denoted by different colors. The zero points of the P − L relations are vertically shifted to visually separate each relation. The solid black lines represent the best-fitting P − L relations.
In the text

	Fig. 7. Comparison of the P − L relations’ slope (α) and intercept (M0) of this paper with those from Laney et al. (2002), McNamara (2011), Ziaali et al. (2019), Jayasinghe et al. (2020), Poro et al. (2021), Barac et al. (2022), and Poro et al. (2024). Our results align well with previous investigations.
In the text

Fig. 8. Comparison of the P − L relations of δ Scuti stars and Cepheids in the J (upper left), H (upper middle), and Ks (upper right) bands, alongside the comparison of δ Scuti stars and RR Lyrae stars in the W1 (lower left), W2 (lower middle), and W3 (lower right) bands. The period range for δ Scuti stars spans 0.02–0.25 day (Chang et al. 2013), for Cepheids 3–100 days (Persson et al. 2004), and for RR Lyrae stars 0.2–1 day (Lafler & Kinman 1965).

In the text

	Fig. 9. Comparisons of the prior and posterior distributions of the distance modulus (left) and E(B − V) (right), with residuals shown in the lower panels.
In the text

	Fig. 10. Comparisons of the residuals of the distance modulus derived from Bailer-Jones et al. (2021) with our posterior distributions (left) and with Gaia DR3 (right).
In the text

	Fig. 11. Violin plots depicting the prior and posterior error distribution for the distance modulus (left) and E(B − V) (right). The substantive depression in posterior distribution peaks relative to prior distributions shows a great error mitigation, particularly in measurements with larger initial uncertainties.
In the text

Fig. 12. Estimation of E(B − V) utilizing multiband P − L relations of δ Scuti stars on held-out test sources. Blue, orange, and green squares represent sources with 9, 10, and 11 photometric measurements, respectively. The x-axis shows the E(B − V) values derived from the 3D dust map from Green et al. (2019), while the y-axis displays the corresponding estimates obtained from the application of our model, using an uninformative color excess prior.

In the text

	Fig. 13. Contour plots for the joint posterior distributions of the distance modulus and E(B − V) for TMTS J00572464+5750191. The expected anticorrelation between extinction and distance is evident.
In the text

	Fig. 14. Comparison between the prior distance modulus distribution (derived from Gaia DR3) and the posterior distribution, demonstrating a substantial improvement in the precision of the latter.
In the text

	Fig. 15. E(B − V) values between 1 and 3 kpc estimated from the dust map of Green et al. (2019) (left) and δ Scuti stars in our dataset (right). The size of each data point represents the distance to the source, with larger points corresponding to closer sources.
In the text

	Fig. 16. Intrinsic scatter in the P − L relation (blue) and P − L − Z relation (red) as a function of wavelength. Incorporating metallicity appears to reduce the intrinsic scatter at longer wavelengths.
In the text

Fig. A.1. Contour plots for the joint posterior distributions of α and M0 across the 11 photometric bands (g, r, i, z, y, J, H, Ks, W1, W2, and W3). For each band, the bottom-left panel shows the contour of α versus M0, while the marginal posterior distributions for the two variables are displayed on the top-left and bottom-right panels, respectively. The dashed vertical lines in the marginal distributions represent the 1σ credible intervals.

In the text