Type II Cepheids: period-luminosity-metallicity relations for the Population II distance scale

Type II Cepheids are a class of pulsating variable stars that play a critical role in our understanding of stellar evolution, distance measurement and tracing the structure and kinematics of old stars in nearby galaxies. This review provides a comprehensive summary of the current state of research on Type II Cepheids, including their observed properties, pulsation mechanisms and their distinction from other variable stars. These pulsating variable stars, found primarily in older stellar populations, exhibit well-defined period-luminosity relations but with an added advantage that they exhibit weak or negligible dependence on metallicity in most passbands. We explore their relevance in the context of their role as distance indicators and potential calibrators of the first rung of the extragalactic distance ladder. Finally, the review highlights recent advancements in theoretical models, observations across different wavelengths and ongoing debates concerning their classification.

1 Introduction

Cepheid variables have long served as vital distance indicators due to their well-defined period-luminosity $(PL)$ relations (see, Soszyński et al., 2011; Soszyński et al., 2017, among others). Among these, Type II Cepheids (T2CEPs) belong to the older, low-mass class of classical pulsators, distinct from their more massive and younger classical (Type I) counterparts (see reviews, Gingold, 1985; Wallerstein, 2002; Bhardwaj, 2020; Bono et al., 2024). T2CEPs are predominantly found in old stellar populations $-$ such as the Galactic halo and bulge, globular clusters and the Magellanic Clouds $-$ as reported by the OGLE surveys (Soszyński et al., 2017; 2018) and the Gaia catalogs (Ripepi et al., 2023), and have also been identified in the Andromeda group (Kodric et al., 2018). At a given pulsation period, T2CEPs are 1-2 magnitudes fainter than classical Cepheids. However, their tight $PL$ relations make them extremely important for anchoring the Population II distance scale, especially when used in combination with RR Lyraes and the Tip of the Red Giant Branch (TRGB) (see, Majaess, 2010; Braga et al., 2020; Das et al., 2025b; Lengen et al., 2025). They are particularly important astrophysical objects in metal-poor systems where classical Cepheids are scarce or absent (see reviews, Caputo, 1998; Wallerstein, 2002; Sandage and Tammann, 2006) and in globular clusters with very blue horizontal branch morphology with rare or no RR Lyrae stars (Gingold, 1976; Pritzl et al., 2002; Di Criscienzo et al., 2007).

The role of metallicity in calibrating the $PL$ relations of classical pulsators has gained increasing attention, especially with the advent of high-precision astrometry (e.g., from Gaia ¹ ), photometric surveys (e.g., OGLE ² , VISTA ³ ), and spectroscopic datasets (e.g., APOGEE ⁴ , LAMOST ⁵ ). Classical Cepheids are known to exhibit strong effects of metallicity on their $PL$ relations, especially at shorter wavelengths (see, for example, De Somma et al., 2022; Bhardwaj et al., 2024; Ripepi et al., 2025; Breuval et al., 2025). On the other hand, even with a wide range of metallicities from [Fe/H] = −2.4 to −0.5 (see Appendix A, Bono et al., 2020), T2CEPs seem to exhibit little or no metallicity dependence in their $PL$ relations, even in the optical bands (further details in Section 3).

In this review, we sum up the current theoretical and empirical understanding of the metallicity effects on T2CEP $PL$ relations. We incorporate findings from key observational datasets as well as theoretical pulsation models and thereby explore the behaviour of the period-luminosity-metallcity $(PLZ)$ relations across the different T2CEP subclasses (BL Her, W Vir, RV Tau) and their implications for cosmic distance scaling.

2 Period-based classification and evolution of T2CEPs

Based on their pulsation periods and evolutionary states, T2CEPs are sub-divided into three main classes following the classification suggested by Soszyński et al. (2018): the BL Herculis stars (BL Her), the W Virginis stars (W Vir) and the RV Tauri stars (RV Tau). In addition, there is a fourth sub-class $-$ the peculiar W Vir stars (pW Vir) that are bluer and brighter than W Vir (Soszyński et al., 2008). Understanding these subclasses is crucial because their evolutionary origins, pulsation characteristics and metallicity distributions differ, potentially influencing their $PL$ behaviour and the extent to which metallicity affects their luminosities.

BL Her stars typically have short pulsation periods with $1<P(\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s})<4$ . They are believed to be low-mass stars evolving from the horizontal branch toward the asymptotic giant branch (AGB) (Bono et al., 2020). A pulsation period of 1 day has traditionally been used to distinguish RR Lyrae stars from T2CEPs (Soszyński et al., 2008; Soszyński et al., 2014); however, defining the boundary between RR Lyraes and T2CEPs has long been problematic. Recent studies recommend using a more general, evolution-dependent criterion for this separation (Braga et al., 2020; Bono et al., 2024). W Vir stars have intermediate periods with $4<P(\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s})<20$ and are thought to be stars undergoing blue loops on the AGB or post-AGB evolutionary stages (Bono et al., 2020). They often show more complex light curves and some evidence of binarity (Groenewegen and Jurkovic, 2017a). This is particularly evident in pW Vir stars $-$ about 50% of those identified in the Magellanic Clouds display signatures of binarity (Groenewegen and Jurkovic, 2017a; Soszyński et al., 2018). RV Tau stars are long-period with $P(\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s})>20$ , evolved post-AGB objects (Bono et al., 2020). They exhibit alternating deep and shallow minima in their light curves, suggesting complex pulsation modes, and are frequently associated with circumstellar dust and mass loss.

3 The effect of metallicity on $PL$ relations

Similar to classical Cepheids, T2CEPs obey well-defined $PL$ relations which make them important astrophysical objects, especially for old stellar systems. However, prior to the identification of Populations I and II by Baade (1944), it was assumed that a single $PL$ relation was applicable to all Cepheids, leading to discrepancies in Hubble’s early distance scale measurements (Hubble and Humason, 1931, and references therein) by a factor of two. It was Baade (1956) who concluded that “there was no a priori reason to expect that two Cepheids of the same period, the one a member of Population I, the other of Population II, should have the same luminosity” ⁶ , thereby subsequently resulting in the revised cosmic distance scale.

Since then, there have been quite a few studies dedicated towards studying the different aspects of T2CEPS, on both the theoretical (Carson and Stothers, 1982; Kovacs and Buchler, 1988; Bono et al., 1997; Deka et al., 2024) and the empirical (Gonzalez, 1994; Bersier et al., 1997; Balog et al., 1997; Vinko et al., 1998; Kiss et al., 2007; Wielgórski et al., 2024; Yacob et al., 2025) fronts. For the most recent empirical $PL$ relations of T2CEPs in multiple wavelengths, the interested reader is referred to the works of Sicignano et al. (2024), Cruz Reyes et al. (2025) and Narloch et al. (2025). However, in this review, our primary focus will be on studies that examine the role of metallicity on the $PL$ relations of T2CEPs.

One of the earliest $PLZ$ studies for T2CEPs was carried out by Nemec et al. (1994), with the assumed metal abundance [Fe/H] determined generally on the scale of Zinn and West (1984). However, the paper follows an old classification with $P(d)<10$ as BL Her, $10<P(d)<26$ as W Vir stars and $26<P(d)<100$ as RV Tau stars as was used in Joy (1949); Demers and Harris (1974); Arp (1955); Wallerstein and Cox (1984). Therefore, while the combined $PLZ$ relations remain the same, the relations from individual sub-classes may be different. McNamara (1995) followed up the work of Nemec et al. (1994) using the same observed dataset and obtained different $PL$ relations for the two classes of BL Her and W Vir stars when considered separately and was one of the firsts to rule out the dependence of the absolute magnitudes of the variables on [Fe/H]. They also ruled out the possibility of T2CEPS pulsating in the first-overtone mode, as was earlier suggested by Arp (1955) and Nemec et al. (1994). Note that while this broadly remains true, two first-overtone T2CEPs were recently identified in the Large Magellanic Cloud (LMC) by Soszyński et al. (2019). Although not a study that explores $PLZ$ , Alcock et al. (1998) as part of the MACHO project is interesting because unlike previous studies which explicitly excluded RV Tau stars from T2CEP $PL$ relations, they observed a single period-luminosity-color relation for both T2CEPs and RV Tau stars $(0.9<\log (P)<1.75)$ in the LMC, thereby suggesting a common evolutionary channel.

Almost a decade later, the study of the metallicity dependence (or lack thereof) was re-ignited in a series of papers by Matsunaga et al. (2006), Matsunaga et al. (2009), Matsunaga et al. (2011). The photometric data of T2CEPs analysed in Matsunaga et al. (2006) were obtained from the Infrared Survey Facility (IRSF) 1.4-m telescope and the Simultaneous 3-Colour Imager for Unbiased Survey (SIRIUS) near-infrared camera (Nagashima et al., 1999; Nagayama et al., 2003) while the parameters for the globular clusters $-$ the metallicity [Fe/H], the colour excess E (B-V) and the magnitude of the horizontal branch V(HB) were adopted from Harris (1996). For the reddening corrections, $R_V=3.1$ was used along with extinction law from Cardelli et al. (1989). The distance moduli adopted were based on the magnitudes of horizontal branches of the clusters (using the relation from Gratton et al., 2003), which were then used to obtain the absolute magnitudes of the T2CEPs. While Matsunaga et al. (2006) found the metallicity contribution $(\gamma =-0.10\pm 0.06)$ towards the $K$ -band $PL$ relation from 46 T2CEPs in 26 Galactic Globular Clusters to be hardly significant, Matsunaga et al. (2011) presented evidence of different $PL$ relations obeyed by T2CEPs when three different systems are considered (globular clusters, LMC and SMC). Using the method of estimating the difference in the distance moduli between LMC and SMC, they also concluded that the absolute magnitudes of W Vir stars are free of metallicity effects while BL Her stars are not. In a more recent empirical study, Wielgórski et al. (2022) used photometric data of T2CEPs from the 0.81 m InfraRed Imaging Survey (IRIS) telescope (Hodapp et al., 2010; Ramolla et al., 2016) which were corrected for interstellar extinction using reddening maps from Schlafly and Finkbeiner (2011) to obtain the colour excess E (B-V) for each star, with $R_V=3.1$ and the reddening law from Cardelli et al. (1989) and O’Donnell (1994). The distances to the individual T2CEPs were obtained from Gaia EDR3 parallaxes (Lindegren et al., 2021) while the spectroscopic metallicity determinations [Fe/H] for a fraction of Milky Way T2CEPs were taken from Maas et al. (2007). They obtained a metallicity effect of $-0.2$ mag ${\text{dex}}^{-1}$ , albeit with the caveat that the sample size is rather small with a total of 7 T2CEPs only, BL Her and W Vir stars included. In yet another recent empirical study, Ngeow et al. (2022) provided new $PLZ$ relations for 37 T2CEPs in 18 globular clusters in the $gri$ bands using data from the Zwicky Transient Facility (ZTF, Bellm et al., 2019) and updated $BVIJHK$ band relations for 58 T2CEPS in 24 globular clusters. The distances to the host globular clusters were obtained from Baumgardt and Vasiliev (2021). For the $gri$ bands, reddening map from Green et al. (2019) was used to obtain the colour excess towards each star, followed by extinction correction relations also from Green et al. (2019). Individual reddening values for each T2CEP in the $BVIJHK$ bands were derived from the Schlegel et al. (1998) dust map, with extinction corrections computed using the relations of Schlafly and Finkbeiner (2011); Green et al. (2019). The metallicities [Fe/H] for the host globular clusters were obtained from the GlObular clusTer Homogeneous Abundances Measurements (GOTHAM) survey Dias et al. (2015), Dias et al. (2016a), Dias et al. (2016b); Vásquez et al. (2018). Except for the $B$ -band, the metallicity effect towards the $PL$ relations in all the other bands ( $gri$ and $VIJHK$ ) were found to be negligible within reported uncertainties. In the most recent empirical study, Lengen et al. (2025) employed multi-band photometry for 14 T2CEPs located in anchor galaxies and 7 T2CEPs in host galaxies, spanning 39 globular clusters identified in Gaia DR3 (Gaia Collaboration et al., 2023). The data were converted to reddening-free Wesenheit magnitudes (Madore, 1982; Ripepi et al., 2019) using the reddening coefficient RW G = 1.867 from Cruz Reyes et al. (2024). Cluster metallicities were taken from the Harris (2010) catalog and cluster-level weighted mean parallaxes from Gaia DR3 were adopted following Cruz Reyes et al. (2024). They reported a metallicity dependence of 0.107 ± 0.043 mag dex⁻¹, while noting that the metallicity dependence should be interpreted with caution. We highlight that different studies adopt varying methods for deriving extinction corrections, metallicities and distances. Consequently, some degree of inhomogeneity may arise when comparing the metallicity dependence of PL relations across empirical works. Moreover, assuming that the metallicities of T2CEPs in globular clusters are identical to the mean metallicities of their host clusters can introduce substantial uncertainty in the metallicity term of the $PLZ$ relation.

On the theoretical front, Di Criscienzo et al. (2007) computed pulsation models following Bono and Stellingwerf (1994), with identical nonlinear, nonlocal, time-dependent convective hydrodynamics and the same equation-of-state and opacity prescriptions. In particular, opacity compilations for temperatures higher than 10,000 K were used from Iglesias and Rogers (1996) and for lower temperatures from Alexander and Ferguson (1994). The bolometric light curves were transformed into respective passbands by adopting bolometric corrections and temperature-color transformations from Castelli et al. (1997a); Castelli et al. (1997b). They thereby provided theoretical relations over multiple wavelengths for the short-period T2CEPs. However, for the purpose of this review, we follow the same convention for BL Her stars as those with pulsation periods between 1 and 4 days while for W Vir stars as those with periods between 4 and 20 days for a uniform comparison throughout the literature. Therefore, although Di Criscienzo et al. (2007) presents the results for BL Her models, we include the study in the BL Her + W Vir category for this review since the models span the range of pulsation periods up to 8 days. In a series of papers, Das et al. (2021), Das et al. (2024), Das et al. (2025b) presented theoretical period relations for BL Her stars across multiple wavelength bands, including Johnson–Cousins–Glass $(UBVRIJHKL{L}^{\prime }M)$ , Gaia $(G{G}_{BP}{G}_{RP})$ , and for the first time, in the Rubin-LSST ⁷ $(ugrizy)$ passbands. The relations were derived from a fine grid of non-linear convective models computed with the Radial Stellar Pulsation (RSP, Smolec and Moskalik, 2008; Paxton et al., 2019) tool within the Modules for Experiments in Stellar Astrophysics (MESA, Paxton et al., 2011; Paxton et al., 2013; Paxton et al., 2015; Paxton et al., 2018; Paxton et al., 2019; Jermyn et al., 2023) software suite. MESA-RSP follows the turbulent convection formulation of Kuhfuss (1986) and the stellar pulsation treatment of Smolec and Moskalik (2008). OPAL opacities (Iglesias and Rogers, 1996), supplemented at low temperatures by Ferguson et al. (2005), were adopted. Bolometric light curves were converted to the Johnson–Cousins–Glass $(UBVRIJHKL{L}^{\prime }M)$ system using the pre-computed bolometric correction table of Lejeune et al. (1998), while transformations to the Gaia $(G,G_{BP},G_{RP})$ and Rubin–LSST $ugrizy$ passbands were performed using those from MIST ⁸ (MESA Isochrones & Stellar Tracks) packaged model grids. Theoretical $PL$ , period-Wesenheit $(PW)$ and period-radius $(PR)$ relations showed good agreement with most empirical relations, with no significant metallicity dependence in the $PL$ (except for $U$ and $B$ bands) and $PR$ relations. Finally, in the most recent theoretical study of T2CEPs, Marconi et al. (2025) presented an extensive set of models for BL Her and short period W Vir (up to 10 days) stars, following the methodology as in Di Criscienzo et al. (2007) but with opacity tables as was used in De Somma et al. (2024). In particular, the radiative Rosseland opacity was taken from Iglesias and Rogers (1996) for temperatures higher than $\log (T)=4.0$ and from Ferguson et al. (2005) for lower temperatures. They predicted light and radial velocity curves and thereby obtained $PLZ$ and $PWZ$ relations across a wide range of wavelengths, including $IJHK$ , Gaia, VISTA and Rubin-LSST.

Table 1 presents, to the best of our knowledge, a comprehensive compilation of all available $PLZ$ relations across multiple wavelengths reported in the literature, which are also illustrated in Figure 1. Values of γ that exceed their corresponding 3σ uncertainties are shown in bold. For most theoretical studies, these γ values are intrinsically very small; they appear statistically significant only because the associated uncertainties are extremely low. Cases with γ > 0.1 mag dex⁻¹ and those lying beyond their 3σ uncertainties are discussed further; all others are treated as indicating only weak or minimal metallicity dependence in this review. The significant metallicity effects in the U and B bands from Das et al. (2021), Nemec et al. (1994) and Ngeow et al. (2022) are possibly due to the heightened sensitivity of bolometric corrections to metallicity at wavelengths shorter than the V band (Gray, 2005; Kudritzki et al., 2008). From Marconi et al. (2025), we find significant metallicity effects in the individual Gaia bands; however, the metallicity dependence is negligible when GaiaWesenheit magnitudes are considered. Lastly, we find significant metallicity effect in results from Wielgórski et al. (2022); however, we note that the sample size in this study is rather small. A homogeneous determination of metal abundances for the full sample of field T2Ceps, based on high-resolution spectroscopy, is essential for a more robust and precise assessment of the metallicity effect. While the set of studies included here can be regarded as comprehensive, the coverage of photometric passbands may nevertheless remain incomplete and the interested reader is encouraged to refer to the original works for more details. This is especially true for the case of theoretical studies where we have included results for the most commonly used passbands and/or for the simplest convection parameters only. In particular, the $PLZ$ relations from Di Criscienzo et al. (2007) included here were obtained for the particular case of mixing length parameter 1.5. For the theoretical relations from Das et al. (2021), Das et al. (2024), Das et al. (2025b), we have only included the results computed using the simplest convection parameter set while the papers also contain results as a function of different convection parameter sets. In addition, we only highlight the relations involving the Wesenheit indices $W(i,g-i)$ , $W(z,i-z)$ , and $W(y,g-y)$ in the Rubin-LSST passbands, owing to their minimal sensitivity to metallicity. These filter combinations are therefore considered optimal for future Rubin–LSST observations of T2CEPs, supporting their use as reliable standard candles. The interested reader is referred to Das et al. (2025b) for individual $PLZ$ relations in the Rubin-LSST $ugrizy$ passbands. Lastly, Marconi et al. (2025) also provides $PLZ$ relations for the VISTA $JY{K}_s$ bands, in addition to the passbands included here in this review. Note that while Di Criscienzo et al. (2007) and Marconi et al. (2025) used the convection formulation outlined in Stellingwerf (1982a), Stellingwerf (1982b), Das et al. (2021), Das et al. (2024), Das et al. (2025b) used MESA-RSP which follows the turbulent convection theory from Kuhfuss (1986). Therefore, the minimal or negligible effect of metallicity towards the $PL$ relations of T2CEPs appears promising and remains robust across various theoretical stellar pulsation codes, even when different theories of convection are employed.

Table 1

Table 1. Compilation of all $PLZ$ relations of the mathematical form, $M_{\lambda }=\alpha +\beta \log (P)+\gamma [\mathrm{F}\mathrm{e}/\mathrm{H}]$ for T2CEPs available in the literature at different wavelengths.

Figure 1

Figure 1. Empirical (in circles) and theoretical (in star-shaped symbols) estimates of the T2CEP metallicity dependence $\gamma$ from the literature across different passbands. The top three panels present results from the individual T2CEP subclasses- BL Her, W Vir and RV Tau stars. The bottom two panels demonstrate the effect of metallicity when different subclasses are combined- BL Her + W Vir and T2CEP (BL Her + W Vir + RV Tau), respectively. Passbands include the bolometric (Bol), Johnson-Cousins-Glass $(UBVRIJHKLM{W}_{VI}{W}_{JK})$ , ZTF $(gri)$ , Gaia $(G{G}_{BP}{G}_{RP}{W}_{\mathit{gaia}})$ and the Rubin–LSST $(W_{gi},W_{gy},W_{iz})$ filters. The x-axis is in increasing order of the central effective wavelengths $({\lambda }_{\text{eff}})$ of the respective passbands as provided by the SVO Filter Profile Service (Rodrigo et al., 2012; Rodrigo and Solano, 2020). The y-scale is same in each panel for a relative comparison.

4 Applications in distance scale and precision cosmology

Classical Cepheids have been extensively used in the first rung of the distance ladder scale of the SH0ES program (Supernovae and $H_0$ for the Equation of State of dark energy, Riess et al., 2009) for the subsequent estimation of the local Hubble constant. As an alternative route, the Carnegie-Chicago Hubble Program (CCHP, Beaton et al., 2016) uses the independent approach of using only Population II distance indicators (RR Lyraes and TRGBs) to estimate the same. The geometric calibration of the $PL$ relations of classical pulsators therefore not only serve in estimating extragalactic distances but is also crucial to provide direct measurements of the $H_0$ using the distance versus redshift approach as opposed to the indirect measurement from an early Universe calibration of the $\mathrm{\Lambda }$ cold dark matter model ( $\mathrm{\Lambda }$ CDM) model-dependent inference from the Planck CMB observations (Planck et al., 2020). The present values of the Hubble constant derived from the early Universe approach is $H_0=67.4\pm 0.5$ km $s^{-1}$ Mpc (Planck et al., 2020) while from the local Universe approach is $H_0=73.18\pm 0.88$ km $s^{-1}$ Mpc (Riess et al., 2022; Bhardwaj et al., 2023; Riess et al., 2024; Riess et al., 2025), resulting in the Hubble tension with a $\sim 6\sigma$ discrepancy (see also Verde et al., 2019). With the recent improvements in the precision of $H_0$ measurements and its critical role in testing extensions to $\mathrm{\Lambda }$ CDM, addressing and reducing previously unrecognized sources of uncertainty and bias in $H_0$ has become increasingly crucial.

In this context, it is essential to validate distances derived from classical Cepheids by employing independent populations of pulsating variables. T2CEPs are particularly promising for this purpose due to their minimal metallicity dependence in the $PL$ relations across different passbands, as discussed in the previous section. Several theoretical and empirical studies have also shown that RR Lyraes and T2CEPs follow a common $PL$ relation, especially at near-infrared bands (for example, Majaess, 2010; Bhardwaj et al., 2017a; Braga et al., 2020; Das et al., 2025b; Lengen et al., 2025). Consequently, when combined with RR Lyrae stars and the TRGB, they could serve as an alternative to classical Cepheids for extragalactic distance measurements. This methodology not only allows for rigorous cross-validation of classical Cepheid-based distances but also extends the applicability of pulsating variables as robust standard candles across a range of stellar populations and metallicity regimes, ultimately contributing to more accurate $H_0$ determinations (see e.g., Beaton et al., 2016).

Despite their promise, T2CEPs currently face a few challenges when applied to precision cosmology. The intrinsically lower luminosity of T2CEPs restricts their effective use as distance indicators to approximately only a few Mpc with present-day instrumentation and the population of known T2CEPs beyond the Local Group therefore remains small. Studies of T2CEPs in the Milky Way have largely focused on globular clusters and the Galactic bulge (e.g., Bhardwaj et al., 2017c; Bhardwaj et al., 2022; Narloch et al., 2025; Cruz Reyes et al., 2025). Investigations in the Magellanic Clouds have examined their luminosities and $PL$ relations in detail (e.g., Bhardwaj et al., 2017b; Groenewegen and Jurkovic, 2017b; Sicignano et al., 2024). Applications to nearby Local Group galaxies beyond the Magellanic Clouds remain limited, with only a few efforts addressing their potential as extragalactic distance indicators (e.g., Majaess et al., 2009). The metallicity calibration of T2CEPs therefore remains less well-established compared to that of Classical Cepheids. However, with the advent of next-generation surveys such as Rubin-LSST and the Roman Space Telescope ⁹ , the potential applications of T2CEPs are expected to increase substantially. Although not suitable for direct determinations of the Hubble constant $H_0$ , T2CEPs could provide important cross-checks on Cepheid-based distances, especially in galaxies hosting both Cepheid and T2CEP populations and in anchoring TRGB distances, which are used in recent independent measurements of $H_0$ . Their relative insensitivity to metallicity even at optical wavelengths, combined with their prevalence in older stellar populations, complements the younger, more metallicity-dependent classical Cepheid distance scale.

5 Summary

This work provides a comprehensive review of the effect of metallicity on the $PL$ relation of T2CEPs. Most empirical (Nemec et al., 1994; Matsunaga et al., 2006; Ngeow et al., 2022) and theoretical (Di Criscienzo et al., 2007; Das et al., 2021; Das et al., 2024; Das et al., 2025b; Marconi et al., 2025) studies to date suggest that metallicity has a weak or negligible effect on the T2CEP $PL$ relation across a broad range of wavelengths, with the exception of the shorter-wavelength $U$ and $B$ bands (Nemec et al., 1994; Das et al., 2021). For distance scale applications, this potentially provides a significant advantage over other commonly used classical pulsators, such as RR Lyraes and classical Cepheids, which are known to exhibit strong metallicity effects, particularly at shorter wavelengths. In addition, RR Lyraes and T2CEPs, which have been shown to follow similar $PL$ relations (Majaess, 2010; Bhardwaj et al., 2017a; Braga et al., 2020; Das et al., 2025b; Lengen et al., 2025), can be combined with the TRGB to provide an independent cross-check of Cepheid-based distances and serve as a complementary Population II distance scale.

Despite significant progress, the field of T2CEPs still presents many open challenges. On the theoretical side, current stellar pulsation codes are able to reliably reproduce the behavior of BL Her and short-period W Vir stars, but they face substantial difficulties in modeling long-period W Vir and RV Tau variables. These complications arise from the unstable outer layers of such stars, where the radiation-diffusion approximation breaks down and pulsation-driven mass loss may also need to be included to adequately describe their variability (Smolec, 2016; Paxton et al., 2019). Empirical $PLZ$ relations for T2CEPs currently show good agreement with theoretical predictions, even when the latter are computed with different convection parameter sets (Das et al., 2021; Das et al., 2024). This consistency may partly reflect the limited precision of current observations, which remain insufficient to decisively distinguish among models based on different convection treatments. The advent of forthcoming large-scale surveys is expected to deliver much higher precision data, making it essential to calibrate convection parameters for T2CEP models, as has recently been done for RR Lyrae stars (Kovács et al., 2023; Das et al., 2024). Beyond calibrating $PLZ$ relations, it is equally important to test whether theoretical light curves can successfully reproduce observed pulsation periods and detailed light-curve morphologies. Such comparisons are vital for ensuring that models capture the underlying physics throughout the entire pulsation cycle, rather than only at mean-light, as recently demonstrated by Das et al. (2025a). Finally, the development of a self-consistent framework that simultaneously incorporates stellar evolution and pulsation physics represents an important step forward, as illustrated by the work of Marconi et al. (2025). On the empirical side, current instrumentation already provides high-quality photometric data for T2CEPs. Nevertheless, a homogeneous determination of metallic abundances from high-resolution spectroscopy across a large sample remains indispensable for achieving a more precise characterization of metallicity effects. Thanks to the Gaia space mission which has provided geometric distance measurements for several dozen field T2CEPs, it will soon be possible to calibrate the $PLZ$ relations of these stars with unprecedented precision and accuracy, once homogeneous photometry and metallicity data become available.

In summary, T2CEPs are emerging as reliable and complementary distance indicators, owing to the minimal impact of metallicity on their $PL$ relations. Their presence in older stellar populations and the shared $PL$ relation with RR Lyraes provide an independent distance scale that can cross-validate classical Cepheid distances. Forthcoming large-scale photometric and spectroscopic surveys will enable more precise calibrations of their $PLZ$ relations, detailed comparisons of theoretical and observed light curves, and improved assessments of metallicity effects. As a result, T2CEPs are poised to play an increasingly important role in refining the extragalactic distance scale and constraining $H_0$ .

Author contributions

SD: Data curation, Formal Analysis, Writing – original draft, Writing – review and editing. AB: Writing – review and editing. MM: Writing – review and editing.

Funding

The authors declare that financial support was received for the research and/or publication of this article. This research was supported by the International Space Science Institute (ISSI) in Bern/Beijing through ISSI/ISSI-BJ International Team project ID #24-603 - “EXPANDING Universe” (EXploiting Precision AstroNomical Distance INdicators in the Gaia Universe). SD acknowledges the KKP-137523 ‘SeismoLab’ Élvonal grant of the Hungarian Research, Development and Innovation Office (NKFIH). AB thanks the funding from the Anusandhan National Research Foundation (ANRF) under the Prime Minister Early Career Research Grant scheme (ANRF/ECRG/2024/000675/PMS).

Acknowledgements

The authors are grateful to the referee for useful suggestions that improved the quality of the manuscript. This research has made use of NASA’s Astrophysics Data System.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

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Footnotes

¹ https://gea.esac.esa.int/archive/

² https://ogle.astrouw.edu.pl/

³ https://www.eso.org/sci/facilities/paranal/telescopes/vista.html

⁴ https://www.sdss4.org/dr17/irspec/

⁵ https://www.lamost.org

⁶ The explanation for the different $PL$ relations for type I and type II Cepheids is as follows: two Cepheids with equal mean radius and mean effective temperature would have the same luminosities (Stefan-Boltzmann law). If these two Cepheids obey Ritter’s pulsation equation $P\sqrt{<\rho >}=Q$ such that they have similar pulsation constants $Q$ and assuming typical masses for type I and type II Cepheids, we find that the expected pulsation period of a 0.6 M type II Cepheid is three times longer than that of a 6 M type I Cepheid (Catelan and Smith, 2015).

⁷ https://rubinobservatory.org/

⁸ https://waps.cfa.harvard.edu/MIST/index.html

⁹ https://roman.gsfc.nasa.gov/

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