2025MNRAS.536.2749M

We present new $P -\phi _{31}-{\rm [Fe/H]}$ and $P -\phi _{31}- A_2 - {\rm [Fe/H]}$ relations for fundamental mode (RRab) and first-overtone mode (RRc) RR Lyrae stars (RRLs), respectively. The relations were calibrated based on pulsation periods and Fourier parameters of the RRL light curves in the GaiaG-band published in the Gaia Data Release 3 (DR3), and accurate spectroscopically measured metallicities available in the literature. We apply the feature selection algorithm to identify the most relevant parameters for the determination of metallicity. To fit the relations, we used the Bayesian approach, which allowed us to carefully take into account uncertainties in various parameters and the intrinsic scatter of the relations. The root mean squared errors of the predicted metallicity values in the training samples are 0.28 and 0.21 dex for RRab and RRc stars, respectively, comparable with the typical uncertainty of low/intermediate resolution spectroscopic metallicity measurements. We applied the new relations to measure individual metallicities and distances to $\sim$134 000 RRLs from the Gaia DR3 catalogue, as well as mean metallicities and distances to 38 Milky Way globular clusters. We also estimate the mean metallicity and distance to the Large Magellanic Cloud and Small Magellanic Cloud: ${\rm [Fe/H]_{LMC} = -1.63\pm 0.36}$ and $\mu _{\rm LMC}=18.55\pm 0.18$ mag, and ${\rm [Fe/H]_{SMC}=-1.86\pm 0.36}$ dex and $\mu _{\rm SMC}=19.01\pm 0.17$ mag, respectively, in excellent agreement with previous measurements.