Classical Cepheid period-Wesenheit-metallicity relation in the Gaia bands

Context. Classical Cepheids (DCEPs) represent a fundamental tool to calibrate the extragalactic distance scale. However, they are also powerful stellar population tracers in the context of Galactic studies. The forthcoming Data Release 3 of the Gaia mission will allow us to study, with unprecedented detail, the structure, the dynamics, and the chemical properties of the Galactic disc, and in particular of the spiral arms, where most Galactic DCEPs reside. Aims. In this paper we aim to quantify the metallicity dependence of the Galactic DCEPs'period-Wesenheit (PWZ) relation in the Gaia bands. Methods. We adopted a sample of 499 DCEPs with metal abundances from high-resolution spectroscopy, in conjunction with Gaia Early Data Release 3 parallaxes and photometry to calibrate a PWZ relation in the Gaia bands. Results. We find a significant metallicity term, of the order of-0.5 mag dex-1, which is larger than the values measured in the near-infrared (NIR) bands by different authors. Our best PWZ relation is W = (-5.988 ± 0.018)-(3.176 ± 0.044)(log P-1.0)-(0.520 ± 0.090)[Fe/H]. We validated our PWZ relations by using the distance to the Large Magellanic Cloud as a benchmark, finding very good agreement with the geometric distance provided by eclipsing binaries. As an additional test, we evaluated the metallicity gradient of the young Galactic disc, finding-0.0527 ± 0.0022 dex kpc-1, which is in very good agreement with previous results.