The influence of metallicity on the Leavitt Law from geometrical distances of Milky Way and Magellanic Clouds Cepheids

The Cepheid Period-Luminosity (PL) relation is the key tool for measuring astronomical distances and for establishing the extragalactic distance scale. In particular, the local value of the Hubble constant ($H_0$) strongly depends on Cepheid distance measurements. The recent Gaia Data Releases and other parallax measurements from the Hubble Space Telescope (HST) already enabled to improve the accuracy of the slope ($\alpha$) and intercept ($\beta$) of the PL relation. However, the dependence of this law on metallicity is still largely debated. In this paper, we combine three samples of Cepheids in the Milky Way (MW), the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC) in order to derive the metallicity term (hereafter $\gamma$) of the PL relation. The recent publication of extremely precise LMC and SMC distances based on late-type detached eclipsing binary systems (DEBs) provides a solid anchor for the Magellanic Clouds. In the MW, we adopt Cepheid parallaxes from the early third Gaia Data Release. We derive the metallicity effect in $V$, $I$, $J$, $H$, $K_S$, $W_{VI}$ and $W_{JK}$. In the $K_S$ band we report a metallicity effect of $-0.221 \pm 0.051$ mag/dex, the negative sign meaning that more metal-rich Cepheids are intrinsically brighter than their more metal-poor counterparts of the same pulsation period.