Title: Host star properties of hot, warm and cold Jupiters in the solar neighborhood from \textit{Gaia} DR3: clues to formation pathways

Abstract: Giant planets exhibit diverse orbital properties, hinting at their distinct formation and dynamic histories. In this paper, using $\textit{Gaia}$ DR3, we investigate if and how the orbital properties of Jupiters are linked to their host star properties, particularly their metallicity and age. We obtain metallicities for main sequence stars of spectral type F, G, and K, hosting hot, warm, and cold Jupiters with varying eccentricities. We compute the velocity dispersion of host stars of these three groups using kinematic information from $\textit{Gaia}$ DR3 and obtain average ages using velocity dispersion-age relation. We find that host stars of hot Jupiters are relatively metal-rich ([Fe/H]=$0.18 \pm 0.13$) and young ( median age $3.97 \pm 0.51$ Gyr) compared to the host stars of cold Jupiters in nearly circular orbits, which are relatively metal-poor ($0.03 \pm 0.18$) and older (median age $6.07 \pm 0.79$ Gyr). Host stars of cold Jupiters in high eccentric orbits, on the other hand, show metallicities similar to that of the hosts of hot Jupiters, but are older, on average (median age $6.25 \pm 0.92$ Gyr). The similarity in metallicity between hosts of hot Jupiters and hosts of cold Jupiters in high eccentric orbits supports high eccentricity migration as the potential origin of hot Jupiters, with the latter serving as the progenitors. However, the average age difference between them suggests that the older hot Jupiters may have been engulfed by the star in a timescale of $\sim 6$ Gyr. This allows us to estimate the value of stellar tidal quality factor $Q'_\ast\sim10^{6\pm1}$.