Context. The birth environments of planetary systems are thought to influence planet formation and orbital evolution, through external photoevaporation and stellar flybys. Recent work has claimed observational support for this, in the form of a correlation between the properties of planetary systems and the local Galactic phase space density of the host star. In particular, Hot Jupiters are found overwhelmingly around stars in regions of high phase space density, which may reflect a formation environment with high stellar density. Aims. We instead investigate whether the high phase density may have a galactic kinematic origin: Hot Jupiter hosts may be biased towards being young and therefore kinematically cold, because tidal inspiral leads to the destruction of the planets on Gyr timescales, and the velocity dispersion of stars in the Galaxy increases on similar timescales. Methods. We use 6D positions and kinematics from Gaia for the Hot Jupiter hosts and their neighbours, and construct distributions of the phase space density. We investigate correlations between the stars local phase space density and peculiar velocity. Results. We find a strong anticorrelation between the phase space density and the host stars peculiar velocity with respect to the Local Standard of Rest. Therefore, most stars in high-density regions are kinematically cold, which may be caused by the aforementioned bias towards detecting Hot Jupiters around young stars before the planets tidal destruction. Conclusions. We do not find evidence in the data for Hot Jupiter hosts preferentially being in phase space overdensities compared to other stars, nor therefore for their originating in birth environments of high stellar density.
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