The $δ$ Scuti instability strip and period-luminosity relation as seen by TESS and Gaia DR3
We present an asteroseismic survey of 1,526,218 A- and F-type stars in and around the $δ$ Scuti instability strip using TESS 10-minute cadence data, which resolve the full $δ$ Scuti pulsation spectra across nearly the entire sky. We derive homogeneous stellar parameters from Gaia DR3 parallaxes and archival photometry, with temperatures that agree with interferometry to within ~300 K. A Positive--Unlabelled classifier identifies 156,706 $δ$ Scuti pulsators, which on average rotate significantly faster than non-variable A-type stars. Within the instability strip in the $T_{eff}$--$\log L$ plane, the pulsator fraction peaks at 30% over the full sample and rises to 80% in the brightest magnitude bins. Detection sensitivity therefore accounts for much, but not all, of the apparent impurity of the strip. Our period--luminosity relation agrees with previous determinations, and we recast it as a scale-invariant period--density relation that connects directly to the pulsation constant. We find that the secondary ridge in these relations is probably a combination of several overtone modes. The $δ$ Scuti stars closest to the fundamental ridge have systematically lower rotational velocities and occupy a distinct region of the instability strip. Finally, a cross-match with a Gaia catalog of open clusters reveals a correlation between the mean pulsation frequency of $δ$ Scuti stars and cluster age, suggesting a new ensemble asteroseismic age indicator for young stellar populations.