Population synthesis of delta Scuti stars: the instability strip, period-luminosity relation, and large-separation distribution
Interpreting the observed properties of intermediate-mass stars requires accounting for rotation, binarity, intrinsic population diversity, and measurement uncertainty. We combine these effects in population-synthesis models of 1.4-2.5 M$_{\odot}$ stars that include pulsation properties of $\delta$ Sct variables. The synthetic populations successfully reproduce the distributions of stars observed in young associations, validating the mapping between intrinsic and observed stellar properties. Using these models, we infer a semi-empirical $\delta$ Sct instability strip by matching synthetic and observed populations. The resulting instability strip is systematically hotter than widely used theoretical prescriptions, implying that $\delta$ Sct stars are on average 0.1 M$_{\odot}$ more massive than previously assumed, with 95% of the population spanning approximately 1.50-2.30 M$_{\odot}$. We then investigate the $\delta$ Sct period-luminosity relation and reproduce its recently identified second ridge with mixture models in which roughly one-third of stars have the fifth or sixth radial overtone as their dominant mode; the fraction increases in younger populations. In contrast, unresolved binarity does not account for the second ridge. Finally, we predict the population-wide distribution of asteroseismic large separations, $\Delta\nu$, finding a peak between 6 and 7 d$^{-1}$, in agreement with recent observations. We also demonstrate that $\Delta\nu$ is most reliably measured with the \'echelle technique, while autocorrelation methods can produce spurious detections when used in isolation. These results show that population synthesis provides a powerful framework for interpreting populations of $\delta$ Sct stars and for linking stellar evolution, pulsation, and observation.