The DESI Y1 RR Lyrae catalog II: The metallicity dependency of pulsational properties, the shape of the RR Lyrae instability strip, and metal rich RR Lyrae

RR Lyrae stars (RRLs) are valuable probes of both Milky Way assembly and stellar-evolution physics. Using a sample 6,240 RRLs obtained in the first year of the Dark Energy Spectroscopic Instrument (DESI) survey, we investigate the metallicity of RRLs and its correlation with their pulsation properties. We find that (1) a clear correlation between period and [Fe/H] reinforces the view that the long-standing Oosterhoff dichotomy arises from the scarcity of intermediate-metallicity Galactic globular clusters hosting sizeable RRL samples; (2) high-amplitude short-period and small-amplitude short-period variables are comparatively metal-rich, with mean [Fe/H] = $-1.39 \pm 0.27$ and $-1.30 \pm 0.28$, respectively; (3) in double-mode pulsators (RRd) the metallicity declines smoothly with increasing fundamental-mode period, and anomalous RRd stars occupy a remarkably narrow [Fe/H] range relative to classical RRd stars; (4) this spectroscopic sample let us, for the first time, place empirical constraints on the metallicity-dependent topology of the instability strip using phase-corrected effective temperatures and a large number of RRLs, where we observe an instability strip that moves towards cooler $T_{\rm eff}$ with declining [Fe/H] with a width roughly consistent with stellar-evolution models; and (5) a subset of metal-rich RRLs exhibits orbits consistent with disk membership and halo kinematics. Our results confirm the tantalizing potential of DESI for Galactic and stellar astrophysics and highlight the importance of the even larger samples of RRLs and data-processing improvements forthcoming in future DESI data releases.