The DESI Y1 RR Lyrae catalog I: Empirical modeling of the cyclic variation of spectroscopic properties and a chemodynamical analysis of the outer halo

We present the catalog of RR Lyrae stars (RRLs) observed in the first year of operations of the Dark Energy Spectroscopic Instrument (DESI) survey. This catalog contains 6,240 RRLs out to $\sim120$\,kpc from the Galactic center and over 12,000 individual epochs with homogeneously-derived stellar atmospheric parameters. We introduce a novel methodology to model the cyclical variation of the spectroscopic properties of RRLs from single-epoch measurements. We employ this method to infer the systemic velocities and mean temperatures of fundamental and first-overtone mode RRLs in our sample (without distinguishing between individual spectral lines). For fundamental mode pulsators, we obtain radial velocity curves with amplitudes of $\sim$30--80\,km\,s$^{-1}$ and effective temperature curves with 300--1,000\,K variations, whereas for first-overtone pulsators these amplitudes are $\sim20$\,km\,s$^{-1}$ and $\sim 600$\,K, respectively. We use our sample to study the metallicity distribution of the halo and its dependence on Galactocentric distance ($R_{\rm GC}$). Using a radius-dependent mixture model, we split the data into chemodynamically distinct components and find that our inner halo sample ($R_{\rm GC}\lesssim50$\,kpc) is predominantly composed of stars with [Fe/H] $\sim-1.5$ and largely radial orbits (with an anisotropy parameter $\beta\sim0.94$), that we associate with the Gaia-Sausage-Enceladus merger. Stars in the halo field exhibit a broader and more metal-poor [Fe/H] distribution with more circular orbits ($\beta\sim0.39$). The metallicity gradient of the metal-rich and the metal-poor components is found to be $0.005$ and $0.010$\,dex\,kpc$^{-1}$, respectively. Our catalog highlights DESI's tantalizing potential for studying the Milky Way and the pulsation properties of RRLs in the era of large spectroscopic surveys.