Towards Understanding the Milky Way's Typicality: Assessing the Chemodynamics of M31's Bulge & Bar, Thick & Thin Discs

We describe a novel framework to model galaxy spectra with two cospatial stellar populations, such as may represent a bulge & bar or thick & thin disc, and apply it to APOGEE spectra in the inner $\sim$2 kpc of M31, as well as to stacked spectra representative of the northern and southern parts of M31's disc ($R\sim4-7$ kpc). We use a custom M31 photometric decomposition and A-LIST spectral templates to derive the radial velocity, velocity dispersion, metallicity, and $\alpha$ abundance for both components in each spectrum. In the bulge, one component exhibits little net rotation, high velocity dispersion ($\sim$170 km s$^{-1}$), near-solar metallicity, and high $\alpha$ abundance ([$\alpha$/M] = 0.28), while the second component shows structured rotation, lower velocity dispersion ($\sim$121 km s$^{-1}$), and slightly higher abundances ([M/H] = 0.09, [$\alpha$/M] = 0.3). We tentatively associate the first component with the classical bulge and the second with the bar. In the north disc we identify two distinct components: the first with hotter kinematics, lower metallicity, and higher $\alpha$ abundance than the second ([M/H] = 0.1 and 0.39, [$\alpha$/M] = 0.29 and 0.07). These discs appear comparable to the Milky Way's ''thick'' and ''thin'' discs, providing the first evidence that M31's inner disc has a similar chemodynamical structure. We do not identify two distinct components in the south, potentially due to effects from recent interactions. Such multi-population analysis is crucial to constrain galaxy evolution models that strive to recreate the complex stellar populations found in the Milky Way.