The initial mass functions of M31 and M32 through far red stellar absorption features

Using the Oxford Short Wavelength Integral Field specTrograph (SWIFT), we investigate radial variations of several initial mass function (IMF) dependent absorption features in M31 and M32. We obtain high signal-to-noise spectra at six pointings along the major axis of M31 out to ~ 700" (2.7 kpc) and a single pointing of the central 10 pc for M32. In M31 the sodium NaI {\lambda}8190 index shows a flat equivalent width profile at ~ 0.4 {\AA} through the majority of the bulge, with a strong gradient up to 0.8 {\AA} in the central 10" (38 pc); the Wing-Ford FeH {\lambda}9916 index is measured to be constant at 0.4 {\AA} for all radii; and calcium triplet CaT {\lambda}8498, 8542, 8662 shows a gradual increase through the bulge towards the centre. M32 displays flat profiles for all three indices, with FeH at ~ 0.5 {\AA}, very high CaT at ~ 0.8 {\AA} and low NaI at ~ 0.1 {\AA}. We analyse these data using stellar population models. We find that M31 is well described on all scales by a Chabrier IMF, with a gradient in sodium enhancement of [Na/Fe] ~ +0.3 dex in the outer bulge, rising within the central 10" to perhaps [Na/Fe] ~ +1.0 dex in the nuclear region. We find M32 is described by a Chabrier IMF and young stellar age in line with other studies. Models show that CaT is much more sensitive to metallicity and [{\alpha}/Fe] than to IMF. We note that the centres of M31 and M32 have very high stellar densities and yet we measure Chabrier IMFs in these regions.