[Mg/Fe] and variable initial mass function: Revision of [$\alpha$/Fe] for massive galaxies

Observations of nearby massive galaxies have revealed that they are older and richer in metals and magnesium than their low-mass counterparts.

In particular, the overabundance of magnesium compared to iron, [Mg/Fe], is interpreted to reflect the short star formation history that the current massive galaxies underwent early in the Universe.

We present a systematic revision of the [Mg/Fe] - velocity dispersion ($\sigma$) relation based on stacked spectra of early-type galaxies with a high signal-to-noise ratio from the Sloan Digital Sky Survey (SDSS).

Using the penalized pixel-fitting (pPXF) method of Cappellari & Emsellem 2004 and the Vazdekis et al. 2015 MILES single stellar population (SSP) models, we fit a wide optical wavelength range to measure the net $\alpha$-abundance.

The combination of pPXF and $\alpha$-enhanced MILES models incorrectly leads to an apparently decreasing trend of [$\alpha$/Fe] with velocity dispersion.

We interpret this result as a consequence of variations in the individual abundances of the different $\alpha$-elements.

This warrants caution for a na\"ive use of full spectral fitting algorithms paired with stellar population models that do not take individual elemental abundance variations into account, especially when deriving averaged quantities such as the mean [$\alpha$/Fe] of a stellar population.

In addition, and based on line-strength measurements, we quantify the impact of a non-universal initial mass function on the recovered abundance pattern of galaxies.

In particular, we find that a simultaneous fit of the slope of the initial mass function and the [Mg/Fe] results in a shallower [Mg/Fe]-$\sigma$ relation.

Therefore, our results suggest that star formation in massive galaxies lasted longer than what has been reported previously, although it still occurred significantly faster than in the solar neighbourhood.

;Comment: 7 pages, 5 figures, accepted for publication in A&A