An important aspect of understanding the formation of bulges of galaxies, including that of the Milky Way, is understanding their metallicity trends. The most notable feature of the metallicity distribution is that when the bulge is viewed edge-on, it appears more vertically pinched than the density. In the Milky Way this manifests as an X-shape when viewed from the Solar perspective. Simulations that include gas, star formation and the chemistry of stars have managed to reproduce the pinched appearance of the metallicity (Debattista et al. 2017). However these simulations are expensive and do not permit an easy detailed matching to the Milky Way. In the past this has forced researchers to paint metallicity on particles in N-body simulations (i.e. without any gas or star formation). This type of painting has assigns a metallicity based on the position of a star (both its distance from the galactic centre and its height above the mid-plane). However this is equivalent to assuming that all stars at a particular position are born at the same time with the same metallicity, which is highly implausible. Using a suite of N-body simulations, we have shown that where a star ends up on a bulge depends strongly on its initial dynamical actions, which measure its angular momentum as well as its radial and vertical random motion. This allowed us to develop a novel technique for painting metallicity onto star particles using the dynamical actions. We showed that this technique produces metallicity maps comparable to those of star forming simulations. This more realistic metallicity assignment allows us to much more rapidly model the formation of the bulge of the Milky Way. The figure shows an example of such a metallicity mapping: the metallicity distribution (colours and black contours) is more pinched along the minor axis than is the density (white contours), as observed in the Milky Way and in external galaxies.

Moreover, using the actions to understand bulge formation has lead us to a number of important new insights. We showed for instance that the vertical gradient of the vertical action is largely erased by the formation of the bulge, whereas the very weak gradient of the radial action is vastly enhanced during the same time. This means that the vertical gradients observed in galaxies are not the result of the superposition of a metal-rich thin disc and a metal-poor thick disc. Another, quite surprising, result of our use of dynamical actions is that there are populations that vertically cool during bulge formation. We are investigating the properties of these populations and hope to be able to use them to constrain what the Milky Way looked like before the bulge formed.