Mass and shape of the Milky Way's dark matter halo with globular clusters from Gaia and Hubble

We estimate the mass of the inner ($<20$ kpc) Milky Way and the axis ratio of its dark matter halo using globular clusters as tracers. At the same time, we constrain the phase-space distribution of the globular cluster system. We use the Gaia DR2 catalogue of 75 globular clusters' proper motions and recent measurements of the proper motions of another 20 distant clusters obtained with the Hubble Space Telescope. We describe the globular cluster system with a 2-component distribution function (DF), with a flat, rotating disc and a rounder, more extended halo. While fixing the Milky Way's disc and bulge, we let the mass and shape of the dark matter halo and we fit these two parameters, together with other six describing the DF, with a Bayesian method. We find the mass of the Galaxy within 20 kpc to be $M(<20{\,\rm kpc})=1.91^{+0.18}_{-0.17} \times 10^{11} M_\odot$, of which $M_{\rm DM}(<20{\,\rm kpc})=1.37^{+0.18}_{-0.17}\times 10^{11}M_\odot$ is in dark matter, and the density axis ratio of the dark matter halo to be $q=1.30 \pm 0.25$. This implies a virial mass $M_{\rm via} = 1.3 \pm 0.3 \times 10^{12} M_\odot$. Our analysis rules out oblate ($q<0.8$) and strongly prolate halos ($q>1.9$) with 99\% probability. Our preferred model reproduces well the observed phase-space distribution of globular clusters and has a disc component that closely resembles that of the Galactic thick disc. The halo component follows a power-law density profile $\rho \propto r^{-3.3}$, has a mean rotational velocity of $V_{\rm rot}\simeq -14\,\rm km\,s^{-1}$ at 20 kpc, and has a mildly radially biased velocity distribution ($\beta\simeq 0.2 \pm 0.07$, fairly constant outside 15 kpc). We also find that our distinction between disc and halo clusters resembles, although not fully, the observed distinction in metal-rich ([Fe/H]$>-0.8$) and metal-poor ([Fe/H]$\leq-0.8$) cluster populations.