Comprehensive assessment of the too big to fail problem

We use a semi-analytical model for the substructure of dark matter haloes to assess the too big to fail (TBTF) problem. The model accurately reproduces the average subhalo mass and velocity functions, as well as their halo-to-halo variance, in N-body simulations. We construct thousands of realizations of Milky Way (MW)-size host haloes, allowing us to investigate the TBTF problem with unprecedented statistical power. We examine the dependence on host halo mass and cosmology, and explicitly demonstrate that a reliable assessment of TBTF requires large samples of hundreds of host haloes. We argue that previous statistics used to address TBTF suffer from the look-elsewhere effect and/or disregard certain aspects of the data on the MW satellite population. We devise a new statistic that is not hampered by these shortcomings, and, using only data on the nine known MW satellite galaxies with V_max > 15 km s^-1, demonstrate that 1.4^{+3.3}_{-1.1} per cent of MW-size host haloes have a subhalo population in statistical agreement with that of the MW. However, when using data on the MW satellite galaxies down to V_max = 8 km s^-1, this MW consistent fraction plummets to <5 × 10^-4 (at 68 per cent confidence level). Hence, if it turns out that the inventory of MW satellite galaxies is complete down to 8 km s^-1, then the maximum circular velocities of MW satellites are utterly inconsistent with Λ cold dark matter predictions, unless baryonic effects can drastically increase the spread in V_max values of satellite galaxies compared to that of their subhaloes.