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We have used the hydrodynamical adaptive mesh refinement code  enzo  to investigate the dynamical evolution of the gas at the centre of dark matter haloes with virial velocities of  ∼20−30 km s −1   and virial temperatures of  ∼13 000–30 000 K  at   z ∼ 15  in a cosmological context. The virial temperature of the dark matter haloes is above the threshold where atomic cooling by hydrogen allows the gas to cool and collapse. We neglect cooling by molecular hydrogen and metals, as may be plausible if  H 2   cooling is suppressed by a metagalactic Lyman–Werner background or an internal source of Lyman–Werner photons, and metal enrichment has not progressed very far. The gas in the haloes becomes gravitationally unstable and develops turbulent velocities comparable to the virial velocities of the dark matter haloes. Within a few dynamical times, it settles into a nearly isothermal density profile over many decades in radius losing most of its angular momentum in the process. About 0.1–1 per cent of the baryons, at the centre of the dark matter haloes, collapse into a self‐gravitating, fat, ellipsoidal, centrifugally supported exponential disc with scalelength of  ∼0.075–0.27  pc and rotation velocities of  25–60 km s −1  . We are able to follow the settling of the gas into centrifugal support and the dynamical evolution of the compact disc in each dark matter halo for a few dynamical times. The dynamical evolution of the gas at the centre of the haloes is complex. In one of the haloes, the gas at the centre fragments into a triple system leading to strong tidal perturbations and eventually to the infall of a secondary smaller clump into the most massive primary clump. The formation of centrifugally supported self‐gravitating massive discs is likely to be an important intermediary stage en route to the formation of a massive black hole seed.

The formation of compact massive self‐gravitating discs in metal‐free haloes with virial temperatures of ∼13 000–30 000 K