PULSAR rotation rates generally decrease steadily owing to external electromagnetic braking torques, but occasionally show sudden increases ('glitches') followed by gradual recoveries that may last days or years. These events are thought to be consequences of angular momentum transfer between a solid crust, which rotates at the measured pulsar periodicity, and a more rapidly rotating "loose' component of the neutron star interior. Sudden braking of the differential rotation between the two components will cause a glitch1, and the subsequent re-establishment of rotational equilibrium between the two components represents the recovery2. Earlier studies, using particular models for the coupling between crust and interior, showed that the loose component carries ∼2.8% and ≳1% of the total moment of inertia of the Vela pulsar3 and PSR 1737 – 30 (ref. 4) respectively. Here, we analyse post-glitch recovery in four pulsars, and deduce that the loose component carries at least 0.8% of the total moment of inertia, independent of the form of the coupling. In the context of the 'vortex creep' model of recovery, in which the loose component is the inner-crust neutron superfluid2'5–7, the constraint on the moment of inertia rules out equations of state that are soft at high densities.