  Recent high-resolution simulations demonstrate that disks around primordial
protostars easily fragment in the accretion phase before the protostars accrete
less than a solar mass. To understand why the gravitational instability
generally causes the fragmentation so early, we develop a one-dimensional (1D)
non-steady model of the circumstellar disk that takes the mass supply from an
accretion envelope into account. We also compare the model results to a
three-dimensional (3D) numerical simulation performed with a code employing the
adaptive mesh refinement. Our model shows that the self-gravitating disk,
through which the Toomre $Q$ parameter is nearly constant at $Q \sim 1$,
gradually spreads as the disk is fed by the gas infalling from the envelope. We
further find that the accretion rate onto the star is an order of magnitude
smaller than the mass supply rate onto the disk. This discrepancy makes the
disk more massive than the protostar in an early evolutionary stage. Most of
the infalling gas is used to extend the outer part of the self-gravitating disk
rather than transferred inward toward the star through the disk. We find that
similar evolution also occurs in the 3D simulation, where the disk becomes
three times more massive than the star before the first fragmentation occurs.
Our 1D disk model well explains the evolution of the disk-to-star mass ratio
observed in the simulation. We argue that the formation of such a massive disk
leads to the early disk fragmentation.
