  We study the gravitational instability and fragmentation of primordial
protostellar discs by using high-resolution cosmological hydrodynamics
simulations. We follow the formation and evolution of spiral arms in
protostellar discs, examine the dynamical stability, and identify a physical
mechanism of secondary protostar formation. We use linear perturbation theory
based on the spiral-arm instability (SAI) analysis in our previous studies. We
improve the analysis by incorporating the effects of finite thickness and
shearing motion of arms, and derive the physical conditions for SAI in
protostellar discs. Our analysis predicts accurately the stability and the
onset of arm fragmentation that is determined by the balance between
self-gravity and gas pressure plus the Coriolis force. Formation of secondary
and multiple protostars in the discs is explained by the SAI, which is driven
by self-gravity and thus can operate without rapid gas cooling. We can also
predict the typical mass of the fragments, which is found to be in good
agreement with the actual masses of secondary protostars formed in the
simulation.
