  The thermodynamical evolution of gas during the collapse of the primordial
star-forming cloud depends significantly on the initial degree of rotation.
However, there is no clear understanding of how the initial rotation can affect
the heating and cooling process and hence the temperature that leads to the
fragmentation of the gas during Population III star formation. We report the
results from three\hbox{-}dimensional, smoothed-particle hydrodynamics (SPH)
simulations of a rotating self-gravitating primordial gas cloud with a modified
version of the Gadget-2 code, in which the initial ratio of the rotational to
the gravitational energy ($\beta_0$) is varied over two orders of magnitude. We
find that despite the lack of any initial turbulence and magnetic fields in the
clouds, the angular momentum distribution leads to the formation and build-up
of a disk that fragments into several clumps. We further examine the behavior
of the protostars that form in both idealized as well as more realistic
minihalos from the cosmological simulations. The thermodynamical evolution and
the fragmentation behavior of the cosmological minihalos are similar to that of
the artificial cases, especially in those with a similar $\beta_0$-parameter.
Protostars with a higher rotation support exhibit spiral-arm-like structures on
several scales, and have lower accretion rates. These type of clouds tend to
fragment more, while some of the protostars escape from the cluster with the
possibility of surviving until the present day. They also take much longer to
form compared to their slowly rotating counterparts. We conclude that the use
of appropriate initial conditions of the gas in minihalos is a pivotal and
decisive quantity to study the evolution and final fate of the primordial
stars.
