  The mechanisms which could lead to chemo-thermal instabilities and
fragmentation during the formation of primordial protostars are investigated
analytically. We introduce approximations for H2 cooling rates bridging the
optically thin and thick regimes. These allow us to discuss instabilities up to
densities when protostars become optically thick to continuum radiation
(n~10^16 cm^-3). During the collapse, instability arises at two different
stages: at low density (n~10^8-10^11 cm^-3), it is due to fast 3-body reactions
converting H into H2; at high density (n>10^13 cm^-3), it is due to Collisional
Induced Emission (CIE). In agreement with the 3D simulations, we find that the
instability at low densities cannot lead to fragmentation, because fluctuations
do not survive turbulent mixing, and because their growth is slow. The
situation at high density is similar. The CIE-induced instability is as weak as
the low density one, with similar ratios of growth and dynamical time scales.
Fluctuation growth time is longer than free fall time, and fragmentation seems
unlikely. One then expects the first stars to be massive, not to form binaries
nor harbour planets. Nevertheless, full 3D simulations are required. They could
become possible using simplified estimates of radiative transfer effects, which
we show to work very well in the 1D case. This indicates that the effects of
radiative transfer during the initial stages of formation of primordial
protostars can be treated as local corrections to cooling. (Abridged)
