We have included the chemical rate network responsible for the formation of
molecular Hydrogen in the N-body hydrodynamic code, Hydra, in order to study
the formation of the first cosmological at redshifts between 10 and 50. We have
tested our implementation of the chemical and cooling processes by comparing
N-body top hat simulations with theoretical predictions from a semi-analytic
model and found them to be in good agreement. We find that post-virialization
properties are insensitive to the initial abundance of molecular hydrogen. Our
main objective was to determine the minimum mass ($M_{SG}(z)$) of perturbations
that could become self gravitating (a prerequisite for star formation), and the
redshift at which this occurred. We have developed a robust indicator for
detecting the presence of a self-gravitating cloud in our simulations and find
that we can do so with a baryonic particle mass-resolution of 40 solar masses.
We have performed cosmological simulations of primordial objects and find that
the object's mass and redshift at which they become self gravitating agree well
with the $M_{SG}(z)$ results from the top hat simulations. Once a critical
molecular hydrogen fractional abundance of about 0.0005 has formed in an
object, the cooling time drops below the dynamical time at the centre of the
cloud and the gas free falls in the dark matter potential wells, becoming self
gravitating a dynamical time later.