  Understanding the formation of the first objects in the universe critically
depends on knowing whether the properties of small dark matter structures at
high-redshift (z > 15) are different from their more massive lower-redshift
counterparts. To clarify this point, we performed a high-resolution N-body
simulation of a cosmological volume 1 Mpc/h comoving on a side, reaching the
highest mass resolution to date in this regime. We make precision measurements
of various physical properties that characterize dark matter haloes (such as
the virial ratio, spin parameter, shape, and formation times, etc.) for the
high-redshift (z > 15) dark matter mini-haloes we find in our simulation, and
compare them to literature results and a moderate-resolution comparison run
within a cube of side-length 100 Mpc/h. We find that dark matter haloes at
high-redshift have a log-normal distribution of the dimensionless spin
parameter centered around {\lambda} $\sim$ 0.03, similar to their more massive
counterparts. They tend to have a small ratio of the length of the shortest
axis to the longest axis (sphericity), and are highly prolate. In fact, haloes
of given mass that formed recently are the least spherical, have the highest
virial ratios, and have the highest spins. Interestingly, the formation times
of our mini-halos depend only very weakly on mass, in contrast to more massive
objects. This is expected from the slope of the linear power spectrum of
density perturbations at this scale, but despite this difference, dark matter
structures at high-redshift share many properties with their much more massive
counterparts observed at later times.
