  Population III (Pop. III) binary stars likely produced the first stellar-born
binary black hole (BBH) mergers in the Universe. Here, we quantify the main
sources of uncertainty for the merger rate density evolution and mass spectrum
of Pop. III BBHs by considering four different formation histories and 11
models of the initial orbital properties of Pop. III binary stars. The
uncertainty on the orbital properties affects the BBH merger rate density by up
to two orders of magnitude, models with shorter orbital periods leading to
higher BBH merger rates. The uncertainty on the star formation history has a
substantial impact on both the shape and the normalisation of the BBH merger
rate density: the peak of the merger rate density shifts from $z\sim{8}$ up to
$z\sim{16}$ depending on the assumed star formation rate, while the maximum BBH
merger rate density for our fiducial binary population model spans from
$\sim{2}$ to $\sim{30}$ Gpc$^{-3}$ yr$^{-1}$. The typical BBH masses are not
affected by the star formation rate model and only mildly influenced by the
binary population parameters. The primary black holes born from Pop. III stars
tend to be rather massive ($30-40$ M$_\odot$) with respect to those born from
metal-rich stars ($8-10$ M$_\odot$). We estimate that the Einstein Telescope
will detect $10-10^4$ Pop. III BBH mergers per year, depending on the star
formation history and binary star properties.
