  Dark matter halo demographics and assembly histories are a manifestation of
cosmological structure formation and have profound implications for the
formation and evolution of galaxies. In particular, merger trees provide
fundamental input for several modelling techniques, such as semi-analytic
models (SAMs), sub-halo abundance matching (SHAM), and decorated halo
occupation distribution models (HODs). Motivated by the new ultra-high-redshift
(z > 10) frontier enabled by JWST, we present a new suite of Gadget at
Ultrahigh Redshift with Extra-Fine Timesteps (GUREFT) dark matter-only
cosmological simulations that are carefully designed to capture halo merger
histories and structural properties in the ultra-z universe. The simulation
suite consists of four 1024^3-particle simulations with box sizes of 5, 15, 35,
and 90 Mpc h-1, each with 170 snapshots stored between 40 > z > 6. With the
unprecedented number of available snapshots and strategically chosen dynamic
range covered by these boxes, gureft uncovers the emerging dark matter halo
populations and their assembly histories in the earliest epochs of cosmic
history. In this work, we present the halo mass functions between z ~ 20 to 6
down to log(Mvir/Msun) ~ 5, and show that at high redshift, these robust halo
mass functions can differ substantially from commonly used analytic
approximations or older fitting functions in the literature. We also present
key physical properties of the ultra-z halo population, such as concentration
and spin, as well as their mass growth and merger rates, and again provide
updated fitting functions.
