  During the first three observing runs of the Advanced gravitational-wave
detector network, the LIGO/Virgo collaboration detected several black hole
binary (BHBH) mergers. As the population of detected BHBH mergers grows, it
will become possible to constrain different channels for their formation. Here
we consider the chemically homogeneous evolution (CHE) channel in close
binaries, by performing population synthesis simulations that combine realistic
binary models with detailed cosmological calculations of the chemical and
star-formation history of the Universe. This allows us to constrain population
properties, as well as cosmological and aLIGO/aVirgo detection rates of BHBH
mergers formed through this pathway. We predict a BHBH merger rate at redshift
zero of $5.8 \, \textrm{Gpc}^{-3} \textrm{yr}^{-1}$ through the CHE channel, to
be compared with aLIGO/aVirgo's measured rate of ${53.2}_{-28.2}^{+55.8} \,
\text{Gpc}^{-3} \text{yr}^{-1}$, and find that eventual merger systems have BH
masses in the range $17 - 43 \, \textrm{M}_{\odot}$ below the pair-instability
supernova (PISN) gap, and $>124 \, \textrm{M}_{\odot}$ above the PISN gap. We
investigate effects of momentum kicks during black hole formation, and
calculate cosmological and magnitude limited PISN rates. We also study the
effects of high-redshift deviations in the star formation rate. We find that
momentum kicks tend to increase delay times of BHBH systems, and our magnitude
limited PISN rate estimates indicate that current deep surveys should be able
to detect such events. Lastly, we find that our cosmological merger rate
estimates change by at most $\sim 8\%$ for mild deviations of the star
formation rate in the early Universe, and by up to $\sim 40\%$ for extreme
deviations.
