  The emission of gravitational waves (GWs) during single-single close
encounters in galactic nuclei (GNs) leads to the formation and rapid merger of
highly eccentric stellar-mass black hole (BH) binaries. The distinct
distribution of physical parameters makes it possible to statistically
distinguish this source population from others. Previous studies determined the
expected binary parameter distribution for this source population in single
GNs. Here we take into account the effects of dynamical friction,
post-Newtonian corrections, and observational bias to determine the detected
sources' parameter-distributions from all GNs in the Universe. We find that the
total binary mass distribution of detected mergers is strongly tilted towards
higher masses. The distribution of initial peak GW frequency is remarkably high
between 1-70 Hz, ~50% of GW capture sources form above 10 Hz with e >~ 0.95.
The eccentricity when first entering the LIGO/Virgo/KAGRA band satisfies e_10Hz
> 0.1 for over 92% of sources and e_10Hz > 0.8 for more than half of the
sources. At the point when the pericenter reaches 10 GM/c^2 the eccentricity
satisfies e_10M > 0.1 for over ~70% of the sources, making single-single GW
capture events in GNs the most eccentric source population among the currently
known stellar-mass binary BH merger channels in the Universe. We identify
correlations between total mass, mass ratio, source detection distance, and
eccentricities e_10Hz and e_10M. The recently measured source parameters of
GW190521 lie close to the peak of the theoretical distributions and the
estimated escape speed of the host environment is ~7.5x10^3 km/s - 1.2x10^4
km/s, making this source a candidate for this astrophysical merger channel.
