The $\chi_\mathrm{eff}-z$ correlation of field binary black hole mergers
  and how 3G gravitational-wave detectors can constrain it
  Understanding the origin of merging binary black holes is currently one of
the most pressing quests in astrophysics. We show that if isolated binary
evolution dominates the formation mechanism of merging binary black holes, one
should expect a correlation between the effective spin parameter,
$\chi_\mathrm{eff}$, and the redshift of the merger, $z$, of binary black
holes. This correlation comes from tidal spin-up systems preferentially forming
and merging at higher redshifts due to the combination of weaker orbital
expansion from low metallicity stars given their reduced wind mass loss rate,
delayed expansion and have smaller maximal radii during the supergiant phase
compared to stars at higher metallicity. As a result, these tightly bound
systems merge with short inspiral times. Given our fiducial model of isolated
binary evolution, we show that the origin of a $\chi_\mathrm{eff}-z$
correlation in the detectable LIGO--Virgo binary black hole population is
different from the intrinsic population, which will become accessible only in
the future by third-generation gravitational-wave detectors such as Einstein
Telescope and Cosmic Explorer. Finally, we compare our model predictions with
population predictions based on the current catalog of binary black hole
mergers and find that current data favor a positive correlation of
$\chi_\mathrm{eff}-z$ as predicted by our model of isolated binary evolution.
