  (Reduced) Using the recent cosmological high-resolution zoom-in simulations,
NewHorizon and Galactica, in which the evolution of black hole spin is followed
on the fly, we have tracked the cosmic history of a hundred of black holes
(BHs) with a mass greater than 2x10^4 Ms. For each of them, we have studied the
variations of the three dimensional angle (Psi) subtended between the BH spins
and the angular momentum vectors of their host galaxies. The analysis of the
individual evolution of the most massive BHs suggests that they are generally
passing by three different regimes. First, for a short period after their
birth, low mass BHs (<3x10^4 Ms) are rapidly spun up by gas accretion and their
spin tends to be aligned with their host galaxy spin. Then follows a second
phase in which the accretion of gas onto low mass BHs (<10^5 Ms) is quite
chaotic and inefficient, reflecting the complex and disturbed morphologies of
forming proto-galaxies at high redshifts. The variations of Psi are rather
erratic during this phase and are mainly driven by the rapid changes of the
direction of the galaxy angular momentum. Then, in a third and long phase, BHs
are generally well settled in the center of galaxies around which the gas
accretion becomes much more coherent (>10^5 Ms). In this case, the BH spins
tend to be well aligned with the angular momentum of their host galaxy and this
configuration is generally stable even though BH merger episodes can temporally
induce misalignment. We have also derived the distributions of cos(Psi) at
different redshifts and found that BHs and galaxy spins are generally aligned.
Finally, based on a Monte Carlo method, we also predict statistics for the 2-d
projected spin-orbit angles lambda. In particular, the distribution of lambda
traces well the alignment tendency in the 3-d analysis. Such predictions
provide an interesting background for future observational analyses.
