  (abridged) The evolution of galaxy clustering from z=0 to z=4.5 is analyzed
using the angular correlation function and the photometric redshift
distribution of galaxies brighter than I_{AB}\le 28.5 in the HDF North. The
reliability of the photometric redshift estimates is discussed on the basis of
the available spectroscopic redshifts, comparing different codes and
investigating the effects of photometric errors. The redshift bins in which the
clustering properties are measured are then optimized to take into account the
uncertainties of the photometric redshifts. The results show that the comoving
correlation length has a small decrease in the range 0<z<1 followed by an
increase at higher z. We compare these results with the theoretical predictions
of a variety of cosmological models belonging to the general class of CDM. The
comparison with the expected mass clustering evolution indicates that the
observed high-redshift galaxies are biased tracers of the dark matter with an
effective bias b strongly increasing with redshift. Assuming an Einstein-de
Sitter universe, we obtain b\simeq 2 at z=2 and b\simeq 5 at z=4. A comparison
of the clustering amplitudes that we measured at z=3 with those reported for
LBG suggests that the clustering depends on the abundance of the objects: more
abundant objects are less clustered, as expected in the paradigm of
hierarchical galaxy formation. The strong clustering and high bias measured at
z=3 are consistent with the expected density of massive haloes predicted for
the various cosmologies here considered. At z=4, the strong clustering observed
in the HDF requires a significant fraction of massive haloes to be already
formed by that epoch. This feature could be a discriminant test for the
cosmological parameters if confirmed by future observations.
