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It is known observationally that the major axes of galaxy clusters and theirbrightest cluster galaxies are roughly aligned with each other. To understandthe origin of the alignment, we identify 40 cluster-sized dark matter (DM)haloes with masses higher than $5\times10^{13}~M_{\odot}$ and their centralgalaxies (CGs) at $z\approx 0$ in the Horizon-AGN cosmological hydrodynamicalsimulation. We trace the progenitors at 50 different epochs between $0<z<5$. Wethen fit their shapes and orientations with a triaxial ellipsoid model. Whilethe orientations of both DM haloes and CGs change significantly due to repeatedmergers and mass accretions, their relative orientations are well aligned ateach epoch even at high redshifts, $z>1$. The alignment becomes tighter withcosmic time; the major axes of the CGs and their host DM haloes at present arealigned on average within $\sim 30^{\circ}$ in the three dimensional space and$\sim 20^{\circ}$ in the projected plane. The orientations of the major axes ofDM haloes on average follow one of the eigen-vectors of the surrounding tidalfield that corresponds to the {\it slowest collapsing} (or even stretching)mode, and the alignment with the tidal field also becomes tighter. This impliesthat the orientations of CGs and DM haloes at the present epoch are largelyimprinted in the primordial density field of the Universe, whereas strongdynamical interactions such as mergers are important to explain their mutualalignment at each epoch.

Cosmological evolution of orientations of cluster-sized dark matter haloes and their central galaxies in the Horizon-AGN simulation