We present a new suite of hydrodynamical simulations and use it to study, in detail, black hole and galaxy properties. The high time, spatial and mass resolution, and realistic orbits and mass ratios, down to 1:6 and 1:10, enable us to meaningfully compare star formation rate (SFR) and BH accretion rate (BHAR) timescales, temporal behaviour and relative magnitude. We find that (i) BHAR and galaxy-wide SFR are typically temporally uncorrelated, and have different variability timescales, except during the merger proper, lasting ~0.2-0.3 Gyr. BHAR and nuclear (<100 pc) SFR are better correlated, and their variability are similar. Averaging over time, the merger phase leads typically to an increase by a factor of a few in the BHAR/SFR ratio. (ii) BHAR and nuclear SFR are intrinsically proportional, but the correlation lessens if the long-term SFR is measured. (iii) Galaxies in the remnant phase are the ones most likely to be selected as systems dominated by an active galactic nucleus (AGN), because of the long time spent in this phase. (iv) The timescale over which a given diagnostic probes the SFR has a profound impact on the recovered correlations with BHAR, and on the interpretation of observational data.
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