It is now well established that many galaxies have nuclear star clusters (NCs) whose total masses correlate with the velocity dispersion σ of the galaxy spheroid in a very similar way to the well-known supermassive black hole (SMBH) M−σ relation. Previous theoretical work suggested that both correlations can be explained by a momentum feedback argument. Observations further show that most known NCs have masses ≲108 M⊙, while SMBHs frequently have measured masses ≳108 M⊙, which remained unexplained in earlier treatments. We suggest here that this changeover reflects a competition between the SMBH and nuclear clusters in the feedback they produce. When one of the massive objects reaches its limiting M−σ value, it drives the gas away and hence cuts off its own mass and also the mass of the ‘competitor’. The latter is then underweight with respect to the expected M−σ mass.
More specifically, we find that the bulge dynamical time-scale is a steeply rising function of velocity dispersion, and that the NC–SMBH changeover occurs where the dynamical time is about equal to the Salpeter time. We propose that SMBHs, growing on the Salpeter time-scale, are unable to reach their M−σ mass quickly enough in small bulges. The central regions of these bulges are swamped with gas which fragments into stars, creating the nuclear clusters. The latter then limit their own growth by the feedback they produce, settling on their (offset) M−σ relation. The SMBH in such bulges should be underweight as their growth is curtailed before they reach the M−σ mass. In large bulges, on the other hand, the SMBH catches up quickly enough to settle on its M−σ relation. Nuclear star clusters may also exist in such bulges but they should be underweight with respect to their M−σ sequence.