Invoking the virial theorem to understand the impact of (dry) mergers on the $M_{\rm bh}$-$\sigma$ relation

While dry mergers can produce considerable scatter in the (black hole mass, $M_{\rm bh}$)-(spheroid stellar mass, $M_{\rm *,sph}$) and $M_{\rm bh}$-(spheroid half-light radius, $R_{\rm e,sph}$) diagrams, the virial theorem is used here to explain why the scatter about the $M_{\rm bh}$-(velocity dispersion, $\sigma$) relation remains low in the face of such mergers. Its small scatter has been claimed as evidence of feedback from active galactic nuclei (AGNs). However, it is shown that galaxy mergers also play a significant role. The major merger of two S0 galaxies with $M_{\rm *,sph}\sim10^{11}$ M$_\odot$ advances a system along a slope of $\sim$5 in the $M_{\rm bh}$-$\sigma$ diagram. However, a major E$+$E galaxy merger moves a system (slightly) along a trajectory with a slope of $\sim$9, while mergers of lower-mass S0 galaxies with $M_{\rm *,sph}\sim10^{10}$ M$_\odot$ move (slightly) along a trajectory with a slope of $\sim$3. This produces a steeper distribution for the E (and Es,e) galaxies in the $M_{\rm bh}$-$\sigma$ diagram, reported here to have a slope of 7.27$\pm$0.91, compared to the S0 galaxies which have a slope of 5.68$\pm$0.60. This result forms an important complement to the AGN feedback models like that from Silk and Rees, providing a more complete picture of galaxy/(black hole) coevolution. It also has important implications for nanohertz gravitational wave research. Abridged.