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On the Offset of Barred Galaxies From the Black Hole $M_{BH}$-$σ$ Relationship

We use collisionless $N$-body simulations to determine how the growth of a supermassive black hole (SMBH) influences the nuclear kinematics in both barred and unbarred galaxies. In the presence of a bar, the increase in the velocity dispersion $σ$ (within the effective radius) due to the growth of an SMBH is on average $\lesssim 10%$, whereas the increase is only $\lesssim 4%$ in an unbarred galaxy. In a barred galaxy, the increase results from a combination of three separate factors (a) orientation and inclination effects; (b) angular momentum transport by the bar that results in an increase in the central mass density; (c) an increase in the vertical and radial velocity anisotropy of stars in the vicinity of the SMBH. In contrast the growth of the SMBH in an unbarred galaxy causes the velocity distribution in the inner part of the nucleus to become less radially anisotropic. The increase in $σ$ following the growth of the SMBH is insensitive to a variation of a factor of 10 in the final mass of the SMBH, showing that it is the growth process rather than the actual SMBH mass that alters bar evolution in a way that increases $σ$. We argue that using an axisymmetric stellar dynamical modeling code to measure SMBH masses in barred galaxies could result in a slight overestimate of the derived $M_{BH}$, especially if a constant M/L ratio is assumed. We conclude that the growth of a black hole in the presence of a bar could result in an increase in $σ$ which is roughly of $4-8%$ larger than the increase that occurs in an axisymmetric system. While the increase in $σ$ due to SMBH growth in a barred galaxy might partially account for the claimed offset of barred galaxies and pseudo bulges from the $M_{BH}-σ$ relation obtained for elliptical galaxies and classical bulges in unbarred galaxies, it is inadequate to account for all of the offset.