Constraint on the quadrupole moment of super-massive black hole candidates from the estimate of the mean radiative efficiency of AGN

The super-massive objects at the center of many galaxies are commonly thought to be black holes. In 4-dimensional general relativity, a black hole is completely specified by its mass $M$ and by its spin angular momentum $J$. All the higher multipole moments of the gravitational field depend in a very specific way on these two parameters. For instance, the mass quadrupole moment is $Q = - J^2/M$. If we can estimate $M$, $J$, and $Q$ for the super-massive objects in galactic nuclei, we over-constrain the theory and we can test the black hole hypothesis. While there are many works studying how this can be done with future observations, in this paper a constraint on the quadrupole moment of these objects is obtained by using the current estimate of the mean radiative efficiency of AGN. In terms of the anomalous quadrupole moment $q$, the bound is $-2.01 < q < 0.14$.