Evolution of the spin parameter of accreting compact objects with non-Kerr quadrupole moment

There is robust observational evidence supporting the existence of 5 - 20 M-circle dot compact bodies in X-ray binary systems and of 10(5) - 10(9) M-circle dot bodies at the center of many galaxies. All these objects are commonly interpreted as black holes, even is there is no direct evidence that they have an event horizon. A fundamental limit for a black hole in 4-dimensional general relativity is the Kerr bound vertical bar a(*)vertical bar <= 1, where a(*) is the spin parameter. This is just the condition for the existence of the event horizon. The accretion process can spin a black hole up to a* approximate to 0.998 and some super-massive objects in galactic nuclei could be rapidly rotating black holes with spin parameter close to this limit. However, if these super-massive objects are not black holes, the Kerr bound does not hold and the accretion process can spin them up to a(*) >= 1. In this paper, I consider compact bodies with non-Kerr quadrupole moment. I study the evolution of the spin parameter due to accretion and I find its equilibrium value. Future experiments like the gravitational wave detector LISA will be able to test if the super-massive objects at the center of galaxies are the black holes predicted by general relativity. If they are not black holes, some of them may be super-spinning objects with a(*) > 1.