In this paper, the Kerr black hole is described as having quadrupole moment of $q=J^2/M$ (which means $q=a^2M$ using $J=aM$) whereas in this paper it says in the abstract that the limiting case of Kerr is $q=0$. and finally this paper says $q=-a^2M^3$ (I think this is due to a different definition of $a$ though, as they say $a=J/M^2$). Which one is correct? Perhaps in the second paper $q=0$ in the approximation they take?
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The paper of Bini et al. (2009) discusses a metric where the symbol $q$ stands for a quadrupole parameter. The origin of $q$ is chosen so that it can be understood as parametrizing the deviation from Kerr space-time. On the other hand, the Kerr metric has the magnitude of the quadrupole moment, as precisely defined by Hansen (1974), equal to $Q = a^2 M$.
However, you can use different definitions of the multipoles, and in particular, different sign conventions. This is the case of the paper of Bambi & Barrausse (2011) you refer to, where they use the signature convention where a positive $Q$ refers to a prolate source/field and a negative $Q$ to an oblate source/field. (You have misread their formula which reads $Q \equiv -J^2/M = -a^2 M$).
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$\begingroup$ Thanks, that answers the Bini paper question. Where did you find the formula in the Bambi paper? I was reading the formula just below eqn (10) $\endgroup$ Commented Apr 4, 2019 at 13:39
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1$\begingroup$ @supercoolphysicist In the abstract and, for instance, in equation (1). Note that they use the symbol $a$ in a somewhat different meaning, their $a$ is dimensionless, which is in contrast to the usual definition of the spin parameter that differs by a factor of $M$ and has the dimension of mass (length). $\endgroup$– VoidCommented Apr 4, 2019 at 14:13