Onset of spontaneous scalarization in spinning Gauss-Bonnet black holes

It has recently been proved numerically that spinning black holes in Einstein-scalar theories which are characterized by a non-minimal negative coupling of the scalar field to the Gauss-Bonnet invariant of the curved spacetime may develop exponentially growing instabilities. Intriguingly, it has been demonstrated that this tachyonic instability, which marks the onset of the spontaneous scalarization phenomenon in the Einstein-Gauss-Bonnet-scalar theory, characterizes spinning black holes whose dimensionless angular momentum parameter $\bar{a}\equiv a\mathrm{/}M$ is larger than some critical value ${\bar a}_{$\text{crit}$}\simeq0.505$. In the present paper we prove, using {\it analytical} techniques, that the critical rotation parameter which marks the boundary between bald Kerr black holes and hairy (scalarized) spinning black holes in the Einstein-Gauss-Bonnet-scalar theory is given by the exact dimensionless relation ${\bar{a}}_{\text{crit}}=\frac{1}{2}$.