Mode coupling mechanism for late--time Kerr tails

We consider the decay rate for scalar fields in Kerr spacetime. We consider pure initial multipoles $\ell'$, and focus attention in the decay rate of the multipole $\ell$. We use an iterative method proposed by Gleiser, Price, and Pullin, and identify the mode coupling mechanism that gives rise to a decay rate formula recently proposed by Zengino\u{g}lu, Khanna, and Burko through the iterations in powers of the square of the Kerr black hole's specific angular momentum. We also show that one may identify the dominant channel of mode excitation, and obtain approximate results for the mode of interest by studying the dominant channel. The results of the dominant channel approximation approach the full--mode results at late times, and their difference approaches zero quadratically in inverse time.