FIG. 8: The local power indices of the ℓ = 1 , 3 , 5 multipole projections as functions of time, for initial data of a pure octupole field (same data as in Fig. 7). FIG. 9: The dipole ( ℓ = 1) and octupole ( ℓ = 3) projections, 1 ψ 3 and 1 ψ 3 , correspondingly (upper panel), and their local power indices, 1 n 1 and 1 n 3 , respectively (left and right lower panels), as functions of time, for initial data of a pure dipole ( ℓ ′ = 1) field. time tails for both “up”- and “down”-modes. It is in some loose sense a “partial scattering matrix,” that includes (partial) information on the excitation amplitude for each of the excitation channels corresponding with a preparation state of the initial data. The scattering matrix is only partial, because it includes no information on the pre-factor ψ$_{0}$ = ψ$_{0}$ ( r$_{0}$, profile) of the late-time tail ψ ( t ) ∼ ψ$_{0}$ t $^{$_{−}$n}$, where r$_{0}$ is the evaluation point, and where ψ$_{0}$ also possibly depends on the radial profile of the initial data. The (partial) scattering matrix includes only the information on n . The scattering matrix has a clear shape of the sum of two triangular matrices: letting rows be associated with ℓ ′ and column with ℓ , the upper triangular matrix elements satisfy n = ℓ ′ + ℓ + 3 and the strictly lower triangular matrix elements satisfy n = ℓ ′ + ℓ + 1. Non–azimuthal modes Our proposal (3) is not limited to azimuthal modes, and predicts that the late-time decay rate of the ℓ, m modes that is excited by the initial ℓ $^{′}$, m mode is independent of m , and depends only on ℓ, ℓ $^{′}$. We have tested this hypothesis in numerous cases, that are summarized in Tables I, II, and III. 9 0 0.5 1 1.5 2 2.5 x 10 −3 11 11.1 11.2 11.3 M / t 3 n 5 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 9 9.1 9.2 9.3 M / t 3 n 3 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 5 5.1 5.2 5.3 M / t 3 n 1 0 0.5 1 1.5 2 2.5 3 −15 −10 −5 0 log$_{10}$ t / M log 10 | 1 L |  L=3  0 1 2 3 x 10 −3 7 7.1 7.2 7.3 7.4  M / t 1 n 3 0 1 2 3 4 x 10 −3 5 5.05 5.1 5.15 5.2 5.25 5.3  M / t 1 n 1  L=1 