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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. 

