8 0 0.5 1 1.5 2 2.5 x 10 −3 5 5.5 6 6.5 7 log$_{10}$ M / t 0 n 4 1.6 1.8 2 2.2 2.4 x 10 −3 7 7.05 7.1 7.15 7.2 7.25 100Kx100 80Kx80 64Kx64 0.5 1 1.5 2 2.5 3 3.5 −25 −20 −15 −10 −5 0 5 log$_{10}$ t / M log 10 |  3  $_{ψ}$ L  | L = 3 L = 1 L = 5 FIG. 7: The ℓ = 1 , 3 , 5 multipole projections, for initial data of a pure octupole field as functions of time. The late–time field behavior is computationally found to be given by ψ 1 ∼ t − 5 . $^{002}$, ψ 3 ∼ t − 9 . $^{009}$, and ψ 5 ∼ t − 11 . $^{008}$, respectively. FIG. 6: The local power index 0 n 4 as a function of the time t for three grid resolutions, 64 K × 64 (dash–dotted curve), 80 K × 80 (dashed curve), and 100 K × 100 (solid curve). for these modes as functions of time. Starting with ℓ ′ = 1, we project the dipole and octupole modes and their respective local power indices in Fig. 9. We encounter the same problem finding the decay rate for ℓ = 5 — i.e., find $_{1}$ψ$_{5}$ and $_{1}$n$_{5}$ — as we did when we we calculated $_{0}$ψ$_{4}$ and $_{0}$n$_{4}$ (Figs. 10 and 11). Again, the second–order “up”–excited mode requires high resolution simulations. Our proposal (3) naturally extends the Barack–Ori formula (2) to cases where the initial data do not include the monopole ( ℓ ′ = 0) mode. Recall that the Barack–Ori formula considers the case of generic initial data, which means a mixture of all multipole modes with arbitrary relative amplitudes and radial profiles. In particular, the monopole mode is generically present, so that the slowest decaying contribution of any “up”–excited ℓ mode is generated by the initial monopole. Our proposal degenerates to the Barack–Ori formula when ℓ ′ = 0. When the monopole is not present in the generic initial data set, our proposal will produce the decay rate of a mixture of modes when ℓ ′ is taken to be the smallest multipole present in the generic data set, and as a particular case will also produce the decay rate of the ℓ multipole when the initial data is a pure ℓ $^{′}$( < ℓ ) mode, the non–generic case, in the language of Barack and Ori. In summary, the Barack–Ori formula Eq. (2) is simply the particular case $_{0}$n$_{ℓ}$ of Eq. (3). Table I summarizes our results for the azimuthal, m = 0, case. It shows the power law indices for the late