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related to the question of the decay rate of the full field. Instead, it is the gaining of more insight towards a more complete understanding of the dynamics and mode coupling properties thereof we are interested in here. Denoting the initially excited pure multipole by ℓ $^{′}$, and the multipole moment of the field whose excitation and decay are is of interest by ℓ (so that for “up”-modes ℓ > ℓ ′ and for “down”-modes ℓ < ℓ $^{′}$), we propose that the late–time decay rate for excited ℓ, m modes along a r = const worldline, is given by t $_{−}$n where 

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Notably, this formula is independent of m . Specifically, for all (allowed) values of m the decay rate of the ℓ excited multipole because of an initial ℓ ′ is the same. This formula is simpler than either the Hod or the Barack–Ori formulas, but is fully consistent with them in their domain of applicability. Specifically, when ℓ ′ = 0 , 1, Eq. (3) coincides with the Barack–Ori formula (2). In the Schwarzschild case there is no mode coupling, ℓ = ℓ $^{′}$, and therefore n = ℓ ′ + ℓ + 3 = 2 ℓ + 3. The numerical evidence described below supports Eq. (3) in all cases we have checked. 

## Azimuthal modes 

To test our proposal —and in particular the m -independence of the decay rate, which may be construed as a statement on the generally m -dependent Green’s function— we focus now on pure multipole initial data sets. ′ 

Letting the initial ℓ = 2, we project the multipoles ℓ = 0 , 2 , 4, and the local power indices are shown in Fig. 3. Of particular interest here is the excitation of the azimuthal, hexadecapole mode, ℓ = 4. In this case, ℓ ′ = 2 , ℓ = 4, so that our proposal is that n = 2 + 4 + 3 = 9. In our numerical simulations, we have found the value of $_{2}$n$_{4}$ = 9 . 01, with uncertainly in the last figure. ′ 

Starting with initial data for a pure ℓ = 0 monopole mode, we project the ℓ = 0 , 2 and 4 modes. The local power indices $_{0}$n$_{0}$ and $_{0}$n$_{2}$ are shown in Fig. 4. The hexadecapole mode ℓ = 4, $_{0}$ψ$_{4}$ , is shown in Fig. 5. Figure 5 suggests that our grid density is not sufficiently high to obtain accurately the very late time hexadecapole projection to the required accuracy. Indeed, as Fig. 5 shows, the tail of the field “curves up” at very late times. However, increasing the grid density appears to straighten up the field, from which we infer that this curving is a numerical artifact. We attribute the problem to our calculation of a mode that is a second–order excited “up” mode, i.e., the ℓ = 4 excited mode from an initial ℓ ′ = 0 mode. The practical problems in determining the decay rate accurately are accentuated by Fig. 6, which displays the local power indices for three grid densities. Figure 6 suggests that indeed a higher resolution simulation would be successful at determining the late–time decay rate with sufficient accuracy. Our results do suggest that $_{0}$n$_{4}$ = 7 (notice, this is a numerical result with one significant figure, not an exact value!), as indicated by the reference curve in Fig. 5, but any determination of the decay rate at higher accuracy than that would require higher resolution simulations. ′ 

We have also tested our proposal with odd modes. Specifically, Let ℓ = 3 and study the late time tail for the modes ℓ = 1 , 3 , 5. Figure 7 shows the projections ℓ = 1 , 3 , 5 as functions of time, and figure 8 displays the local power indices 

