Comparing numerical and analytical calculations of post-innermost stable circular orbit ringdown amplitudes

We numerically compute the ringdown amplitudes following the plunge of a particle from the innermost stable circular orbit of a Schwarzschild black hole in the extreme-mass ratio limit. We show that the ringdown amplitudes computed in this way are in good agreement with a recent analytical calculation [S. Hadar and B. Kol, this issue, Phys. Rev. D 84 DM10604 (2011)].].

Figure 1

Visual comparison of two waveforms for $l=m=2$ with differing cutoffs ($ϵ={10}^{-2}$, $5\times {10}^{-3}$). The time axis has been shifted so that the maxima of the wave amplitudes $|X_{lm}(t)|$ (shown in the inset) occur at the same value of $(t-r_{*})/M$.

Figure 2

Amplitude comparison for 16 different modes with $n=1$ and $ϵ=5\times {10}^{-3}$. The horizontal axis represents the decay constant ${\gamma }_{nlm}$ in units where $r_s=1$. Each point represents the residual $\Delta$ value (after subtracting a linear fit or tuning $\mathcal{N}$, $t_0$) for the given $(l,m)$. Different ${\gamma }_{nlm}$ values correspond to different $l$’s: the leftmost points correspond to $l=2$, and ${\gamma }_{nlm}$ grows monotonically with $l$.