= Solution
With $C_{\mathbf k}$ and $a$ from <3/e>, inversion gives
$$
\delta\Phi_{\mathbf k}^{\mathrm{tot}}(t)=\frac1{2\pi}\int_{\operatorname{Im}\omega=\eta>0}d\omega\,e^{-i\omega t}\frac{iC_{\mathbf k}}{(\omega-a)\epsilon_{\mathbf k}(\omega)}.
$$
For $t>0$, deform the contour downwards. Closing below is clockwise, so the <residue theorem> multiplies the sum of enclosed residues by $-i$. The forcing pole has residue $iC_{\mathbf k}e^{-iat}/\epsilon_{\mathbf k}(a+i0^+)$ and therefore contributes
$$
\frac{C_{\mathbf k}e^{-iat}}{\epsilon_{\mathbf k}(a+i0^+)}.
$$
A simple collective root $\omega_j$ of the continued <gravitational dielectric function> contributes
$$
\frac{C_{\mathbf k}e^{-i\omega_jt}}{(\omega_j-a)\epsilon_{\mathbf k}'(\omega_j)}.
$$
These are the <Landau mode> transients excited by introducing the mass abruptly. For a strictly damped background they decay, together with any remaining phase-mixing contour contribution under the usual regularity and decay assumptions. Thus
$$
\boxed{\delta\Phi_{\mathbf k}^{\mathrm{tot}}(t)\sim\frac{\delta\Phi_{\mathbf k}^{\mathrm{ext}}(t)}{\epsilon_{\mathbf k}(\mathbf k\cdot\mathbf v_p+i0^+)},\qquad t\to\infty.}
$$
The causal boundary value is what the paper abbreviates by $\epsilon_{\mathbf k}(\mathbf k\cdot\mathbf v_p)$. Physically, the moving mass polarizes the stellar population and carries a <gravitational wake>. Each spatial harmonic is changed in amplitude and phase by $1/\epsilon_{\mathbf k}$ at the forcing <frequency>. <Self-gravity> can amplify the wake. The induced part alone is $(1/\epsilon_{\mathbf k}-1)\delta\Phi_{\mathbf k}^{\mathrm{ext}}$; the wake's back-reaction on the perturber produces <dynamical friction>.
The waiting time is controlled by the \b[slowest excited collective damping], not just an orbital or crossing time. If $\omega_j=\omega_{rj}-i\Gamma_j$ with $\Gamma_j>0$, then a modal amplitude is suppressed by $e^{-\Gamma_jt}$, and one needs $t\Gamma_j\gg1$ for every appreciably excited mode, together with decay of the phase-mixing remainder. Large transient prefactors can require additional time. Near marginal stability, a relevant $\Gamma_j$ tends to zero and the required waiting time diverges; the stationary wake approximation is then not uniform in the stability parameter.<The linear response of stellar systems does not diverge at marginal stability by Chris Hamilton and Tobias Heinemann>[Sections 2.1–2.2: forcing-pole and Landau-mode contributions, and the nonuniform late-time limit]{ref}
\b[Qualification of stability.] If stable means only non-growing, a persistent neutral mode invalidates the asserted asymptote. A concrete example is a homogeneous <waterbag distribution> with parallel marginal $F(u)=\rho\mathbf1_{|u|<V}/(2V)$. Its <gravitational dielectric function> is
$$
\epsilon_k(\omega)=1-\frac{4\pi G\rho}{k^2V^2-\omega^2}.
$$
For $k^2V^2>4\pi G\rho$, the roots $\omega=\pm\sqrt{k^2V^2-4\pi G\rho}$ are real, so the switch-on transients persist although that harmonic has no growing mode. Taking the smallest box <wavenumber> to satisfy the same inequality makes the whole waterbag model free of growing harmonics. The claimed late-time formula therefore requires damping of all excited modes and $\epsilon_k(a)\ne0$; exact forcing at an undamped root instead produces a secular response.
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