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For a homogeneous collisionless stellar model with its mean field subtracted, the response to an imposed Newtonian gravitational potential is ${\tilde{\Phi }}_{\mathbf{k}}^{\mathrm{tot}}={\tilde{\Phi }}_{\mathbf{k}}^{\mathrm{ext}}/{ϵ}_{\mathbf{k}}$, whereThe phase-space distribution function $f_0$ is mass-normalized, and $\mathbf{k}\ne 0$. Causal analytic continuation uses the Landau contour wherever the equilibrium permits it. Zeros describe collective Landau modes. This response function is a gravitational analogue of the electrostatic dielectric function; its susceptibility sign reflects attraction.

In the periodic cube, $H_0=|\mathbf{v}{|}^2/2=(2\pi /L{)}^2|\mathbf{J}{|}^2/2$, so $\mathbf{\Omega }=(2\pi /L)\mathbf{v}$. The supplied action-angle variables are canonical: $d^3\theta d^{3}J=d^{3}x{d}^{3}v$. For $\mathbf{k}=2\pi \mathbf{n}/L$,Consequently c becomesFor each nonzero wavenumber, the Poisson equation for Newtonian gravity givesSubstitute the preceding response and add the externally imposed Newtonian gravitational potential:ThereforeThis gravitational dielectric function has the opposite susceptibility sign to the electron dielectric function, reflecting attractive rather than repulsive collective interactions. The homogeneous mean field is explicitly subtracted in this fictitious periodic model: the Poisson equation for Newtonian gravity acts on density fluctuations, and the $\mathbf{k}=0$ mode is omitted. Otherwise a nonzero uniform positive mass density would be incompatible with a periodic potential. This is the convention often called the Jeans swindle.

For analytic continuation, resolve velocity parallel to $\mathbf{k}$: let $u=\hat{\mathbf{k}}\cdot \mathbf{v}$ and $F(u)=\int f_0(u,{\mathbf{v}}_{\perp })d^2{v}_{\perp }$. ThenStart at $\mathrm{Im}\omega >0$ and deform the Landau contour continuously below the moving pole. At real $\omega$, the integral isFor $\mathrm{Im}\omega <0$, this causal continuation equals the undeformed real-axis integral plus $2\pi i{F}^{\prime }(\omega /k)$ wherever $F^{\prime }$ has the necessary analytic extension. It must not be replaced by the undeformed integral alone.

Zeros of ${ϵ}_{\mathbf{k}}$ are self-consistent collective Landau modes: with no external forcing, a nonzero Newtonian gravitational potential is possible at such a root. For time dependence $e^{-i\omega t}$, upper-half-plane roots grow, lower-half-plane roots describe Landau damping, and real roots are neutral. Damped Landau modes are poles of the continued response; they need not be regular discrete eigenfunctions of the original collisionless operator.

Regularity qualification. Continuation throughout the complex plane presupposes suitable analytic $F$. Arbitrary smooth $F$ need not allow it. For example, $F(u)=C\exp [-1/(1-u^2)]$ for $|u|<1$ and zero elsewhere is smooth but not analytic at $u=\pm 1$; its Cauchy integral cannot be holomorphically continued across those endpoints. The Landau contour gives the standard causal continuation on the domains where the equilibrium permits it, with possible singularities retained.

With $C_{\mathbf{k}}$ and $a$ from e, inversion givesFor $t>0$, 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 $i{C}_{\mathbf{k}}{e}^{-iat}/{ϵ}_{\mathbf{k}}(a+i{0}^{+})$ and therefore contributesA simple collective root ${\omega }_j$ of the continued gravitational dielectric function contributesThese 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. ThusThe causal boundary value is what the paper abbreviates by ${ϵ}_{\mathbf{k}}(\mathbf{k}\cdot {\mathbf{v}}_p)$. Physically, the moving mass polarizes the stellar population and carries a gravitational wake. Each spatial harmonic is changed in amplitude and phase by $1/{ϵ}_{\mathbf{k}}$ at the forcing frequency. Self-gravity can amplify the wake. The induced part alone is $(1/{ϵ}_{\mathbf{k}}-1)\delta {\Phi }_{\mathbf{k}}^{\mathrm{ext}}$; the wake's back-reaction on the perturber produces dynamical friction.

The waiting time is controlled by the 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 $e^{-{\Gamma }_{j}t}$, and one needs $t{\Gamma }_j\gg 1$ 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.^[ref]

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 $F(u)=\rho 1_{|u|<V}/(2V)$. Its gravitational dielectric function isFor $k^2{V}^2>4\pi G\rho$, the roots $\omega =\pm \sqrt{k^2{V}^2-4\pi G\rho }$ 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 ${ϵ}_k(a)\ne 0$; exact forcing at an undamped root instead produces a secular response.

By Chris Hamilton and Tobias Heinemann. Sections 2.1–2.2 separate the forcing-pole gravitational wake from the collective Landau mode transients and examine the nonuniform late-time limit near marginal stability.