= Gravitational dielectric function
{title2=$\epsilon_{\mathbf k}(\omega)$}
For a homogeneous collisionless stellar model with its mean field subtracted, the response to an imposed <Newtonian gravitational potential> is $\widetilde\Phi^{\mathrm{tot}}_{\mathbf k}=\widetilde\Phi^{\mathrm{ext}}_{\mathbf k}/\epsilon_{\mathbf k}$, where
$$
\epsilon_{\mathbf k}(\omega)=1+\frac{4\pi G}{k^2}\int d^3v\,\frac{\mathbf k\cdot\partial_{\mathbf v}f_0}{\mathbf k\cdot\mathbf v-\omega}.
$$
The <phase-space distribution function> $f_0$ is mass-normalized, and $\mathbf k\ne0$. 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.
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