= Solution
The restriction $k_\varphi=\pm m$ describes an $m$-fold rotating perturbation of a <galactic disk>, for example a transient <spiral density wave> or, for $m=2$, a <galactic bar>. Its <pattern speed> is $\Omega_{\mathrm p}$, its amplitude peaks at $t_{\mathrm{peak}}$, and its <Gaussian function> envelope has duration of order $\tau$. The coefficients $u_{\mathbf k}(\mathbf J)$ describe its strength along different <stellar orbits>. A real perturbing potential requires $u_{-\mathbf k}=u_{\mathbf k}^*$.
Write $g(t)=\exp[-(t-t_{\mathrm{peak}})^2/(2\tau^2)]$. The product entering <3/c> is
$$
\delta\phi_{\mathbf k}(t')\delta\phi_{\mathbf k}^*(t)=|u_{\mathbf k}|^2e^{ik_\varphi\Omega_{\mathrm p}(t-t')}g(t')g(t).
$$
The phase therefore measures detuning from the rotating pattern:
$$
\boxed{\omega_{\mathbf k}(\mathbf J)=\mathbf k\cdot\boldsymbol\Omega-k_\varphi\Omega_{\mathrm p}=k_\varphi(\Omega_\varphi-\Omega_{\mathrm p})+k_R\Omega_R.}
$$
Substitution gives
$$
\boxed{\begin{aligned}
Q_0={}&-2\pi\sum_{k_\varphi=\pm m}\sum_{k_R}k_\varphi\int_0^\infty dJ_R\,|u_{\mathbf k}|^2\\
&\times\int_0^t dt'\,\mathbf k\cdot\partial_{\mathbf J}f_0(\mathbf J,t')\,e^{-i\omega_{\mathbf k}(t-t')}\\
&\times\exp\left[-\frac{(t-t_{\mathrm{peak}})^2}{2\tau^2}-\frac{(t'-t_{\mathrm{peak}})^2}{2\tau^2}\right].
\end{aligned}}
$$
The strongest long-lived response occurs near an orbital <resonance>, $\omega_{\mathbf k}=0$, where successive <force> contributions do not rapidly cancel in phase.
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