= Solution
The <cosmological continuity equation> expresses conservation of dark-matter mass: $\delta$ is the <density contrast>, $\mathbf v$ is the <peculiar velocity>, and a prime denotes a <conformal time> derivative. The <cosmological Euler equation> expresses momentum conservation: $\mathcal H=a'/a$ is the conformal Hubble rate, $\phi$ is the <peculiar gravitational potential>, $\rho$ is the density, and $\sigma_{ij}$ is the <velocity-dispersion tensor of collisionless matter>. The terms are respectively Hubble drag, convective acceleration, gravity, and velocity-dispersion stress.
For curl-free flow, introduce the <peculiar-velocity divergence> $\theta=\nabla\cdot\mathbf v$ and set $\sigma_{ij}=0$. A <Fourier transform> of the nonlinear continuity term gives
$$
\delta'(\mathbf k)+\theta(\mathbf k)
=-\int\frac{d^3q}{(2\pi)^3}
\alpha(\mathbf q,\mathbf k-\mathbf q)
\theta(\mathbf q)\delta(\mathbf k-\mathbf q),
$$
with the <alpha mode-coupling kernel>
$$
\boxed{\alpha(\mathbf k_1,\mathbf k_2)
=\frac{(\mathbf k_1+\mathbf k_2)\cdot\mathbf k_1}{k_1^2}}.
$$
Taking the divergence of the Euler equation gives the quadratic velocity kernel
$$
\boxed{\beta(\mathbf k_1,\mathbf k_2)
=\frac{|\mathbf k_1+\mathbf k_2|^2
(\mathbf k_1\cdot\mathbf k_2)}{2k_1^2k_2^2}}.
$$
This is the <beta mode-coupling kernel>; its symmetry follows from the two velocity factors.
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