= Solution
The <Helmholtz decomposition> separates the <peculiar velocity> into a <gradient field> and a <transverse vector field>:
$$
\mathbf v=\nabla V+\mathbf v_\perp,\qquad \nabla\cdot\mathbf v_\perp=0.
$$
With decaying or periodic spatial boundary conditions this is equivalently the <Fourier transform> projection, for $k\ne0$,
$$
v_{\perp i}(\mathbf k)=\left(\delta_{ij}-\frac{k_ik_j}{k^2}\right)v_j(\mathbf k).
$$
The <transverse projection operator> annihilates a <gradient>, because $(\delta_{ij}-k_ik_j/k^2)k_j=0$. Both forces in the linear <cosmological Euler equation> are <gradients>, with spatially homogeneous coefficients. Applying the <transverse projection operator> therefore gives
$$
\dot{\mathbf v}_\perp+H\mathbf v_\perp=0.
$$
Since $H=\dot a/a$, this is $d(a\mathbf v_\perp)/dt=0$, and hence
$$
\boxed{\mathbf v_\perp(\mathbf x,t)=\frac{a(t_i)}{a(t)}\mathbf v_\perp(\mathbf x,t_i).}
$$
At this order the scalar pressure and gravitational forces cannot generate <vorticity>; an initially vanishing <transverse vector field> remains zero. Its nonzero initial value is damped by <cosmic expansion>.
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