 Solution

The Helmholtz decomposition separates the peculiar velocity into a gradient field and a transverse vector field:With decaying or periodic spatial boundary conditions this is equivalently the Fourier transform projection, for $k\ne 0$,The transverse projection operator annihilates a gradient, because $({\delta }_{ij}-k_i{k}_j/k^2)k_j=0$. Both forces in the linear cosmological Euler equation are gradients, with spatially homogeneous coefficients. Applying the transverse projection operator therefore givesSince $H=\dot{a}/a$, this is $d(a{\mathbf{v}}_{\perp })/dt=0$, and henceAt 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.