 Solution

The linear density contrast is Gaussian, so its cubic moment vanishes. The leading contribution isTwo connected Wick contractions giveThe third pairing vanishes because $F_{2,s}(\mathbf{q},-\mathbf{q})=0$. Isotropy makes $⟨\mu ⟩=0$ and $⟨{\mu }^2⟩=1/3$, so the angular average of the kernel isSince ${\sigma }^2=D^2\int d^{3}q{P}_{\mathrm{lin}}(q)/(2\pi {)}^3$ to leading order, the unsmoothed matter-density reduced skewness isThis assumes finite unsmoothed variance; inserting a smoothing window generally changes $S_3$.