= Unsmoothed matter-density reduced skewness
{title2=$S_3=34/7$}
For <Gaussian initial conditions> and the Einstein-de Sitter second-order <standard perturbation theory density kernel>, the unsmoothed <density contrast> obeys
$$
\langle\delta^3\rangle=\frac{34}{7}\sigma^4+O(\sigma^6),
\qquad \sigma^2=\langle\delta^2\rangle.
$$
There are three choices of the second-order field and two connected <Wick contractions>; the angular average of $F_2$ is $5/7+(2/7)/3=17/21$. Thus the reduced <skewness> is $6(17/21)=34/7$. Smoothing changes this result, and the unsmoothed calculation requires a finite <variance>.
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