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In conformal time the interaction Hamiltonian isThe first-order in-in formalism formula and Wick theorem give the connected primordial trispectrumThe factor $4!$ from the Wick contractions cancels the vertex factor. Writing $K=k_1+k_2+k_3+k_4$, the powers of $\eta$ cancel because $a^4{\prod }_a{f}_{k_a}$ is constant apart from its phase, and the i-epsilon prescription givesConsequentlyThe full four-point function also contains the three disconnected products of the free two-point function found in part c.

The first-order in-in formalism formula at observation time ${\tau }_0$ isFourier transforming the three derivatives giveswhere ${\int }_{\mathbf{p}}=\int d^{3}p/(2\pi {)}^3$. Because the four species are distinct, Wick contraction pairs each vertex field with the external field of the same species and introduces no permutation factor. IfthenCombining this with part i gives the requested time integral:The factor $\mathcal{E}$ is a pseudoscalar, so the resulting primordial trispectrum is parity odd and purely imaginary in this momentum-space convention.

A late-time connected four-point function of a scale-invariant field in three spatial dimensions, after removing its momentum-conserving delta function, must obeyIn both contributions, $s/(k_1{k}_2{k}_3{k}_4)$ scales as ${\Lambda }^{-3}$ and every energy-denominator term scales as ${\Lambda }^{-6}$. Hence the complete $s$-channel primordial trispectrum scales as ${\Lambda }^{-9}$, as required by scale invariance.

The trispectrum is the Fourier transform of a fourth-order cumulant, a fourth-order counterpart of the bispectrum. It describes connected four-point correlations; the primordial trispectrum applies it to cosmological random fields.