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Use $⟨{\phi }_G(\mathbf{k}){\phi }_G({\mathbf{k}}^{\prime })⟩=(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}+{\mathbf{k}}^{\prime })P_{\phi }(k)$. For nonzero external momenta, the quadratic term in local-type primordial non-Gaussianity isInsert it in one of the three external fields. Wick theorem supplies two connected Wick contractions at each insertion, whereas the mean subtraction removes the disconnected zero-mode contribution. Thus the leading primordial bispectrum isLinear propagation by the cosmological Poisson equation gives $P_m(k)=\alpha (k{)}^2{P}_{\phi }(k)$ and multiplies the primordial bispectrum by ${\alpha }_1{\alpha }_2{\alpha }_3$. Defining the matter three-point function by its momentum-conserving Dirac delta function times $B_m$, the primordial contribution to the matter bispectrum isThe nonlinear gravitational contribution from standard perturbation theory in cosmology is excluded here.

In a squeezed bispectrum configuration, $k_1=k_L\ll k_2\simeq k_3=k_S$, soFor a nearly scale-invariant potential, $P_{\phi }(k_L)\gg P_{\phi }(k_S)$, so the second term is subleading andRelative to the product of matter power spectra, the squeezed response is enhanced by $k_L^{-2}$. Its sign is the sign of $f_{\mathrm{NL}}$. This is the three-point counterpart of the conditional short-scale variance modulation by local non-Gaussianity.

The displayed kernel is the growing-mode result of standard perturbation theory in cosmology. Its assumptions are a Newtonian, weak-field, subhorizon treatment of pressureless cold matter; a single-stream, irrotational velocity field; and negligible velocity-dispersion tensor of collisionless matter, so ${\sigma }_{ij}=0$. The background is the Einstein-de Sitter universe, which makes the growing mode proportional to $a$ and permits the separated powers $a^n{F}_n$. The density contrast is assumed perturbatively small and the decaying solutions are discarded. Gaussian initial conditions are not needed to derive $F_2$, but they are needed for the loop contractions in the later parts.

In standard perturbation theory in cosmology, ${\delta }^{(n)}$ is an $n$-leg vertex carrying the symmetrized standard perturbation theory density kernel $F_n$. For Gaussian random fields, a connected four-point diagram with $L$ loops contains $3+L$ linear power spectra, hence $6+2L$ linear fields.

At tree level the required perturbative-order partitions areThe first is a star with one $F_3$ vertex and three linear external legs. The second has two $F_2$ vertices joined by one internal power-spectrum line, with one linear external leg attached to each vertex. At one loop all five partitions of eight into four positive parts are required:These are conventionally denoted $T_{5111}$, $T_{4211}$, $T_{3311}$, $T_{3221}$, and $T_{2222}$; all inequivalent placements of external legs and all Wick contractions are understood. This list is the complete set of one-loop matter trispectrum topologies.

Let $P_a=P_L(k_a)$ and ${\mathbf{k}}_{ab}={\mathbf{k}}_a+{\mathbf{k}}_b$. With ${\sum }_a{\mathbf{k}}_a=0$, the star contribution iswhere $\{a,b,c\}=\{1,2,3,4\}\setminus \{d\}$. For the exchange contribution, sum over the six unordered choices $\{a,b\}$ of linear external legs and let $\{c,d\}$ be the complementary pair:Thus the connected tree trispectrum is $T_{\mathrm{tree}}=T_{3111}+T_{2211}$.

The pressureless single-stream equations cease to be a complete one-loop prediction because loop momenta probe short nonlinear scales and produce ultraviolet-sensitive terms. A consistent result must include the effective field theory of large-scale structure: effective-stress counterterms, including their insertions into the tree topologies, and stochastic terms at the order allowed by mass conservation and momentum conservation. Their coefficients absorb short-scale dependence and must be fitted or matched. If the observable is a biased tracer rather than matter itself, the corresponding renormalized bias expansion is also required.

If the linear density contrast obeys $\delta (\mathbf{k})=\alpha (k)\phi (\mathbf{k})$, then local-type primordial non-Gaussianity contributesIn a squeezed bispectrum configuration and for a nearly scale-invariant potential, the leading term is $4f_{\mathrm{NL}}{P}_m(k_L)P_m(k_S)/\alpha (k_L)$. This excludes the standard perturbation theory in cosmology contribution from nonlinear gravity.