One-loop matter bispectrum

For Gaussian initial conditions, the matter bispectrum through one loop is the sum of the tree diagram $B_{211}$ and the four one-loop topologies $B_{222}$, $B_{321}^I$, $B_{321}^{II}$, and $B_{411}$.

The connected correlation of one fourth-order and two first-order density contrasts contracts a pair of inputs at the fourth-order vertex into a loop. For each choice of the fourth-order external field there are twelve Wick contractions. In one dimension,For the one-dimensional cosmological density kernel, $F_4=-k_3^4/(24k_1{k}_2{q}^2)$ exactly. Its leading cutoff dependence is therefore proportional to $k_3^2{F}_2(k_1,k_2)P_L(k_1)P_L(k_2)\int dq{P}_L(q)/q^2$.

The connected correlation of three second-order density contrasts forms a triangle of three linear power spectra. It has eight Wick contractions. In one dimension, with $k_1+k_2+k_3=0$ and $P_L$ the power at the evaluation time,For the one-dimensional cosmological density kernel, its hard-loop integrand starts as $-k_1^2{k}_2^2{k}_3^2{P}_L(q{)}^3/q^6$.