Write a filled vertex with $n$ incoming linear fields for the $n$th-order <peculiar-velocity divergence>. Gaussian initial conditions require every linear field to be paired. Through sixth order in the linear density there are exactly five connected topologies: the tree diagram $B_{211}$ and the one-loop triangle $B_{222}$, vertex-correction diagram $B_{321}^{I}$, propagator-correction diagram $B_{321}^{II}$, and four-leg diagram $B_{411}$. These are the five diagrams of the <one-loop matter bispectrum>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 3 / iii / a / Solution

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 $B_{411}$.


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 Incoming links: One-loop matter bispectrum