Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 3

Let $\theta =\nabla \cdot \mathbf{v}$. Vanishing vorticity implies in Fourier spaceFourier transforming $\nabla \cdot (\delta \mathbf{v})$ in the continuity equation giveswith the alpha mode-coupling kernelTaking the divergence of $(\mathbf{v}\cdot \nabla )\mathbf{v}$ and symmetrizing its two velocity arguments giveswhere the beta mode-coupling kernel is

The ultraviolet softness of the second-order density kernel requiresfor fixed $\mathbf{k}$ as $q\to \infty$. In this limit the two hard vectors are opposite,The constant part of the stated kernel is thereforeIt must vanish, givingThis recovers the standard standard perturbation theory density kernel $F_2$.

For Gaussian linear perturbations, a connected tree on five external density insertions has four Wick-contraction edges and therefore containsThus $n=4$. The three unlabeled tree-level connected matter five-point function topologies correspond to the perturbative-order partitionsThey are respectively an $F_4$ star with four $F_1$ leaves, an $F_3$ vertex joined to an $F_2$ vertex with three $F_1$ leaves, and a chain with three $F_2$ vertices and two endpoint $F_1$ vertices.

For one labeling of the last topology, let ${\mathbf{k}}_{12}={\mathbf{k}}_1+{\mathbf{k}}_2$ and ${\mathbf{k}}_{123}={\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3$. Its algebraic contribution isThe factor $2^3=8$ counts the interchange of the two linear legs at each symmetric $F_2$ vertex. The full correlator adds all distinct permutations of external labels and the overall momentum-conserving delta function.