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

Liouville transport of the phase-space density along the particle trajectories givesUsing the equations of motion produces the Collisionless dark-matter Vlasov equation

Integrating the Collisionless dark-matter Vlasov equation over momentum makes the force term a vanishing momentum-space boundary term. Since $\rho =m{a}^{-3}\int f{d}^{3}p$, the zeroth moment isMultiplying by $p^i/(am)$ before integrating gives the first moment. Decompose the second velocity moment asusing the velocity-dispersion tensor of collisionless matter. Combining the result with the continuity equation givesThe single-stream pressureless-fluid equations follow when ${\sigma }^{ij}=0$.

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.

Define the standard perturbation theory velocity kernel bywhere ${\int }_{\mathbf{q}}=\int d^{3}q/(2\pi {)}^3$ and $P(q)=⟨|{\delta }_1(\mathbf{q}){|}^2{⟩}^{\prime }$. For ${\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3=0$, one representative external labelling of the tree contribution isThe four one-loop representatives areThe full answer adds the distinct permutations of the external labels; the question asks for only one labelling of each topology.

The ultraviolet part of the one-loop matter power spectrum is renormalized by the leading effective field theory of large-scale structure operator proportional to $k^2{\delta }^{(1)}$. The analogous velocity-divergence counterterm iswhere the time dependence and the nonlinear reference scale may be absorbed into the renormalized coefficient $c_{\theta }^2$. In the bispectrum it suppliesand permutations, with momentum shape $-c_{\theta }^2{k}^2{B}_{211}$ on the corrected leg. This cancels the local $k^2$ ultraviolet dependence of $⟨{\theta }^{(3)}{\theta }^{(2)}{\theta }^{(1)}⟩$.