Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 2

Use the Fourier convention $f(\mathbf{x})={\int }_{\mathbf{k}}f(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}$ with ${\int }_{\mathbf{k}}=\int d^{3}k/(2\pi {)}^3$. Neglect anisotropic stress and assume the pressureless velocity is irrotational, soFourier transforming $\nabla \cdot (\delta \mathbf{v})$ in the nonlinear continuity equation giveswhere the alpha mode-coupling kernel and its symmetrization areTaking the divergence of the Euler equation giveswith the symmetric beta mode-coupling kernel

In the Einstein-de Sitter universe, $a^{\prime }=\mathcal{H}a$ and ${\mathcal{H}}^{\prime }=-{\mathcal{H}}^2/2$. At linear order the ansatz gives ${\tilde{\theta }}^{(1)}={\tilde{\delta }}^{(1)}$. At second order, the continuity and Euler equations becomewhere each integral includes the momentum-conserving measure above. Eliminating ${\tilde{\theta }}^{(2)}$ yieldswith the standard perturbation theory density kernel

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Expand $\delta =a{\delta }_1+a^2{\delta }_2+a^3{\delta }_3+\cdots$. For a Gaussian linear density field, odd linear correlators vanish and Wick theorem reduces the fourth-order terms to products of the linear power spectrum. HenceThe two contractions of the two quadratic fields giveThe three choices for which argument of $F_3$ carries the external momentum giveThus the contribution conventionally called $P_{13}+P_{31}$ is $2P_{13}=6P(k)\int F_{3}P$. These are the two one-loop diagrams of the one-loop matter power spectrum.

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Set $\mathbf{q}={\mathbf{k}}_1$, $\mathbf{k}-\mathbf{q}={\mathbf{k}}_2$, $q=|\mathbf{q}|$, and $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{q}}$. Substitution into $F_2=5{\alpha }_s/7+2\beta /7$ givesThe azimuthal integral in $d^{3}q=q^{2}dqd\mu d\varphi$ and the factor two in $P_{22}$ then give

For the scale-free matter power spectrum $P(k)=A{k}^n$, put $r=q/k$ andAll dimensional factors separate:where

As $r\to 0$, $K_{22}\sim r^n{\mu }^2/(8{\pi }^2)$, so the soft-$q$ integral converges exactly when $n>-1$. For $r\to \infty$, $K_{22}\sim r^{2n-2}$ times an angular function, requiring $n<1/2$. At fixed $\mu <1$, $r\to 1$ is regular. The corner $(r,\mu )=(1,1)$ makes $|\mathbf{k}-\mathbf{q}|$ soft; after resolving that corner in the local soft momentum, its radial behavior is again $p^{n}dp$, requiring $n>-1$. Thereforeis the infrared- and ultraviolet-convergence window for $P_{22}$.

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