Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 4

The displayed kernel is the growing-mode result of standard perturbation theory in cosmology. Its assumptions are a Newtonian, weak-field, subhorizon treatment of pressureless cold matter; a single-stream, irrotational velocity field; and negligible velocity-dispersion tensor of collisionless matter, so ${\sigma }_{ij}=0$. The background is the Einstein-de Sitter universe, which makes the growing mode proportional to $a$ and permits the separated powers $a^n{F}_n$. The density contrast is assumed perturbatively small and the decaying solutions are discarded. Gaussian initial conditions are not needed to derive $F_2$, but they are needed for the loop contractions in the later parts.

For symmetrized standard perturbation theory density kernels, the two one-loop contractions areandThe factors $2$ and $3$ count the relevant Wick contractions. Since the cross-correlation occurs in both orders, the one-loop matter power spectrum is

Put $\mathbf{p}=\mathbf{k}-\mathbf{q}$, $p^2=k^2-2kq\mu +q^2$, and $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{q}}$. Direct substitution into $F_2$ givesUsing $d^{3}q=2\pi q^{2}dqd\mu$ in part ii therefore yieldsFor $q\ll k$, $F_2(\mathbf{q},\mathbf{k}-\mathbf{q})\sim k\mu /(2q)$. Including the equal soft region $|\mathbf{k}-\mathbf{q}|\to 0$ and using ${\int }_{-1}^1{\mu }^{2}d\mu =2/3$ givesFor $q\gg k$, the constant and linear hard-momentum terms cancel, displaying the ultraviolet softness of the second-order density kernel. Since $p\sim q$,The angular integral ${\int }_{-1}^1(3-10{\mu }^2{)}^{2}d\mu =18$ then gives

The leading infrared terms cancel in the observable sum:This infrared cancellation in large-scale structure follows from the Equivalence principle: a sufficiently long-wavelength displacement translates short-scale structure without changing an equal-time power spectrum.

The ultraviolet part of $2P_{13}$ is proportional to $k^{2}P(k)$ and depends on the cutoff $\Lambda$. The leading deterministic effective field theory of large-scale structure counterterm has exactly this shape,Its cutoff-dependent part can be chosen aswhich cancels the stated $2P_{13,\mathrm{UV}}$. The $P_{22}$ ultraviolet contribution begins at $k^4$ and, when cutoff sensitive, is absorbed by higher-derivative or stochastic counterterms.

Physically, coarse graining the matter equations does not justify setting the short-scale stress ${\sigma }_{ij}$ to zero. Unresolved multistreaming and nonlinear motion generate an effective pressure and viscosity. Their leading derivative contribution to the Euler equation is proportional to $-c_{\mathrm{eff}}^2{\nabla }_i\delta$, producing the required $k^2\delta$ correction and making long-distance predictions independent of the arbitrary cutoff.