Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 3

The cosmological continuity equation expresses conservation of dark-matter mass: $\delta$ is the density contrast, $\mathbf{v}$ is the peculiar velocity, and a prime denotes a conformal time derivative. The cosmological Euler equation expresses momentum conservation: $\mathcal{H}=a^{\prime }/a$ is the conformal Hubble rate, $\varphi$ is the peculiar gravitational potential, $\rho$ is the density, and ${\sigma }_{ij}$ is the velocity-dispersion tensor of collisionless matter. The terms are respectively Hubble drag, convective acceleration, gravity, and velocity-dispersion stress.

For curl-free flow, introduce the peculiar-velocity divergence $\theta =\nabla \cdot \mathbf{v}$ and set ${\sigma }_{ij}=0$. A Fourier transform of the nonlinear continuity term giveswith the alpha mode-coupling kernelTaking the divergence of the Euler equation gives the quadratic velocity kernelThis is the beta mode-coupling kernel; its symmetry follows from the two velocity factors.

Expand $\delta ={\delta }^{(1)}+{\delta }^{(2)}+{\delta }^{(3)}+\cdots$, where the standard perturbation theory density kernel $F_n$ convolves $n$ linear fields. Through fourth order in the Gaussian linear field, the only power-spectrum diagrams are the tree contraction $P_{11}$, the loop joining two $F_2$ vertices $P_{22}$, and the two orderings that join an $F_3$ vertex to a linear leg, $P_{13}+P_{31}=2P_{13}$. Their expressions are

The effective field theory of large-scale structure supplements the one-loop prediction by the leading deterministic counterterm and a stochastic term,where the normalization scale may be absorbed into $c_{\mathrm{eff}}^2$ and mass and momentum conservation make $P_{\mathrm{stoch}}=O(k^4)$ at small $k$. The ultraviolet part of the P13 contribution to the one-loop matter power spectrum hasIts cutoff dependence has exactly the $k^2{P}_L(k)$ form of the effective sound-speed counterterm in large-scale structure, so the running of $c_{\mathrm{eff}}^2(\Lambda )$ cancels it. The stochastic and higher-derivative counterterms similarly absorb the allowed analytic ultraviolet dependence of $P_{22}$ and higher orders.

Conservation of tracer number under the map from the Lagrangian coordinate $\mathbf{q}$ to the Eulerian position $\mathbf{x}$ givesMatter conservation gives $[1+{\delta }^{(E)}(\mathbf{x})]d^{3}x=d^{3}q$, and henceAt linear order, ${\delta }_g^{(L)}=b_1^{(L)}\delta$ and thereforeThe Lagrangian-to-Eulerian linear bias relation is