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

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