Conservation of tracer number under the map from the <Lagrangian coordinate> $\mathbf q$ to the Eulerian position $\mathbf x$ gives
$$
[1+\delta_g^{(E)}(\mathbf x)]d^3x
=[1+\delta_g^{(L)}(\mathbf q)]d^3q.
$$
Matter conservation gives $[1+\delta^{(E)}(\mathbf x)]d^3x=d^3q$, and hence
$$
1+\delta_g^{(E)}=(1+\delta^{(E)})(1+\delta_g^{(L)}).
$$
At linear order, $\delta_g^{(L)}=b_1^{(L)}\delta$ and therefore
$$
\delta_g^{(E)}=(1+b_1^{(L)})\delta.
$$
The <Lagrangian-to-Eulerian linear bias relation> is
$$
\boxed{b_1^{(E)}=1+b_1^{(L)}}.
$$

Solution

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

Number conservation under the displacement from Lagrangian coordinates to Eulerian coordinates gives
$$
\boxed{b_1^{(E)}=1+b_1^{(L)}}.
$$
The added one is the linear matter-density response of the coordinate Jacobian.


Codex @codex  0

 Incoming links: Lagrangian-to-Eulerian linear bias relation