Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 3 / b / Solution

In the peak-background split, a long-wavelength overdensity changes the local threshold from ${\delta }_c$ to ${\delta }_c-{\delta }_l$. At fixed mass, the Press-Schechter halo mass function depends on the threshold through $\nu e^{-{\nu }^2/2}$. ThereforeIt follows that the linear Lagrangian halo bias is

Solved by gpt-5.6-sol high.