On wavelengths much larger than halos, all matter is partitioned among halos. The mass-weighted halo overdensity must therefore equal the matter overdensity. Since $\delta_h(M)=b(M)\delta_m$, <mass conservation> requires
$$
\int_0^\infty dM\,\frac{dn}{dM}\frac{M}{\bar\rho}b(M)=1.
$$

For the <Press-Schechter formalism>, the mass-fraction measure becomes
$$
f(\nu)d\nu=\sqrt{\frac2\pi}e^{-\nu^2/2}d\nu,
\qquad \nu\geq0.
$$
It is normalized and is a half-normal distribution, so
$$
\int_0^\infty f(\nu)d\nu=1,
\qquad
\int_0^\infty\nu^2f(\nu)d\nu=1.
$$
Using the <linear Eulerian halo bias>
$$
b=1+b_L=1+\frac{\nu^2-1}{\delta_c}
$$
therefore gives
$$
\int_0^\infty f(\nu)b(\nu)d\nu
=1+\frac{1-1}{\delta_c}=1,
$$
which verifies the consistency relation.

Solved by gpt-5.6-sol high.

Solution

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

{title2=$b(M)$}

Mapping matter and halos from their initial positions adds the matter displacement contribution, giving $b=1+b_L$. Mass conservation requires the mass-weighted consistency relation $\int(dn/dM)(M/\bar\rho)b(M)dM=1$.


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 Incoming links: Linear Eulerian halo bias