Write the matter density as a sum over normalized halo profiles,
$$
\rho(\mathbf x)=\sum_iM_i u(\mathbf x-\mathbf x_i|M_i),
\qquad
\int d^3x\,u(\mathbf x|M)=1.
$$
For $\mathbf k\ne0$, its density contrast is
$$
\delta(\mathbf k)=\frac1{\bar\rho}
\sum_iM_i\widetilde u(k|M_i)e^{-i\mathbf k\cdot\mathbf x_i}.
$$
If halo locations form an uncorrelated Poisson process, only equal-halo terms survive after subtracting the homogeneous contribution. Replacing the sum per unit volume by the halo abundance gives the <one-halo term>
$$
\boxed{P(k)=P_{1h}(k)
=\int_0^\infty dM\,\frac{dn}{dM}
\left(\frac{M}{\bar\rho}\right)^2
|\widetilde u(k|M)|^2}.
$$

Solved by gpt-5.6-sol high.

Solution

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

{title2=$P_{1h}$}

The one-halo term comes from pairs of mass elements in the same halo:
$$
P_{1h}(k)=\int dM\,\frac{dn}{dM}
\left(\frac{M}{\bar\rho}\right)^2|\widetilde u(k|M)|^2.
$$


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