Taking the <divergence> of the linear Euler equation and defining the <peculiar-velocity divergence> $\theta=\nabla\mathbin\cdot\mathbf v$ gives
$$
\boxed{
\dot\theta+H\theta
=-\frac1{a\bar\rho}\nabla^2\delta P
-\frac1a\nabla^2\Phi}.
$$
The assumption of negligible <vorticity> ensures that this longitudinal variable captures the velocity perturbation relevant to scalar density growth.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 4 / a / Solution

{title2=$\theta$}

The peculiar-velocity divergence is $\theta=\nabla\mathbin\cdot\mathbf v$. For irrotational linear perturbations it captures the scalar part of the peculiar velocity field.


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