Linearizing the <cosmological Euler equation> gives
$$
\mathbf v'+\mathcal H\mathbf v=-\nabla\phi.
$$
Taking the <curl> removes the gradient force, so the <vorticity> $\mathbf w=\nabla\mathbin\times\mathbf v$ obeys
$$
\boxed{\mathbf w'+\mathcal H\mathbf w=0}.
$$
Since $\mathcal H=a'/a$, its solution is
$$
\boxed{\mathbf w\propto a^{-1}}.
$$
Thus expansion dilutes linear vorticity; in an <Einstein-de Sitter universe>, $a\propto\tau^2$ and $\mathbf w\propto\tau^{-2}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 4 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-312/4/i/solution