= Cosmological Euler equation
In the Newtonian pressureless approximation, let $\mathbf x$ denote comoving positions, $\eta$ <conformal time>, $a$ the <scale factor>, and $\mathbf v=d\mathbf x/d\eta$ the physical peculiar <velocity>. With $\mathcal H=a'/a$ and <gravitational potential> $\phi$, the <Euler equation> is
$$
\mathbf v'+\mathcal H\mathbf v+(\mathbf v\cdot\nabla_x)\mathbf v=-\nabla_x\phi.
$$
At linear order, define the comoving-coordinate curl $\boldsymbol\omega_c=\nabla_x\times\mathbf v$. The gradient force has zero <curl>, so
$$
\boldsymbol\omega_c'+\mathcal H\boldsymbol\omega_c=0,\qquad
\boldsymbol\omega_c\propto a^{-1}.
$$
The physical spatial derivative is $\nabla_{\rm phys}=a^{-1}\nabla_x$. Therefore the physical <vorticity> is $\boldsymbol\omega_{\rm phys}=a^{-1}\boldsymbol\omega_c\propto a^{-2}$. Stating which derivative defines the <vorticity> prevents confusing these two dilution laws.
This smooth single-stream model omits <pressure> forces and velocity dispersion. Shell crossing or additional nongravitational forces can invalidate its simple description; the linear decay law is not a universal assertion about every rotating cosmic structure.<Cosmology by David Tong>[Chapter 3, section 3.1.3, peculiar-velocity evolution and density perturbations in an expanding space]{ref}
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