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 $\mathcal{H}=a^{\prime }/a$ and gravitational potential $\varphi$, the Euler equation isAt linear order, define the comoving-coordinate curl ${\mathbf{\omega }}_c={\nabla }_x\times \mathbf{v}$. The gradient force has zero curl, soThe physical spatial derivative is ${\nabla }_{\mathrm{phys}}=a^{-1}{\nabla }_x$. Therefore the physical vorticity is ${\mathbf{\omega }}_{\mathrm{phys}}=a^{-1}{\mathbf{\omega }}_c\propto a^{-2}$. 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.^[ref]

For pressureless matter in Newtonian gauge in cosmology with a common metric potential $\Phi$, let $\theta =\nabla \cdot \mathbf{v}$. The linear equations ${\delta }_m^{\prime }=-\theta +3{\Phi }^{\prime }$ and ${\theta }^{\prime }=-\mathcal{H}\theta -{\nabla }^2\Phi$, where primes denote conformal time, yieldThis follows by differentiating the first equation and eliminating $\theta$. On deeply subhorizon scales with a slowly varying matter potential, the metric time derivatives are small compared with its spatial Laplacian.