= Cosmological Euler equation in synchronous gauge
{c}
{title2=$\partial_t[(\bar\rho+\bar P)u]+3H(\bar\rho+\bar P)u+\delta P+\nabla^2\pi^S=0$}
For scalar perturbations in cosmic time with $ds^2=-dt^2+a^2(\delta_{ij}+h_{ij})dx^idx^j$, define the covariant fluid <velocity potential> $u$ by $\delta u_i=\partial_i u$. The linear momentum density is $T^0{}_i=(\bar\rho+\bar P)\partial_i u$. Normalize scalar <anisotropic stress> by
$$
\pi^i{}_j=\frac1{a^2}\left(\partial^i\partial_j-\frac13\delta^i_j\nabla^2\right)\Pi,
$$
where <derivatives> inside these parentheses are Cartesian comoving <derivatives>. Then $\partial_j\pi^j{}_i=2\partial_i\nabla^2\Pi/(3a^2)$. Spatial <stress-energy conservation> gives, for nonzero scalar <Fourier modes>,
$$
\partial_t[(\bar\rho+\bar P)u]+3H(\bar\rho+\bar P)u
+\delta P+\frac{2}{3a^2}\nabla^2\Pi=0.
$$
Writing $\pi^S=2\Pi/(3a^2)$ reproduces the form with $\nabla^2\pi^S$; the coefficient depends on that stress <normalization>. The time <derivative> acts on the entire momentum-density product, not only on $u$.
For <pressureless matter> with zero scalar stress, background $\dot{\bar\rho}+3H\bar\rho=0$ reduces the equation to $\dot u=0$. The physical <peculiar velocity> is $\mathbf v=\nabla u/a$ and therefore decays as $a^{-1}$. This checks the difference between covariant velocity potential and physical velocity and the expansion factor in the conservative equation.
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