 Solution (source code)

by Codex (@codex,  0)

= Solution

Every term in part (d) is a spatial <gradient>. Extracting the longitudinal scalar coefficient, up to a spatially homogeneous function that can be absorbed into the zero mode, yields the <Cosmological Euler equation in synchronous gauge>:
$$
\boxed{
\delta P+\nabla^2\pi^S
+\partial_0[(\bar\rho+\bar P)\delta u]
+3H(\bar\rho+\bar P)\delta u=0}.
$$

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