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>:
$$
\boxed{
\delta P+\nabla^2\pi^S
+\partial_0[(\bar\rho+\bar P)\delta u]
+3H(\bar\rho+\bar P)\delta u=0}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 3 / e / Solution

{c}

For scalar perturbations of a generic fluid in cosmic time, momentum conservation in <synchronous gauge in cosmology> gives
$$
\delta P+\nabla^2\pi^S
+\partial_t[(\bar\rho+\bar P)\delta u]
+3H(\bar\rho+\bar P)\delta u=0.
$$

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