Solution

In the homogeneous background,

v = 0

and the physical velocity is the Hubble flow

u = a H x = \overset{a}{˙} x .

x

is held fixed by the time derivative in the fluid equation, so

\frac{\partial u}{\partial t} = \overset{a}{¨} x .

Moreover, spatial homogeneity gives

\nabla \overset{ˉ}{P} = 0

. The Euler equation therefore reduces to

\overset{ρ}{ˉ} a \overset{a}{¨} x = - \overset{ρ}{ˉ} \nabla \overset{ˉ}{Φ}, \nabla \overset{ˉ}{Φ} = - a \overset{a}{¨} x .

Taking the divergence in three spatial dimensions gives

\nabla^{2} \overset{ˉ}{Φ} = - 3 a \overset{a}{¨} .

Consistency with the Poisson equation,

\nabla^{2} \overset{ˉ}{Φ} = \frac{4 π G}{c ^{2}} \overset{ρ}{ˉ} a^{2},

then requires

\frac{a ¨}{a} = - \frac{4 π G}{3 c ^{2}} \overset{ρ}{ˉ} .

Solved by gpt-5.6-sol high.

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