Solution

Write the action density as

L = a^{3} [\frac{1}{2} \dot{ϕ}^{2} - \frac{c ^{2}}{2 a ^{2}} \nabla ϕ \cdot \nabla ϕ - V (ϕ)] .

Under a compactly supported variation

ϕ \mapsto ϕ + ϵη

δ S = \int d^{3} x d t [a^{3} \dot{ϕ} \overset{η}{˙} - a c^{2} \nabla ϕ \cdot \nabla η - a^{3} V^{'} (ϕ) η] .

Integrating the first term by parts in time and the second in space, and discarding the boundary terms, gives

δ S = \int d^{3} x d t [- \frac{\partial}{\partial t} (a^{3} \dot{ϕ}) + a c^{2} \nabla^{2} ϕ - a^{3} V^{'} (ϕ)] η .

The principle of stationary action requires this to vanish for every

η

. Since

\frac{\partial}{\partial t} (a^{3} \dot{ϕ}) = a^{3} (\ddot{ϕ} + 3 H \dot{ϕ}),

division by

- a^{3}

\ddot{ϕ} + 3 H \dot{ϕ} - \frac{c ^{2}}{a ^{2}} \nabla^{2} ϕ + \frac{d V}{d ϕ} = 0 .

Solved by gpt-5.6-sol high.

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