Inflaton action in an expanding universe
= Inflaton action in an expanding universe
For a canonical scalar field in a spatially flat expanding universe,
$$
S[\phi]=\int d^3x\,dt\,a^3
\left[
\frac12\dot\phi^2
-\frac{c^2}{2a^2}|\nabla\phi|^2
-V(\phi)
\right].
$$
The field <Euler-Lagrange equation> is
$$
\ddot\phi+3H\dot\phi-\frac{c^2}{a^2}\nabla^2\phi+V'(\phi)=0.
$$