Scalar field equation of motion ($\square \phi -V^{\prime }(\phi )=0$)

For a canonical real scalar field with metric signature $(-,+,+,+)$ and scalar potential $V$, variation of its minimally coupled action gives the displayed equation. A homogeneous field in an FLRW metric therefore obeys $\ddot{\phi }+3H\dot{\phi }+V^{\prime }(\phi )=0$ in natural units. Its energy density is ${\dot{\phi }}^2/2+V$ and its pressure is ${\dot{\phi }}^2/2-V$, which satisfy the cosmological continuity equation.