= Frozen scalar fields and the Friedmann constraint
For a homogeneous canonical <scalar field>, the <cosmological perfect-fluid continuity equation> reduces to $\dot\phi(\ddot\phi+3H\dot\phi+V')=0$. It implies the <inflaton equation of motion> wherever $\dot\phi\ne0$, and by continuity at isolated turning points. It cannot by itself exclude a frozen field on a time interval. Indeed, choose $\phi=\phi_0$ with $V(\phi_0)>0$ but $V'(\phi_0)\ne0$; a <de Sitter scale factor in flat slicing> with $H^2=V(\phi_0)/(3M_{\rm Pl}^2)$ satisfies the homogeneous <Friedmann equation> and <Friedmann acceleration equation>, but not the scalar equation. Varying the <inflaton action in an expanding universe> supplies the missing equation also for a frozen field, requiring $V'(\phi_0)=0$.
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