Homogeneous massive-scalar equation in an expanding universe ($\ddot{\varphi }+3H\dot{\varphi }+m^2\varphi =0$)

A canonical spatially homogeneous field with $V=m^2{\varphi }^2/2$ obeys $\ddot{\varphi }+3H\dot{\varphi }+m^2\varphi =0$. The $3H\dot{\varphi }$ term expresses dilution of field momentum by expansion. Conservation of its stress energy gives this equation wherever $\dot{\varphi }\ne 0$; the canonical field equation is needed to fix an identically stationary interval, while continuity extends a regular solution across isolated turning points.