 Solution

Holding the background metric fixed, the linearized homogeneous-field equation isConsequently each comoving Fourier coefficient obeysThe unscaled Laplacian in the starting equation must be interpreted as the physical spatial Laplacian; for the explicitly comoving wavevector it gives $k^2/a^2$, not $k^2$. Metric fluctuations have been discarded as permitted. The mode written later is a massless, minimally coupled approximation, requiring $|m_{\mathrm{eff}}^2|\ll H^2$; it is not a solution for an arbitrary potential curvature.