 Solution

For a minimally coupled massless scalar field, the covariant wave operator isThe Fourier mode $\varphi =u(\eta )e^{i\mathbf{k}\cdot \mathbf{r}}$ therefore satisfiesSubstituting $u=(A+B\eta )e^{-i\omega \eta }$ leaves $2(i\omega A-B)e^{-i\omega \eta }/\eta$. ThusThis is an unrescaled de Sitter massless scalar mode. For $k\ne 0$, the opposite-frequency time solution supplies the second independent mode; a real scalar field combines appropriate conjugate spatial modes. For $k=0$ the particular ansatz gives only the constant solution, while the full homogeneous solution is $u=C+D{\eta }^3$. The requested ansatz was not a general zero-mode parametrization.