 Unrescaled de Sitter massless scalar mode

In the flat slicing of de Sitter spacetime, $a\propto -1/\eta$, $\eta <0$, a Fourier mode of a minimally coupled massless scalar field obeys $f_k^{\prime \prime }-2f_k^{\prime }/\eta +c^2{k}^2{f}_k=0$. Substitution verifies the displayed positive-frequency branch. The Canonically rescaled de Sitter scalar mode is $u_k=a{f}_k$, a different variable satisfying $u_k^{\prime \prime }+(c^2{k}^2-2/{\eta }^2)u_k=0$. For nonzero $k$, the opposite-frequency branch completes the time solution; at $k=0$ the solutions are $1$ and ${\eta }^3$. This separation avoids losing the homogeneous decaying mode by taking a restricted ansatz literally at zero wave number.