= Solution
For a <minimally coupled massless scalar field>, the <covariant wave operator> is
$$
\Box\phi=\frac1{\sqrt{-g}}\partial_\mu(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi)
=\frac1{a^2}\left[-\partial_0^2\phi-2\mathcal H\partial_0\phi+\nabla^2\phi\right].
$$
The <Fourier mode> $\phi=u(\eta)e^{i\mathbf k\cdot\mathbf r}$ therefore satisfies
$$
u_{\eta\eta}-\frac2\eta u_\eta+\omega^2u=0,\qquad\omega=c|k|.
$$
Substituting $u=(A+B\eta)e^{-i\omega\eta}$ leaves $2(i\omega A-B)e^{-i\omega\eta}/\eta$. Thus
$$
\boxed{B=i\omega A=ic|k|A,\qquad
\phi=A(1+ic|k|\eta)e^{-ic|k|\eta+i\mathbf k\cdot\mathbf r}.}
$$
This is an <unrescaled de Sitter massless scalar mode>. For $k\ne0$, 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.
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