= Solution
For the <FLRW metric> in <conformal time>, $\sqrt{-g}=a^4$, $g^{00}=-a^{-2}$ and $g^{ij}=a^{-2}\delta^{ij}$. The equation for a <minimally coupled massless scalar field> is consequently
$$
-(a^2\delta\phi')'+a^2\nabla^2\delta\phi=0.
$$
For a <Fourier mode> of comoving wavevector $\mathbf k$, the <Laplacian> contributes $-k^2$, where $k=|\mathbf k|$. Thus
$$
\delta\phi_k''+2\frac{a'}a\delta\phi_k'+k^2\delta\phi_k=0.
$$
Since $a'/a=-1/\tau$, this yields
$$
\boxed{\delta\phi_k''-\frac2\tau\delta\phi_k'=-k^2\delta\phi_k.}
$$
The absence of a curvature-coupling term is material: a <conformally coupled scalar field> obeys a different mode equation.
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