= Solution
Denote the comoving box volume by $\mathcal V$, keeping it distinct from the <inflaton> potential $V(\phi)$. For the <Canonically rescaled de Sitter scalar mode> $u_k=a\delta\phi_k$, the equation in part (b) becomes
$$
u_k''+\left(k^2-\frac2{\tau^2}\right)u_k=0.
$$
Direct differentiation verifies the positive-frequency solution
$$
u_k=\frac{1-i/(k\tau)}{\sqrt{2k\mathcal V}}e^{-ik\tau},\qquad\boxed{\delta\phi_k^{(+)}=\frac{H(-\tau+i/k)}{\sqrt{2k\mathcal V}}e^{-ik\tau}.}
$$
For $|k\tau|\gg1$, $u_k$ tends to $e^{-ik\tau}/\sqrt{2k\mathcal V}$, the positive-frequency flat-space mode. It therefore selects the incoming <Bunch-Davies vacuum>. The <Bunch-Davies mode normalization in a finite comoving volume> is fixed by the <Wronskian> condition
$$
a^2\mathcal V\left(\delta\phi_k^{(+)}\delta\phi_k^{(+)*\prime}-\delta\phi_k^{(+)*}\delta\phi_k^{(+)\prime}\right)=i.
$$
After <cosmological horizon exit>, $|k\tau|\ll1$, the mode tends to the time-independent value $iH/\sqrt{2k^3\mathcal V}$. The complex mode and its <Hermitian conjugate> together make the real quantum field.
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