= Bunch-Davies mode normalization in a finite comoving volume
{c}
{title2=$f_k=H(-\tau+i/k)e^{-ik\tau}/\sqrt{2k\mathcal V},\quad a^2\mathcal V(f_kf_k^{*\prime}-f_k^*f_k')=i$}
For a <minimally coupled massless scalar field> in flat <de Sitter spacetime>, $a=-1/(H\tau)$ and the <Canonically rescaled de Sitter scalar mode> is $u_k=af_k$. The equation $u_k''+(k^2-2/\tau^2)u_k=0$ and the positive-frequency condition $u_k\sim e^{-ik\tau}/\sqrt{2k\mathcal V}$ select the displayed mode. Direct differentiation verifies its <Wronskian> normalization, which ensures the equal-time <canonical commutation relations> with box-normalized creation and annihilation operators. A conformally coupled massless scalar has a different mode equation.
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