Bunch-Davies mode normalization in a finite comoving volume ($f_k=H(-\tau +i/k)e^{-ik\tau }/\sqrt{2k\mathcal{V}},a^2\mathcal{V}(f_k{f}_k^{\ast \prime }-f_k^{\ast }{f}_k^{\prime })=i$)

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 $u_k=a{f}_k$. The equation $u_k^{\prime \prime }+(k^2-2/{\tau }^2)u_k=0$ 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.