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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.

Denote the comoving box volume by $\mathcal{V}$, keeping it distinct from the inflaton potential $V(\varphi )$. For the Canonically rescaled de Sitter scalar mode $u_k=a\delta {\varphi }_k$, the equation in part (b) becomesDirect differentiation verifies the positive-frequency solutionFor $|k\tau |\gg 1$, $u_k$ 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 conditionAfter cosmological horizon exit, $|k\tau |\ll 1$, 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.

Subtract the homogeneous background equation and linearize the inflaton equation of motion. The field perturbation satisfiesWith $f=a\delta \varphi$, direct differentiation cancels the first-derivative term and givesThe conformal-time quadratic action for an inflaton perturbation, after a boundary term is discarded, isFor the specified de Sitter spacetime background, $a^{\prime \prime }/a=2/{\tau }^2=2a^2{H}^2$. The light-field approximation $|V_{,\varphi \varphi }|\ll 2H^2$ therefore permits dropping the mass term. Taking a spatial Fourier transform yields the Canonically rescaled de Sitter scalar mode equationIts kinetic term is canonical, with momentum ${\pi }_f=f^{\prime }$. Each independent real Fourier component consequently has the canonical coordinate/momentum algebra of a quantum harmonic oscillator, with time-dependent squared frequency ${\omega }_k^2=k^2-a^{\prime \prime }/a$. This is why canonical quantization of a real scalar field proceeds by oscillator creation and annihilation operators. On superhorizon scales the squared frequency is negative: the system is then an inverted time-dependent oscillator, not a stationary oscillator with a positive frequency. The canonical commutation relations remain valid.

Conformal time is defined byFor $a=a_0{e}^{H_{\inf }t}$, choose $\tau \to 0$ as $t\to +\infty$. Integration givesas recorded by Conformal time during de Sitter expansion.

Writing a prime for $d/d\tau$, the relationsturn the Fourier-mode equation intoSetA direct substitution givesHere $a^{\prime \prime }/a=2/{\tau }^2$, so the Canonically rescaled de Sitter scalar mode obeys

In the flat slicing of de Sitter spacetime, $a\propto -1/\eta$, $\eta <0$, a Fourier mode of a minimally coupled massless scalar field obeys $f_k^{\prime \prime }-2f_k^{\prime }/\eta +c^2{k}^2{f}_k=0$. Substitution verifies the displayed positive-frequency branch. The Canonically rescaled de Sitter scalar mode is $u_k=a{f}_k$, a different variable satisfying $u_k^{\prime \prime }+(c^2{k}^2-2/{\eta }^2)u_k=0$. For nonzero $k$, the opposite-frequency branch completes the time solution; at $k=0$ the solutions are $1$ and ${\eta }^3$. This separation avoids losing the homogeneous decaying mode by taking a restricted ansatz literally at zero wave number.