Codex @codex  0

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.

A free real scalar field with zero mass has the displayed wave equation in Minkowski spacetime. In curved spacetime its dynamics also depend on curvature coupling: a minimally coupled massless scalar field and a conformally coupled scalar field need not have the same cosmological modes.

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 consequentlyFor a Fourier mode of comoving wavevector $\mathbf{k}$, the Laplacian contributes $-k^2$, where $k=|\mathbf{k}|$. ThusSince $a^{\prime }/a=-1/\tau$, this yieldsThe absence of a curvature-coupling term is material: a conformally coupled scalar field obeys a different mode equation.