Massless scalar field ($m=0,\square \varphi =0$)

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.

The displayed action functional contains no mass term and no $R{\varphi }^2$ curvature coupling. Its Euler-Lagrange field equation is ${\partial }_{\mu }(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\varphi )=0$. In a flat FLRW metric, a Fourier mode obeys ${\varphi }_k^{\prime \prime }+2(a^{\prime }/a){\varphi }_k^{\prime }+k^2{\varphi }_k=0$ in four dimensions. The expansion can therefore freeze these modes; the curvature term of a conformally coupled scalar field changes that conclusion.

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.