= Unrescaled de Sitter massless scalar mode
{title2=$f_k(\eta)=A(1+ick\eta)e^{-ick\eta}$}
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 $f_k''-2f_k'/\eta+c^2k^2f_k=0$. Substitution verifies the displayed positive-frequency branch. The <Canonically rescaled de Sitter scalar mode> is $u_k=af_k$, a different variable satisfying $u_k''+(c^2k^2-2/\eta^2)u_k=0$. 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.
Back to article page
