Zero mode of a radiation-era massless scalar (${\varphi }_0=C+D/a$)

For a minimally coupled massless scalar field in a flat radiation-dominated universe, conformal time gives $a=\alpha \tau$ and $(a\varphi {)}^{\prime \prime }-{\nabla }^2(a\varphi )=0$. Every nonzero Fourier mode is therefore a free oscillator divided by $a$. The zero-wavenumber equation instead gives $a{\varphi }_0=A+B\tau$, hence the displayed constant-plus-decaying solution. Substituting $k=0$ into a fixed-amplitude oscillatory formula retains only the $1/a$ term and loses the constant field solution. A truly general solution must include this zero mode or explicitly exclude a homogeneous background.