= Zero mode of a radiation-era massless scalar
{title2=$\phi_0=C+D/a$}
For a minimally coupled <massless scalar field> in a flat radiation-dominated universe, <conformal time> gives $a=\alpha\tau$ and $(a\phi)''-\nabla^2(a\phi)=0$. Every nonzero <Fourier mode> is therefore a free oscillator divided by $a$. The zero-wavenumber equation instead gives $a\phi_0=A+B\tau$, 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.
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