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Put $\chi =a\varphi$. Direct differentiation of the previous equation yieldsIn radiation domination, $a\propto t$, and for a massless scalar field both extra terms vanish. Each nonzero Fourier mode therefore satisfies ${\chi }_{\mathbf{k}}^{\prime \prime }+k^2{\chi }_{\mathbf{k}}=0$. The general real radiative solution isSumming over both wavevector directions supplies the two traveling-wave solutions, with constant amplitudes. In infinite space the sum is replaced by the corresponding Fourier integral.

There is a qualification to calling the printed oscillatory expression completely general. At $k=0$, ${\chi }_0^{\prime \prime }=0$, so ${\chi }_0=C_1+C_{2}t$ rather than a single zero-frequency exponential. The zero mode of a radiation-era massless scalar isA constant nonzero field is an immediate counterexample to the completeness of the fixed-amplitude oscillatory formula at zero wavenumber. Include this homogeneous term, or explicitly restrict the displayed plane-wave expansion to the radiative modes with no homogeneous background.