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After a spatial Fourier transform, the preceding equation isThe constant particular solution is ${\delta }_r=-4\Phi$. Hence the Sachs-Wolfe combinationcontains only the homogeneous harmonic oscillator modes and satisfiesFor adiabatic initial conditions on large scales, $S=\Phi /3=\mathcal{R}/5$ and $S^{\prime }=0$ at the start of matter domination. Up to an irrelevant choice of the phase origin,Thus the radiation acoustic transfer function is flat at $1/5$ as $k\to 0$ and oscillates on smaller scales. Projection onto the sky produces a large-angle Sachs-Wolfe plateau and, after the full photon-baryon physics is included, the acoustic structure of the Cosmic microwave background power spectrum.

The baryon transfer function should also oscillate soon after cosmological recombination: before decoupling, Thomson scattering forced baryons to share the acoustic motion of the photon-baryon fluid, and recombination does not instantly erase the resulting density pattern.

Yes. The baryons carry the oscillatory baryon transfer function inherited from the pre-recombination photon-baryon fluid. Their later infall toward cold-dark-matter overdensities suppresses the relative mode and drives ${\delta }_b/{\delta }_c\to 1$, but gravitational evolution does not remove every scale-dependent acoustic phase. The total cosmological density power spectrum therefore contains the small wiggles known as a baryon acoustic oscillation in the matter power spectrum.