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A blackbody photon density perturbation gives an intrinsic temperature fluctuation ${\Theta }_0={\delta }_{\gamma }/4$. The baryon/electron bulk velocity produces a Doppler effect at last scattering. In the common-potential convention of part (ii),Ignoring the slowly varying damping derivative at leading acoustic order, differentiating the density cosine produces a velocity sine. Thus density and velocity acoustic sources are approximately a quarter cycle out of phase. For regular adiabatic initial conditions the density oscillation has approximately a common cosine phase across modes; this phase coherence is what makes a sequence of peaks instead of a smooth incoherent superposition.

Let $D_{\ast }={\tau }_0-{\tau }_{\ast }$ be the comoving distance to last scattering and $r_s({\tau }_{\ast })={\int }^{{\tau }_{\ast }}{c}_{s}d\tau$ its sound horizon. In the present limit $r_s\simeq {\tau }_{\ast }/\sqrt{3}$. Density extrema have $k{r}_s\simeq m\pi$, whereas velocity extrema have $k{r}_s\simeq (m+1/2)\pi$. The projection from spatial wavenumber to high angular multipoles is roughly $\ell \simeq k{D}_{\ast }$, giving an acoustic spacingThe Doppler effect fills some density troughs and modifies the detailed peak shapes; its different projection prevents treating it as an identical second series of narrow peaks.

For clarity, choose the sightline to point outward from the observer and let its angle to $\mathbf{k}$ have cosine $\mu$. The intrinsic-plus-emitter-Doppler source is then $[{\delta }_{\gamma }/4-ik\mu {\theta }_b]e^{ik\mu D_{\ast }}$. Expanding its angular dependence gives the angular projection of density and Doppler acoustic sourcesThe second sign follows from the chosen outward sightline and $v_b^i=i{k}^i{\theta }_b$; it changes with the sightline convention together with the velocity projection. The $j_{\ell }$ are Spherical Bessel functions. Both sources are proportional to the primordial curvature. If ${\mathcal{T}}_{\ell }={\Theta }_{\ell }/{\zeta }_{\ast }$, their angular spectrum in the usual continuum Fourier normalization isThis formula includes the density-velocity cross term, not just a sum of independent source powers.

Photon diffusion multiplies both leading source amplitudes by $e^{-k^2/k_D^2}$, so their power carries $e^{-2k^2/k_D^2}$. The approximate angular damping scale isThis produces the damping tail in the Cosmic microwave background power spectrum. Finite last-scattering thickness adds further small-scale averaging. Gravity, baryon loading and later propagation alter detailed peak locations and heights, but the oscillatory peak sequence, phase-shifted velocity contribution and diffusion damping follow directly from the simplified solution obtained here.