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Let $q_{\ast }=k{\tau }_{\mathrm{dec}}/\sqrt{3}$ and $E_D=\exp (-k^2/k_D^2)$ at last scattering. For random oscillator amplitudes, define spectra $P_A,P_B$ and cross-spectrum $P_{AB}$ using the same Fourier normalization as the density spectrum. Averaging the solution's squared modulus givesConsequently the intrinsic-temperature source has variance spectrum $P_{{\delta }_{\gamma }}/16$. A common acoustic phase is necessary for pronounced peaks in the variance. Equal independent sine/cosine amplitudes would erase them. A regular growing adiabatic mode begins with negligible velocity outside the sound horizon and selects approximately the cosine phase, $B\simeq 0$, preserving a ${\cos }^2{q}_{\ast }$ sequence. This illustrates coherent initial phases are needed for acoustic variance peaks.

From continuity, ${\theta }_{\gamma }=3{\delta }_{\gamma }^{\prime }/(4k^2)$. Ignoring the small damping correction in this velocity relation, the coherent cosine solution givesThe Doppler CMB anisotropy source therefore has directional variance $3{\mu }^2{P}_A{E}_D^2{\sin }^2{q}_{\ast }/16$, where $\mu =\hat{\mathbf{n}}\cdot \hat{\mathbf{k}}$. Its velocity maxima lie between density maxima: compression and bulk motion are a quarter acoustic period apart.

Angular projection must also be included. Let ${\chi }_{\ast }={\tau }_0-{\tau }_{\mathrm{dec}}$, assume a thin last-scattering surface, and choose the emission phase $e^{ik{\chi }_{\ast }\mu }$. The coherent intrinsic-plus-Doppler radial response is proportional toThe derivative follows from $i\mu e^{ix\mu }={\partial }_x{e}^{ix\mu }$ and the velocity sign above. A reversed line-of-sight convention reverses the corresponding Doppler sign consistently. The CMB angular power spectrum is $C_{\ell }=(2/\pi )\int k^{2}dk{P}_A(k)|{\Theta }_{\ell }(k){|}^2$ in this simplified model. Squaring the combined response keeps its correlated density–velocity cross term; adding two independent angular powers would discard that correlation. The derivative Bessel projection also spreads the Doppler contribution differently from the density contribution.

The sound horizon here is $r_s\simeq {\tau }_{\mathrm{dec}}/\sqrt{3}$. Intrinsic acoustic extrema occur near $k{r}_s=n\pi$, mapping roughly to ${\ell }_n\simeq n\pi {\chi }_{\ast }/r_s$, with interleaved velocity extrema and projection broadening. Cosmic microwave background diffusion damping suppresses both contributions exponentially at $k\gtrsim k_D$, or multipoles $\ell \gtrsim k_D{\chi }_{\ast }$. Modes outside the sound horizon have no completed acoustic oscillation.

The metric integral represents gravitational redshift, including contributions that in another gauge are separated into endpoint Sachs-Wolfe effects and Integrated Sachs-Wolfe effects. It is not correct to call the entire synchronous metric integral only a late-time integrated effect. Large-scale metric/curvature fluctuations remain important where photon pressure cannot redistribute matter causally and produce the broad large-angle signal rather than just the pressure-oscillation sequence. Potential evolution during radiation-to-matter transition also drives and shifts acoustic amplitudes near horizon entry; late accelerating-era evolution adds a large-angle integrated signal. These effects are excluded by the oscillator's instruction to ignore metric perturbations. Even if the Newtonian-gauge potential is constant in matter domination, synchronous $h_{ij}^{\prime }$ need not vanish, and the complete observable requires the consistent density, velocity and metric combination.

The comoving distance to the Cosmic microwave background last-scattering surface is of order the present horizon, ${\chi }_{\ast }\sim 1.3\times 10^4\mathrm{Mpc}$. At recombination the particle horizon is much smaller, so replacing ${\chi }_{\ast }$ by today's horizon is adequate for the requested estimate. A transverse Fourier mode projects approximately to angular multipole $\ell \sim k{\chi }_{\ast }$. If an angular feature's width is estimated by $\theta \sim 1/\ell$, the resolution ${\theta }_{\min }\sim 10^{-3}$ givesUsing the common half-wavelength convention $\theta \sim \pi /\ell$ instead gives ${\ell }_{\max }\sim 3000$ and $k_{\max }\sim 0.2$--$0.3\mathrm{Mp}{\mathrm{c}}^{-1}$. These express the same order-of-magnitude resolution estimate with different angular-scale conventions. On the graph, a useful indicative band is therefore from about $10^{-4}\mathrm{Mp}{\mathrm{c}}^{-1}$ to order $10^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, potentially a few times $10^{-1}$; it is not a sharp instrumental boundary. The lowest useful multipoles are of order a few, with substantial cosmic variance.

The cosmic microwave background constrains the matter power spectrum indirectly through the primordial spectrum and radiation/matter cosmological transfer functions. Beam resolution, Cosmic microwave background diffusion damping, foreground emission, and projection of three-dimensional fluctuations into angular anisotropy limit the recoverable small-scale information. Resolving a given angle alone does not guarantee a precise matter-spectrum measurement there.

The conventional vertical variable is $D_{\ell }=\ell (\ell +1)C_{\ell }/(2\pi )$ in $\mu {\mathrm{K}}^2$, plotted against the dimensionless angular multipole $\ell$. The Sachs-Wolfe effect supplies a nearly flat large-angle plateau for $\ell \lesssim 30$. The total spectrum has its first acoustic peak near $\ell \simeq 220$ with $D_{\ell }$ of order $5\times 10^3\mu {\mathrm{K}}^2$, followed by further Cosmic microwave background acoustic peaks; the Doppler CMB anisotropy is phase shifted relative to the photon-density oscillation. At $\ell \gtrsim 1000$, Cosmic microwave background diffusion damping lets photons random-walk across perturbations during recombination and produces the rapidly falling damping tail.