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Consider a logarithmic band of wavenumbers around $k\sim R^{-1}$, where $R$ is a comoving scale. The dimensionless cosmological power spectrum of the density contrast is ${\Delta }_{\delta }^2(k)=k^3{P}_{\delta }(k)/(2{\pi }^2)$. For $P_{\delta }=Ak$ this scales as $k^4$, so the rms density amplitude in such a band scales as $R^{-2}$.

The cosmological Poisson equation gives $-k^2{\Phi }_{\mathbf{k}}=4\pi G{a}^2{\bar{\rho }}_m{\delta }_{\mathbf{k}}$, with $\Phi$ a dimensionful peculiar gravitational potential. Its spectrum and its potential fluctuations per logarithmic wavenumber are thereforeindependent of $k$. Equivalently, ${\Phi }_R\sim G{\bar{\rho }}_m(aR{)}^2{\delta }_R$ is independent of $R$ because ${\delta }_R\propto R^{-2}$. This is the scale-local meaning of the Harrison-Zeldovich spectrum. An uncut spectrum has a divergent total potential variance; equal amplitude per logarithmic band does not imply a finite unsmoothed variance over all scales. For wavelengths beyond the Hubble radius, the statement is formulated using the corresponding conserved primordial curvature mode and its adiabatic matter-era potential, rather than treating the subhorizon Poisson approximation as universally valid.

On angular scales larger than roughly a degree, the Sachs-Wolfe effect relates the last-scattering potential to the observed Cosmic microwave background anisotropy: $\Delta T/T\simeq \Phi /(3c^2)$ for adiabatic modes in matter domination. Thus the observed large-angle amplitude constrains $A$. Scale-invariant potential bands produce a Sachs-Wolfe plateau, $\ell (\ell +1)C_{\ell }$ approximately constant, rather than constant $C_{\ell }$. Its amplitude and any tilt test the normalization and spectral shape. Cosmic variance, Integrated Sachs-Wolfe effect and assumptions about adiabatic initial conditions limit this inference; the density spectrum's late-time transfer must also be included when comparing it with galaxy clustering.

Let $q=\mathbf{k}\cdot \hat{\mathbf{n}}$ and retain the negative-spatial-perturbation convention of the preceding part. Since $h=h_S$, the isotropic pieces of the supplied Fourier mode contraction cancel:For $h=A(\mathbf{k}){\tau }^2{k}^2$, the factor $1/2$ in the line-of-sight integral cancels the derivative's factor two. ThusAn antiderivative convenient even at $q=0$ isIt follows directly, or by two integrations by parts, thatThe source sign and both endpoint coefficients are fixed by the metric convention; no division by $q$ is needed.

To interpret the large-angle Cosmic microwave background anisotropy, restore the observer-centered propagation phase $e^{-iq{\tau }_0}$, which can equivalently be absorbed into the definition of each Fourier mode amplitude. Put $D={\tau }_0-{\tau }_{\mathrm{dec}}$. One mode contributesThe first term is a local angular monopole plus a dipole, so it is removed when discussing observed angular multipoles $\ell \ge 2$. For modes outside the horizon at decoupling, $k{\tau }_{\mathrm{dec}}\ll 1$, the higher-multipole emission signal is dominated byIts transfer amplitude has no leading positive power of $k$: the two explicit powers in $h$ have canceled in the integral. Projection onto the sky supplies Spherical Bessel functions $j_{\ell }(kD)$, with modes of order $kD\sim \ell$ contributing to each angular scale. This is the large-angle Sachs-Wolfe effect; the emission velocity-like correction is relatively of order $k{\tau }_{\mathrm{dec}}$. For wavelengths much larger than the entire observed region, $kD\ll 1$, removing the monopole and dipole leaves a signal starting at order $A(kD{)}^2$, rather than a finite observable anisotropy from a spatially uniform mode.

The stochastic scale dependence of $A(\mathbf{k})$ is not specified by the deterministic metric solution. If one additionally assumes a scale-invariant primordial spectrum, $k^3{P}_A(k)=\text{constant}$, thenfor $\ell \ge 1$, giving the familiar large-angle Sachs-Wolfe plateau, $\ell (\ell +1)C_{\ell }\simeq \text{constant}$, for the measured $\ell \ge 2$ multipoles. A tilted primordial spectrum changes that trend. The full numerical temperature transfer also includes intrinsic last-scattering and Doppler terms, omitted from the stipulated metric-only integral.

Use Newtonian gauge with scalar metric perturbationsLet $\hat{n}$ point from the observer toward the emitter, $\Theta =\delta T/T$, and evaluate the unperturbed ray at $\mathbf{x}(\tau )=({\tau }_0-\tau )\hat{n}$. In the sudden-decoupling approximation the Sachs-Wolfe effect and its linear companions giveHere ${\Theta }_0={\delta }_{\gamma }/4$ is the intrinsic photon monopole; ${\Psi }_{\ast }$ is the emission gravitational redshift; the velocity term is the Doppler CMB anisotropy of the baryon-photon plasma; and the integral is the Integrated Sachs-Wolfe effect from evolving potentials during travel. The last two local terms are an observer monopole and dipole, removed when discussing the higher measured multipoles. Changing the direction convention reverses both velocity signs. A finite cosmological visibility function replaces the sharp emission surface by a weighted source integral.

For scale-invariant primordial potentials, ${\mathcal{P}}_{\Psi }=k^3{P}_{\Psi }/(2{\pi }^2)$ is constant before transfer effects. In matter domination, the adiabatic superhorizon relation is ${\Theta }_0=-2\Psi /3$, so the observable Sachs-Wolfe combination is ${\Theta }_0+\Psi =\Psi /3$. Its amplitude per logarithmic scale is therefore approximately independent of angle. In a flat geometry $\ell \sim k{D}_{\ast }$ and $\theta \sim {\ell }^{-1}$, and projection gives the Sachs-Wolfe plateauBoth the intrinsic and gravitational emission terms have this leading large-angle scaling; they must be combined, not treated as unrelated random sources.

The Euler equation gives $v_b\sim k{\tau }_{\ast }\Psi$ for the growing large-scale matter mode. Thus the Doppler amplitude grows roughly as ${\theta }^{-1}$ relative to the large-scale potential amplitude, while its logarithmic power grows as ${\theta }^{-2}$, until acoustic dynamics and damping invalidate that simple estimate. On subhorizon pressureless scales the Poisson equation gives ${\delta }_m\sim (k\tau {)}^2\Psi$: the corresponding density amplitude has the often-used ${\theta }^{-2}$ scaling. This is a matter-density estimate, not a universal small-angle photon-temperature law; the actual intrinsic photon source undergoes the oscillations in part(b). On superhorizon scales it instead has the constant-potential behavior just described.

The integrated source vanishes for constant linear potentials in pure matter domination. Potential decay during radiation domination or late accelerated expansion makes it nonzero. Its broad, slowly evolving sources favor large angular scales and give a potential-sized contribution there; crossings of many structures cancel progressively on small scales. Scale invariance alone fixes no universal power of angle for the integrated term: its time evolution and transfer functions are also required. Consequently the large-angle plateau, velocity-gradient enhancement and subhorizon density-gradient estimate describe distinct physical regimes, rather than one extrapolated power law across the whole spectrum.

Let $⟨{\Phi }_{\mathbf{k}}{\Phi }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }^3(\mathbf{k}-{\mathbf{k}}^{\prime })P_{\Phi }(k)$. Statistical isotropy and $\int |Y_{\ell m}{|}^{2}d\Omega =1$ applied to the preceding expression give the CMB angular power spectrumFor $n=1$, define the constant dimensionless potential power ${\mathcal{A}}_{\Phi }=k^3{P}_{\Phi }(k)/(2{\pi }^2)$. ThenThe needed logarithmic spherical Bessel square integral can be proved directly. Put $I_{\ell }={\int }_0^{\infty }{j}_{\ell }^{2}dx/x$ and $J_{\ell }={\int }_0^{\infty }{j}_{\ell }{j}_{\ell -1}dx$. From the Spherical Bessel function recurrences,integration of the squares' derivatives gives $J_{\ell }=(\ell +1)I_{\ell }$, and for $\ell >1$ also $J_{\ell }=(\ell -1)I_{\ell -1}$. At $\ell =1$, $j_0(0)=1$ and $j_0(\infty )=0$ give $J_1=1/2$, hence $I_1=1/4$. Induction now proves $I_{\ell }=1/[2\ell (\ell +1)]$. ConsequentlyThis is the flat large-angle Sachs-Wolfe plateau. The useful integral printed in the PDF is missing the square on $j_{\ell }$: without that square the formula is false, since at $\ell =1$ its left-hand side is $\pi /4$, whereas its asserted value would be $1/4$. The squared identity above is the one required by the variance calculation. The plateau describes the matter-era large-angle approximation, not acoustic peaks or late integrated-potential corrections.