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For an isotropic, full-sky temperature map with a fixed bispectrum template $b=f_{\mathrm{NL}}{b}^1$, the inverse-variance-weighted cubic statistic has normalization $F=\frac{1}{6}\sum (b^1{)}^2{\mathcal{G}}^2/(C_1{C}_2{C}_3)$ over ordered triples. It is unbiased under the linear template relation and has Gaussian cosmic variance $1/F$. The factor $1/6$ counts the six cross-triple Wick contractions. A removed monopole cancels the nine internal-pairing terms; incomplete sky coverage or anisotropic noise generally require a linear correction.

Primordial density contrasts perturb the intrinsic photon temperature, the gravitational potential, and the velocity of the last-scattering material. Intrinsic photon-density fluctuations obey $\delta T/T={\delta }_{\gamma }/4$ because ${\rho }_{\gamma }\propto T^4$; adiabatic initial conditions relate these to matter density contrasts. Subsequent acoustic evolution, Doppler effect and gravitational frequency shifts determine the observed Cosmic microwave background anisotropy. There is no scale-independent identification of the final sky map with matter density contrast alone.

The spherical harmonic coefficients are random variables inherited from these perturbations. After subtracting the mean sky temperature, $⟨a_{lm}⟩=0$ for the anisotropy modes. Gaussian primordial perturbations give approximately Gaussian random variables; this extra Gaussian assumption is not needed for diagonal covariance. Reality of the sky map implies $a_{l,-m}=(-1{)}^m{a}_{lm}^{\ast }$.

For a statistically isotropic sky, its two-point function depends only on $\hat{n}\cdot \hat{m}$. Rotations mix the $m$ components at fixed $l$ and do not mix different irreducible $l$ spaces. A rotation-invariant covariance matrix is therefore a multiple of the identity in each $l$ block:On our one sky, average the $2l+1$ angular components at fixed $l$, or equivalently suitable rotated sky pairs, rather than treating different universes as observed samples. The CMB angular power spectrum is estimated byUnder the Gaussian, full-sky, noiseless assumptions, there are $2l+1$ independent real degrees of freedom and $\mathrm{Var}({\hat{C}}_l)=2C_l^2/(2l+1)$. This cosmic variance remains even with perfect detectors; a finite sky mask and detector noise add further uncertainty. The estimator is unbiased, $⟨{\hat{C}}_l⟩=C_l$, but does not turn one realization into the exact ensemble mean.

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.

The given large-angle temperature transfer relation gives the low-multipole tensor fraction in quartic hilltop inflationThis is not the primordial tensor-to-scalar ratio $r=16{ϵ}_V$, because the scalar and tensor contributions to low CMB multipoles have different transfer functions. With $\sigma \ll m_{\mathrm{pl}}$, the tensor contribution is far below the scalar temperature signal and its cosmic variance, and the corresponding primordial gravitational-wave or polarization signal is extremely small.

For COBE normalization of quartic hilltop inflation, the historical scalar horizon amplitude is of order ${\delta }_H\simeq 2\times 10^{-5}$, or ${\mathcal{P}}_S\simeq 25{\delta }_H^2/4\sim 2.5\times 10^{-9}$ in the usual matter-era convention. The slow-roll spectrum simplifies toHenceLate-time transfer and tilt corrections to the COBE normalization change the order-one coefficient, not the conclusion that the dimensionless coupling must be extraordinarily small. The leading normalization is independent of $\sigma$: reducing the height of the potential also changes the slope and field value at fixed $N$, causing the scale factors to cancel.

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.

A survey depth $R\sim 400\mathrm{Mpc}$ makes wavelengths substantially larger than its volume poorly measurable. A characteristic inverse depth is $1/R\simeq 2.5\times 10^{-3}\mathrm{Mp}{\mathrm{c}}^{-1}$; a Fourier wavelength that fits across a dimension $L$ instead requires $k\gtrsim 2\pi /L$. For $L$ between $R$ and $2R$, this is roughly $0.008$--$0.016\mathrm{Mp}{\mathrm{c}}^{-1}$. A limited sky patch can have smaller transverse dimensions, so its survey window function raises the practical minimum in those directions. An indicative useful band for a broad survey is around $10^{-2}\mathrm{Mp}{\mathrm{c}}^{-1}$ and above, including scales near equality if its volume is large enough.

There is no upper wavenumber determined by depth alone. Mean separation of observed galaxies, galaxy power spectrum shot noise, angular/redshift resolution, and nonlinear clustering set it. Straightforward linear inference is typically restricted to roughly $k\lesssim 0.1\mathrm{Mp}{\mathrm{c}}^{-1}$ at low redshift, with smaller scales requiring modelling. The magnitude limit does not specify the galaxy comoving number density, so a more precise sampling cutoff cannot be deduced from these data.

The two measurements are complementary. The cosmic microwave background samples a large volume at an early epoch when relevant perturbations are mostly linear, and is free of galaxy bias; its matter-spectrum inference is indirect and depends on projection, cosmological parameters, foregrounds, and cosmic variance. A galaxy survey gives three-dimensional late-time information and can reach smaller scales, but observes biased, discrete tracers: galaxy bias, nonlinear gravitational growth, redshift-space distortions, selection effects, and galaxy power spectrum shot noise must be controlled. Combining both constrains the initial spectrum and its subsequent growth more strongly than either alone, while their overlapping modes test the inferred evolution.

Let $b_{{\ell }_1{\ell }_2{\ell }_3}^1$ denote the template at unit $f_{\mathrm{NL}}$, and define $A_{123}=b_{{\ell }_1{\ell }_2{\ell }_3}^1{\mathcal{G}}_{m_1{m}_2{m}_3}^{{\ell }_1{\ell }_2{\ell }_3}$. All sums below are over ordered triples of the retained multipoles, with the monopole removed and $C_{\ell }>0$. A finite maximum multipole makes these expressions ordinary finite sums. Taking the expectation of the cubic numerator and using the template relation givesThus the full-sky cubic bispectrum estimator is unbiased forThe assumed template must have nonzero support, so $F>0$. Using the Gaunt sum rule, writeThe factor $1/6$ belongs to the ordered sum; an unordered triangle sum would instead use multiplicity factors for equal multipoles. This normalization is also the Gaussian Fisher information for the template amplitude.

For the cosmic variance, use the reality condition ${\Theta }_{\ell m}^{\ast }=(-1{)}^m{\Theta }_{\ell ,-m}$ and Gaussian covarianceThere are $15$ pairings of the six multipoles in the squared cubic numerator. Six pairings connect every factor in the first triple to a factor in the second. Each gives $\sum A_{123}^2/(C_{{\ell }_1}{C}_{{\ell }_2}{C}_{{\ell }_3})=6F$. The covariance phases cancel: nonzero Gaunt integrals have $m_1+m_2+m_3=0$, and simultaneous reversal of the three $m$ indices multiplies the Gaunt integral by $(-1{)}^{{\ell }_1+{\ell }_2+{\ell }_3}=1$.

The other nine pairings contain one contraction within each triple. To see why their sums vanish, contract two legs of a Gaunt integral and use the spherical harmonic addition theorem:The template and inverse-covariance weights are independent of $m$, so they do not spoil this cancellation. The only possible surviving unpaired mode is the monopole, and ${\Theta }_{00}=0$ removes it. This is monopole cancellation of internal cubic-estimator contractions. It explains why no linear correction is needed in this ideal full-sky isotropic problem; masks or anisotropic noise would spoil the argument.

The cubic numerator therefore has Gaussian variance $6(6F)=36F$. Dividing by $(6F{)}^2$ givesThe result concerns the Gaussian-limit covariance. Non-Gaussian connected four- and six-point terms can change the variance at finite amplitude. Unbiasedness uses the assumed linear template relation for the observed three-point function.

Write $S_{\ast }(\mathbf{x})=({\Theta }_0+\psi )({\eta }_{\ast },\mathbf{x})$ and $E=e^{-{\tau }_{\mathrm{re}}}$. Under the stated matter-era growing-mode approximation, $\dot{\varphi }+\dot{\psi }=0$, and ignoring velocities removes the Doppler CMB anisotropy. The two peaks of the cosmological visibility function give the Cosmic microwave background line-of-sight solutionApply the same solution with observation time ${\eta }_{\mathrm{re}}$, just before the thin reionization screen. Between recombination and that screen there is no further scattering, so free propagation givesA direction-independent potential commutes with the angular average. Thus the incoming photon monopole plus the potential at the screen isSubstitution givesThis derivation inherits the neglected scattering quadrupole and polarization assumptions of part (ii); the screen is an idealized isotropizing approximation.

For a plane Fourier mode $S_{\ast }(\mathbf{x})=S_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{x}}$, the angular average is explicit:where $j_0$ is the zeroth Spherical Bessel function. The mode's observed transfer isIf $k\Delta \chi \gg 1$, the rescattered monopole is suppressed by at least $1/(k\Delta \chi )$, because the incoming directions sample incoherent phases. If $k\Delta \chi \ll 1$, $j_0=1+O((k\Delta \chi {)}^2)$ and the two propagation phases agree to leading order. The two weights then sum to one. ConsequentlyThese are the small-scale damping and large-scale coherence limits of reionization damping of cosmic microwave background temperature anisotropy.

Each small-scale primary temperature amplitude is multiplied by $e^{-{\tau }_{\mathrm{re}}}$, so its Cosmic microwave background power spectrum is multiplied by $e^{-2{\tau }_{\mathrm{re}}}$. Holding transfer parameters fixed,Many measured high-$\ell$ modes determine this combination accurately. Their derivatives $\partial \log C_{\ell }/\partial \log A_s=1$ and $\partial \log C_{\ell }/\partial {\tau }_{\mathrm{re}}=-2$ are degenerate: a change $d\log A_s=2d{\tau }_{\mathrm{re}}$ leaves the leading temperature spectrum unchanged. The undamped large-scale modes can distinguish the parameters, but there are few of them and cosmic variance gives fractional full-sky uncertainty $\sqrt{2/(2\ell +1)}$ per multipole. This explains the primordial-amplitude–optical-depth degeneracy. Polarization from reionization, lensing, and more complete late-time effects partly break it. The TeX's final ${\tau }_e$ is a transcription mismatch; the PDF uses ${\tau }_{\mathrm{re}}$ consistently.

The metric source is a gravitational frequency shift. photons following null geodesics through a time-dependent spatial metric change momentum relative to the background comoving frame. The directional projection of that metric change gives the ${\mu }^2$ factor. It is not a collision term or continuing photon scattering after recombination. The normalization and sign belong to the specified definition of $h$ and $\Delta$; a distribution perturbation should not silently be identified with temperature without fixing that normalization.

Define $\mathcal{H}=\dot{R}/R$, where dots here mean conformal-time derivatives. In an Einstein-de Sitter background,A power law $h\propto {\tau }^p$ therefore satisfies $p(p-1)+2p=6$, or $(p-2)(p+3)=0$. Thuswith the required growing mode $h=A{\tau }^2$. This is a solution in the perturbation convention used in the question, rather than a statement that every gauge-invariant gravitational potential grows as ${\tau }^2$.

Use an integrating factor for the photon equation:For the growing mode $\stackrel{...}{h}=0$. For the polynomial-source photon transport solution, a particular solution is $B_h=-2i\mu \dot{h}/k+2\ddot{h}/k^2$: direct substitution gives ${\dot{B}}_h+ik\mu B_h=2{\mu }^2\dot{h}$. The initial-value solution is consequentlyThis also follows by integrating the source twice by parts. The apparent inverse powers of $k$ cancel in the complete long-wavelength limit; the integral form remains regular. At $\mu =0$, the source vanishes and the complete expression reduces to $\Delta ={\Delta }_R$.

Physically, the exponential transports the recombination pattern along photon trajectories; its direction-dependent phase projects a Fourier perturbation onto different angular multipoles. The extra terms are the accumulated gravitational contribution, here expressible as endpoint terms because the source has this special time dependence. For endpoint monopole and dipole in photon transport, at the observer the last two terms are respectively a directional dipole and an angular monopole for each mode; they do not supply a new sky pattern at multipoles $\ell \ge 2$. The propagating recombination combination contains the intrinsic perturbation and gravitational/velocity endpoint contributions. Individual terms depend on the perturbation gauge; observable temperature anisotropy requires the consistent emitter and observer contributions.

In pure matter domination the gauge-invariant scalar potentials are constant for a growing linear mode: the matter contrast grows as $R$ while the potential scales as its contrast divided by $R$. Thus growth of this $h$ is compatible with no late Integrated Sachs-Wolfe effect in Einstein-de Sitter. When dark energy suppresses matter growth, the potentials generally evolve; photons gain and lose unequal energies entering and leaving a potential well. In a convention where $\Phi ,\Psi$ are the dimensionless Newtonian-gauge potentials, the integrated temperature source is $\int (\dot{\Phi }+\dot{\Psi })d\tau$ along the light path.

Large-angle cosmic microwave background data can constrain that late contribution, but low-multipole power alone has large cosmic variance and depends on the primordial radiation spectrum, geometry and foreground treatment. Cross-correlating the temperature map with low-redshift galaxies or other matter tracers isolates a late-time connection; decaying wells typically correlate overdensities with hotter cosmic microwave background directions. Boughn and Crittenden compared WMAP temperature data with the HEAO1 hard X-ray background and examined possible foreground origins of the correlation. This tests potential evolution under a cosmological model, not dark energy independently of all gravity and curvature assumptions.^[ref]

The primary small-scale Cosmic microwave background power spectrum measures the amplitude $A_s{e}^{-2{\tau }_{\mathrm{re}}}$, because reionization damps each temperature amplitude once. The unaffected large-scale temperature modes are few and limited by cosmic variance. Polarization and gravitational lensing provide additional information that partly breaks this degeneracy.