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Write ${\mathbf{x}}_j={\mathbf{x}}_0+({\eta }_0-{\eta }_j)\hat{\mathbf{n}}$. The delta functions in $g$ evaluate the local last-scattering sources, while the piecewise $e^{-\tau }$ weights the integrated gravitational source. ThusThe first two lines are weighted Sachs-Wolfe combination and Doppler sources at the two last-scattering surfaces; the final lines are the corresponding Integrated Sachs-Wolfe effect.

To predict these quantities from primordial perturbations, one solves the coupled linear Einstein field equations and Boltzmann equation hierarchy for photons, baryons, cold dark matter, and neutrinos, together with the ionization history that determines $\Gamma$ and hence $g$. The solutions provide ${\Theta }_0$, ${\mathbf{v}}_e$, $\Phi$, and $\Psi$ at both surfaces and along the line of sight.

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]

A cosmological constant has fixed density and equation of state $w=p/(\rho c^2)=-1$. A separately conserved dynamical component obeysIf $w$ departs from $-1$, its history changes the Hubble parameter, luminosity distances and angular-diameter distances. Type Ia supernovae probe a relative distance-redshift relation, baryon acoustic oscillations provide standard-ruler distances and expansion information, and the cosmic microwave background fixes early-time scales and geometric combinations. Combining these breaks some degeneracies with matter density, curvature and calibration. Distance observables integrate the expansion history, so they smooth rapid variations in $w$ and cannot determine an arbitrary function exactly.

Structure-growth measurements add a dynamical test. Galaxy clustering, redshift-space distortions and weak gravitational lensing constrain the growth of matter perturbations and metric potentials. The late Integrated Sachs-Wolfe effect probes potential evolution through CMB-matter correlations. A scalar-field dark-energy model can have perturbations and an effective sound speed different from other models with a similar background expansion. Comparisons of geometry and growth can therefore distinguish some models and test gravity assumptions.

A statistically significant, robust departure from $w=-1$ or from constant dark-energy density would disfavor a pure cosmological constant within the stated framework. Consistency with $w=-1$ over a finite redshift range does not prove exact constancy: a slowly rolling field can mimic it. Supernova calibration/evolution, galaxy bias, photometric redshift errors, intrinsic galaxy alignments and foregrounds must be controlled; modified gravity can also change growth. Parameterizations such as $w(a)=w_0+w_a(1-a)$ are useful finite-dimensional tests rather than a complete classification of all dark-energy histories.

Let $q=\mathbf{k}\cdot \mathbf{n}$, $E=e^{iq(\tau -{\tau }_0)}$ and $T=h_S^{\prime }/(2k^2)$. The synchronous-to-Newtonian transformation with positive spatial perturbations gives $(h^{\prime }-h_S^{\prime })/6={\Phi }^{\prime }+{\Psi }^{\prime }+T^{\prime \prime }$. Since $[E(T^{\prime }-iqT){]}^{\prime }=E(T^{\prime \prime }+q^{2}T)$, the displayed identity follows by integration. The emission endpoint combines intrinsic and velocity shifts into the ordinary Sachs-Wolfe effect; the local observer endpoint is a monopole plus a dipole. Without anisotropic stress the remaining integral is the Integrated Sachs-Wolfe effect, $2\int E{\Phi }^{\prime }d\tau$. The angular term in the synchronous contraction is positive for this metric convention.

Robert Crittenden co-authored an analysis of the cross-correlation between cosmic microwave background maps from WMAP and cosmic X-ray background maps from HEAO 1. Such a comparison tests whether angular variations measured in different radiation bands share a physical origin.

The paper examined possible foreground explanations for the correlation before interpreting it as evidence consistent with the Integrated Sachs-Wolfe effect. This is an example of testing a cosmological signal through independently measured maps and explicit alternative sources of correlation.^[ref]

Stephen Boughn co-authored work investigating the connection between the cosmic microwave background and the cosmic X-ray background. A cross-correlation between their sky maps can probe the Integrated Sachs-Wolfe effect, providing a cosmological test that uses more than the microwave map alone.

The cited study specifically investigated foreground sources that might mimic the correlation. Its interpretation depended on testing those alternatives, illustrating why detection of a statistical association and identification of its physical origin are separate scientific steps.^[ref]

With instantaneous photon decoupling and tight-coupling initial brightness, integrate the synchronous photon brightness equation along the unperturbed ray $\mathbf{x}(\tau )={\mathbf{x}}_0-\mathbf{n}({\tau }_0-\tau )$. The direction $\mathbf{n}$ here is photon propagation, opposite to the observer-to-source sky direction. Emission terms are evaluated at the emitter, not the observer. The synchronous metric integral combines endpoint gravitational redshifts and evolving-potential effects when expressed in Newtonian gauge; it is not solely the Integrated Sachs-Wolfe effect.

// Target: cosmology.bigb