Sachs-Wolfe combination

In Newtonian gauge, ${\Theta }_0+\Psi$ combines the intrinsic photon temperature perturbation with the gravitational redshift at emission. With constant potentials and negligible baryon loading it oscillates as $\cos (k{r}_s)$ for adiabatic initial conditions.

For a constant growing adiabatic mode with curvature convention $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/[4\pi G{a}^2(\bar{\rho }+\bar{P})]$, ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. The superhorizon photon continuity equation conserves ${\Theta }_0-\varphi$. With radiation initial value ${\Theta }_0=\mathcal{R}/3$, it gives ${\Theta }_{0,\mathrm{mat}}=2\mathcal{R}/5$ and $({\Theta }_0+\psi {)}_{\mathrm{mat}}=-\mathcal{R}/5$ when $\psi =\varphi$.

The Sachs-Wolfe effect is the large-angle CMB temperature anisotropy produced by intrinsic photon-temperature perturbations and gravitational redshift at last scattering. For an adiabatic mode during matter domination, ${\Theta }_0+\Psi =\Psi /3$ on superhorizon scales.

The Doppler CMB anisotropy is the line-of-sight velocity contribution from the last-scattering plasma. With the convention used here it contributes $-\hat{\mathbf{n}}\cdot {\mathbf{v}}_e$ to the observed fractional temperature perturbation.

The integrated Sachs-Wolfe effect is the CMB temperature shift produced when the gravitational potentials traversed by a photon evolve with time. Its line-of-sight source is proportional to ${\Phi }^{\prime }+{\Psi }^{\prime }$.

Cross-correlating a cosmic microwave background temperature map with a low-redshift matter tracer can isolate a late gravitational contribution from the much larger primordial anisotropy. Decaying potential wells generally produce a positive correlation of matter overdensity with temperature. Tracer bias, overlapping redshift distributions, foreground emission, masks and chance correlations limit the inference. Potential evolution also depends on curvature and gravity assumptions, so this is not a dark-energy measurement independent of a cosmological model.

The Rees-Sciama effect is the microwave temperature shift produced by time-dependent gravitational potentials in nonlinear structure. A photon gains energy entering a potential well and loses it leaving; the shifts do not cancel when the well changes during transit. To leading order in its effect on the photon spectrum, this is a frequency-independent thermodynamic temperature shift, unlike the thermal Sunyaev-Zeldovich effect. A nonlinear potential can evolve even during matter domination, when linear growing-mode potentials are constant.