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 ${\Phi }^{\prime }+{\Psi }^{\prime }$.

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.