Relativity and gravitational theory Relativistic cosmology Fact

Sachs–Wolfe effect

Builds on Cosmological perturbation theory Friedmann–Lemaître–Robertson–Walker metric

Statement

Consider a spatially flat Friedmann–Lemaître–Robertson–Walker universe. Take scalar cosmological perturbations in the Newtonian gauge, $ds^2 = a(\eta)^2 \left[ -(1 + 2\Psi)\, d\eta^2 + (1 - 2\Phi)\, \delta_{ij}\, dx^i dx^j \right],$ where $\eta$ is conformal time and $\Psi, \Phi$ are small functions of $(\eta, x)$ . Assume that the photons decouple instantaneously at conformal time $\eta_*$ and then travel freely to the observer at time $\eta_0$ . Neglect the Doppler term from the velocity of the emitting matter. Let $\Theta_0 = \delta T / T$ be the fractional temperature perturbation of the photons at the emission point. Drop the terms at the observer, which only add a monopole and a dipole. Then the fractional temperature perturbation seen in the direction $\hat n$ is, to first order, $\frac{\delta T}{T}(\hat n) = \left[ \Theta_0 + \Psi \right](\eta_*, x_*) + \int_{\eta_*}^{\eta_0} \left( \partial_\eta \Psi + \partial_\eta \Phi \right)(\eta, x(\eta))\, d\eta,$ where $x(\eta) = (\eta_0 - \eta)\, \hat n$ is the photon path and $x_* = x(\eta_*)$ . For adiabatic perturbations on scales larger than the horizon at decoupling, in a matter-dominated era, $\Theta_0 = -\tfrac{2}{3} \Psi$ , so the first term becomes $\tfrac{1}{3} \Psi(\eta_*, x_*)$ .

Remarks

The formula comes from the geodesic equation for photons in the perturbed metric. The photon energy changes when it climbs out of the potential $\Psi$ and when the potentials change in time along its path. The first term is the ordinary Sachs–Wolfe effect, and the integral is the integrated Sachs–Wolfe effect. In an overdense region $\Psi < 0$ , so the photon loses more energy climbing out of the well than it had gained in intrinsic temperature, and the region appears colder by $\abs{\Psi}/3$ . In a matter-dominated universe with no anisotropic stress, $\Phi = \Psi$ is constant in time on linear scales, so the integrated term vanishes. It becomes nonzero when radiation is still important just after decoupling, and at late times when a cosmological constant or spatial curvature makes the potentials decay. The effect dominates the temperature anisotropy of the cosmic microwave background at large angular scales.

Going deeper

For a student

In linear cosmological perturbation theory the observed temperature anisotropy is a sum of a few terms along each line of sight. The Sachs–Wolfe terms are the gravitational ones: the potential at last scattering and the time change of the potentials afterwards. On large angular scales they give a nearly flat angular power spectrum $\ell(\ell + 1) C_\ell$ for a nearly scale-invariant spectrum of initial perturbations. The late integrated term correlates with the large-scale distribution of galaxies, and this correlation is a test for dark energy. Sign conventions for $\Phi$ and $\Psi$ differ between textbooks, so the formula must be read together with the metric above.

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Cosmological perturbation theory concept
Cosmological principle concept
Space form concept
Spacetime concept
Friedmann–Lemaître–Robertson–Walker metric example
Sachs–Wolfe effect statement

Tag	`KNJ2Y146KA`
URI	`m2://KNJ2Y146KA`
Kind	statement, fact
Writing	W0 stub
MSC	83F05, 85A40
Status	This copy of the site has no build data. Run `m2 status KNJ2Y146KA`.
About	Cosmological perturbation theory
Uses	Cosmological perturbation theory, Friedmann–Lemaître–Robertson–Walker metric
Same as	Wikidata Q1617362 Wikipedia Sachs–Wolfe effect
Changed	3 November 2026