The Sachs-Wolfe effect is the gravitational redshift of photons in the cosmic microwave background due to density fluctuations.In 1967, Rainer K. Sachs and Arthur M. Wolfe showed that gravitational redshift explains the main anisotropies of the CMB. Before that, cosmologists puzzled over the origin of the microwave sky's patchiness. The idea is simple: a photon leaving a gravitational well loses energy, while one falling into a well gains it. But at the time of CMB emission, the wells were shallow, and photons escaped almost freely, slightly cooled. The total temperature change is proportional to the difference in gravitational potential at the emission point and on Earth. Since Earth is in a very weak field, the main contribution comes from the potential at the last scattering surface. It's like launching balls from different depths of a pool: the deeper, the slower the ball emerges to the surface.
In 1967, Rainer Sachs and Arthur Wolfe first described the mechanism of temperature fluctuations in the microwave background due to gravitational redshift. Their paper 'Perturbations of a Cosmological Model and Angular Variations of the Microwave Background' laid the foundation of modern CMB physics. Later, in the 1980s, with the development of inflationary cosmology, the effect became key to predicting the primordial perturbation spectrum. Measurements by satellites COBE, WMAP, and Planck confirmed the contribution of the Sachs–Wolfe effect with high precision, including the integrated version linked to dark energy.
The effect is used to extract information about dark matter distribution at recombination and constrain dark energy models via the integrated Sachs–Wolfe effect. It works within linear perturbation theory in GR under the condition of small fluctuation amplitudes. On small angular scales, other effects dominate (acoustic oscillations, Silk damping). The integrated effect is noticeable only in the presence of dynamical dark energy, making it a sensitive probe of cosmic expansion.
CMB maps use the Sachs–Wolfe effect to look back to the recombination epoch and test inflation models.
Thanks to the Sachs–Wolfe effect, astronomers can 'weigh' the most distant galaxy clusters without even seeing them: from the temperature of spots, they estimate the gravitational potential at recombination.
| ΔT/T | variable | Relative temperature fluctuation of CMB — temperature fluctuation |
| Φ | variable | Gravitational potential perturbation — gravitational potential metre squared per second squared |
| η | variable | Conformal time — conformal time second |
| ∫ | operator | Integrates over conformal time along the photon path — integral |
| dot | operator | Derivative with respect to conformal time — time derivative |
















