Adapted from P. Coles, 1999, The Routledge Critical Dictionary of the New Cosmology, Routledge Inc., New York. Reprinted with the author's permission. To order this book click here:

On large angular scales, the most important of various physical processes by which the primordial density fluctuations should have left their imprint on the cosmic microwave background radiation in the form of small variations in the temperature of this radiation in different directions on the sky. It is named after Rainer Kurt Sachs (1932- ) and Arthur Michael Wolfe (1939- ). The effect is essentially gravitational in origin. Photons travelling from the last scattering surface to an observer encounter variations in the metric which correspond to variations in the gravitational potential in Newtonian gravity. These fluctuations are caused by variations in the matter density rho from place to place. A concentration of matter, in other words an upward fluctuation of the matter density, generates a gravitational potential well. According to general relativity, photons climbing out of a potential well will suffer a gravitational redshift which tends to make the region from which they come appear colder. There is another effect, however, which arises because the perturbation to the metric also induces a time-dilation effect: we see the photon as coming from a different spatial hypersurface (labelled by a different value of the cosmic scale factor a(t) describing the expansion of the Universe).

For a fluctuation phi in the gravitational potential, the effect of gravitational redshift is to cause a fractional variation of the temperature DeltaT/T = phi / c2, where c is the speed of light. The time dilation effect contributes DeltaT/T = -deltaa/a (i.e. the fractional perturbation to the scale factor). The relative contributions of these two terms depend on the behaviour of a(t) for a particular cosmological model. In the simplest case of a flat universe described by a matter-dominated Friedmann model, the second effect is just -2/3 times the first one. The net effect is therefore given by DeltaT/T = phi/3c2. This relates the observed temperature anisotropy to the size of the fluctuations of the gravitational potential on the last scattering surface.

It is now generally accepted that the famous ripples seen by the Cosmic Background Explorer (COBE) satellite were caused by the Sachs-Wolfe effect. This has important consequences for theories of cosmological structure formation, because it fixes the amplitude of the initial power spectrum of the primordial density fluctuations that are needed to start off the gravitational Jeans instability on which these theories are based.

Any kind of fluctuation of the metric, including gravitational waves of very long wavelength, will produce a Sachs-Wolfe effect. If the primordial density fluctuations were produced in the inflationary Universe, we would expect at least part of the COBE signal to be due to the very-long-wavelength gravitational waves produced by quantum fluctuations in the scalar field driving inflation.


Sachs, R.K. and Wolfe, A.M., `Perturbations of a cosmological model and angular variations of the cosmic microwave background', Astrophysical Journal, 1967, 147, 73.