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:
http://www.routledge-ny.com/books.cfm?isbn=0415923549

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 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 in the
gravitational potential, the effect of
gravitational redshift is to cause a fractional variation of the
temperature *T/T* = / *c*^{2}, where *c*
is the speed of light. The time
dilation effect contributes *T/T* =
-*a/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
*T/T* =
/3*c*^{2}. 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.

FURTHER READING:

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.