Physics of temperature fluctuations 223involves both a quantum mechanical scale (the interaction cross section) and a
cosmological scale (the time scale for the expansion of the universe). Conversion
between these cosmological units and physical (cgs) units can be achieved by
inserting needed factors ofh ̄,c,andkb. The standard textbook by Kolb and
Turner (1990) contains an extremely useful appendix on units.
7.2.1 Causes of temperature fluctuations
Blackbody radiation in a perfectly homogeneous and isotropic universe, which
is always adopted as a zeroth-order approximation, must be at a uniform
temperature, by assumption. When perturbations are introduced, three elementary
physical processes can produce a shift in the apparent blackbody temperature of
the radiation emitted from a particular point in space. All temperature fluctuations
in the microwave background are due to one of the following three effects.
The first is simply a change in the intrinsic temperature of the radiation
at a given point in space. This will occur if the radiation density increases
via adiabatic compression, just as with the behaviour of an ideal gas. The
fractional temperature perturbation in the radiation just equals the fractional
density perturbation.
The second is equally simple: a Doppler shift if the radiation at a particular
point is moving with respect to the observer. Any density perturbations within the
horizon scale will necessarily be accompanied by velocity perturbations. The
induced temperature perturbation in the radiation equals the peculiar velocity
(in units ofc, of course), with motion towards the observer corresponding to a
positive temperature perturbation.
The third is a bit more subtle: a difference in gravitational potential between
a particular point in space and an observer will result in a temperature shift of
the radiation propagating between the point and the observer due to gravitational
redshifting. This is known as the Sachs–Wolfe effect, after the original paper
describing it (Sachs and Wolfe, 1967). This paper contains a completely
straightforward general relativistic calculation of the effect, but the details are
lengthy and complicated. A far simpler and more intuitive derivation has been
given by Hu and White (1997) making use of gauge transformations. The Sachs–
Wolfe effect is often broken into two parts, the usual effect and the so-called
Integrated Sachs–Wolfe effect. The latter arises when gravitational potentials are
evolving with time: radiation propagates into a potential well, gaining energy
and blueshifting in the process. As it climbs out, it loses energy and redshifts,
but if the depth of the potential well has increased during the time the radiation
propagates through it, the redshift on exiting will be larger than the blueshift on
entering, and the radiation will gain a net redshift, appearing cooler than it started
out. Gravitational potentials remain constant in time in a matter–dominated
universe, so to the extent the universe is matter dominated during the time the
microwave background radiation freely propagates, the Integrated Sachs–Wolfe
effect is zero. In models with significantly less than critical density in matter (i.e.