258 The cosmic microwave background
alocalcondition on the spacetime. Its global structure is still unspecified.
It is possible to construct spacetimes which at every point have the usual
homogeneous and isotropic metric, but which are spatially compact (have finite
volumes). The most familiar example is the construction of a three-torus from a
cubical piece of the flat spacetime by identifying opposite sides. Classifying the
possible topological spaces which locally have the metric structure of the usual
cosmological spacetimes (i.e. have the Friedmann–Robertson–Walker spacetimes
as a topological covering space) has been studied extensively. The zero-curvature
and positive-curvature cases have only a handful of possible topological spaces
associated with them, while the negative curvature case has an infinite number
with a very rich classification. See Weeks (1998) for a review.
If the topology of the universe is non-trivial and the volume of the universe
is smaller than the volume contained by a sphere with radius equal to the distance
to the surface of last scattering, then it is possible to detect the topology. Cornish
et al(1998) pointed out that because the last scattering surface is always a
sphere in the covering space, any small topology will result in matched circles of
temperature on the microwave sky. The two circles represent photons originating
from the same physical location in the universe but propagating to us in two
different directions. Of course, the temperatures around the circles will not match
exactly, but only the contributions coming from the Sachs–Wolfe effect and the
intrinsic temperature fluctuations will be the same; the velocity and Integrated
Sachs–Wolfe contributions will differ and constitute a noise source. Estimates
show the circles can be found efficiently via a direct search of full-sky microwave
background maps. Once all matching pairs of circles have been discovered, their
number and relative locations on the sky strongly overdetermine the topology of
the universe in most cases. Remarkably, the microwave background essentially
allows us to determine the size of the universe if it is smaller than the current
horizon volume in any dimension.
7.7 Finale: testing inflationary cosmology
In summary, the CMB radiation is a remarkably interesting and powerful source
of information about cosmology. It provides an image of the universe at an early
time when the relevant physical processes were all very simple, so the dependence
of anisotropies on the cosmological model can be calculated with high precision.
At the same time, the universe at decoupling was an interesting enough place
that small differences in cosmology will produce measurable differences in the
anisotropies.
The microwave background has the ultimate potential to determine
fundamental cosmological parameters describing the universe with percent-level
precision. If this promise is realized, the standard model of cosmology would
compare with the standard model of particle physics in terms of physical scope,
explanatory power and detail of confirmation. But in order for such a situation