252 The cosmic microwave background
are attractive from an aesthetic point of view: all add significant complexity and
freedom to the models without any corresponding restrictions on the original
parameter space. The principle of Occam’s Razor should cause us to be sceptical
about any such additions to the space of models.
However, it is possible that some elementismissing from the model space,
or that the actual cosmological model is radically different in some respect. The
microwave background is the probe of cosmology most tightly connected to
the fundamental properties of the universe and least influenced by astrophysical
complications, and thus the most capable data source for deciding whether
the universe actually is well described by some model in the usual model
space. An interesting question is the extent to which the microwave background
can determine various properties of the universe independent from particular
models. While any cosmological interpretation of temperature fluctuations in
the microwave sky requires some kind of minimal assumptions, all of the
conclusions outlined later can be drawn without invoking a detailed model of
initial conditions or structure formation. These conclusions are in contrast to
precision determination of cosmological parameters, which does require the
assumption of a particular space of models and which can vary significantly
depending on the space.
7.6.1 Flatness
The Friedmann–Robertson–Walker spacetime describing homogeneous and
isotropic cosmology comes in three flavours of spatial curvature: positive,
negative and flat, corresponding to>1,<1and=1 respectively. One of
the most fundamental questions of cosmology, dating to the original relativistic
cosmological models, is the curvature of the background spacetime. The fate
of the universe quite literally depends on the answer: in a cosmology with only
matter and radiation, a positively curved universe will eventually recollapse in a
fiery ‘Big Crunch’ while flat and negatively curved universes will expand forever,
meeting a frigid demise. Note these fates are at least 40 billion years in the future.
(A cosmological constant or other energy density component with an unusual
equation of state can alter these outcomes, causing a closed universe eventually
to enter an inflationary stage.)
The microwave background provides the cleanest and most powerful probe
of the geometry of the universe (Kamionkowskiet al 1994). The surface of
last scattering is at a high enough redshift that photon geodesics between the
last scattering surface and the Earth are significantly curved if the geometry of
the universe is appreciably different than flat. In a positively curved space, two
geodesics will bend towards each other, subtending a larger angle at the observer
than in the flat case; likewise, in a negatively curved space two geodesics bend
away from each other, resulting in a smaller observed angle between the two.
The operative quantity is the angular diameter distance; Weinberg (2000) gives
a pedagogical discussion of its dependence on. In a flat universe, the horizon