In summary, the cosmic microwave background 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 to come about, we must first choose a model space which includes the correct model for the universe. The accuracy with which cosmological parameters can be determined is of course limited by the accuracy with which some model in the model space represents the actual universe.

The space of models discussed in Sec. 6.1 represents universes which we would expect to arise from the mechanism of inflation. These models have become the standard testing ground for comparisons with data because they are simple, general, and well-motivated. So far, these types of models fit the data well, much better than any competing theories. Future measurements may remain perfectly consistent with inflationary models, may reveal inconsistencies which can be remedied via minor extensions or modifications of the parameter space, or may require more serious departures from these types of models.

For the sake of a concluding discussion about the power of the
microwave background, assume that the universe actually is well
described by inflationary cosmology, and that it can be modelled by
the parameters in Sec. 6.1.
For an overview of inflation and the problems it solves, see
Kolb and Turner
(1990,
chapter 8) or the lectures of A. Linde in this volume.
To what extent can we hope to
verify inflation, a process which likely would have occurred at an
energy scale of 10^{16} GeV when the universe was
10^{-38} seconds
old? Direct tests of physics at these energy scales are unimaginable,
leaving cosmology as the only likely way to probe this physics.

Inflation is not a precise theory, but rather a mechanism for exponential expansion of the universe which can be realized in a variety of specific physical models. Cosmology in general and the cosmic microwave background in particular can hope to test the following predictions of inflation (see Kamionkowski and Kosowsky 1999 for a more complete discussion of inflation and its observable microwave background properties):

- The most basic prediction of inflation is a spatially flat
universe. The flatness problem was one of the fundamental motivations
for considering inflation in the first place. While it is possible to
construct models of inflation which result in a non-flat universe,
they all must be finely tuned for inflation to end at just the right
time for a tiny but non-negligible amount of curvature to remain. The
geometry of the universe is one of the fundamental pieces of physics
which can be extracted from the microwave background power
spectra. Recent measurements make a strong case that the universe is
indeed flat.
- Inflation generically predicts primordial perturbations which
have a gaussian statistical distribution. The microwave background is
the only precision test of this prediction. Primordial gaussian
perturbations will still be almost precisely gaussian at
recombination, whereas they will have evolved significant
nongaussianity by the time the local large-scale structure forms, due
to gravitational collapse. Other methods of probing gaussianity, like
number densities of galaxies or other objects, inevitably depend
significantly on astrophysical modelling.
- The simplest models of inflation, with a single dynamical
scalar field, give adiabatic primordial perturbations. The only real
test of this prediction comes from the microwave background power
spectrum. More complex models of inflation with multiple dynamical
fields generically result in dominant adiabatic fluctuations with some
admixture of isocurvature fluctuations. Limits on isocurvature
fluctuations obtained from microwave background measurements could be
used to place constraints on the size of couplings between different
fields at inflationary energy scales.
- Inflation generically predicts primordial perturbations
on all scales, including scales outside the horizon. Of course
we can never test directly whether perturbations on scales larger than
the horizon exist, but the microwave background can reveal
perturbations at recombination on scales comparable to the horizon scale.
Zaldarriaga and
Spergel (1997)
have argued that inflation
generically gives a peak in the polarization power spectrum at
angular scales larger than 2°, and that no causal perturbations
at the epoch of last scattering can produce a feature at such large
scales. Inflation further predicts that the primordial power spectrum
should be close to a scale-invariant power law (e.g.
Huterer and Turner
2000),
although complicated models can lead to
power spectra with features or significant departures from
scale invariance. The microwave background can probe the primordial
power spectrum over three orders of magnitude.
- Inflationary perturbations result in phase-coherent
acoustic oscillations. The coherence arises because on any given
scale, the perturbations start in the same state determined only
by their character outside the horizon. For a discussion in
the language of squeezed quantum states, see
Albrecht (2000).
It is extremely difficult to produce coherent oscillations by any
mechanism other than perturbations outside the horizon. The microwave
background temperature and polarization power spectra will together
clearly reveal coherent oscillations.
- Inflation finally predicts potentially measurable relationships
between the amplitudes and power law indices of the primordial density
and gravitational wave perturbations (see
Lidsey
*et al.*1997 for a comprehensive overview), and measuring a*C*_{l}^{C}power spectrum appears to be the only way to obtain precise enough measurements of the tensor perturbations to test these predictions, thanks to the fact that the density perturbations don't contribute to*C*_{l}^{C}. Detection of inflationary tensor perturbations would reveal the energy scale at which inflation occurred, while confirming the inflationary relationships between scalar and tensor perturbations would provide a strong consistency check on inflation.

The potential power of the microwave background is demonstrated by the fact that inflation, a theoretical mechanism which likely would occur at energy scales not too different from the Planck scale, would result in several distinctive signatures in the microwave background radiation. Current measurements beautifully confirm a flat universe and are fully consistent with gaussian perturbations; the rest of the tests will come into clearer view over the coming years. If inflation actually occurred, we can expect to have very strong circumstantial supporting evidence from the above signatures, along with precision measurements of the cosmological parameters describing our universe. On the other hand, if inflation did not occur, the universe will likely look different in some respects from the space of models in Sec. 6.1. In this case, we may not be able to recover cosmological parameters as precisely, but the microwave background will be equally important in discovering the correct model of our universe.

I thank the organizers for a stimulating and enjoyable Summer School. The preparation of these lectures has been supported by the NASA Astrophysics Theory Program and the Cotrell Scholars program of the Research Corporation.