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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 1016 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 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.

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