Inflation has revolutionized the way cosmologists view the Universe and provides the current working hypothesis for extending the standard cosmology. It explains how a region of size much, much greater than our Hubble volume could have become smooth and flat without recourse to special initial conditions (Guth 1981), as well as the origin of the density inhomogeneities needed to seed structure (Hawking, 1982; Starobinsky, 1982; Guth & Pi, 1982; and Bardeen et al, 1983). Inflation is based upon well defined, albeit speculative physics - the semi-classical evolution of a weakly coupled scalar field - and this physics may well be connected to the unification of the particles and forces of Nature.
It would be nice if there were a standard model of inflation, but there isn't. What is important, is that almost all inflationary models make three very testable predictions: flat Universe, nearly scale-invariant spectrum of Gaussian density perturbations, and nearly scale-invariant spectrum of gravitational waves. These three predictions allow the inflationary paradigm to be decisively tested. While the gravitational waves are an extremely important and challenging test, I do not have space to mention them again here (see e.g., Turner 1997).
The tremendous expansion that occurs during inflation is key to its beneficial effects and robust predictions: A small, subhorizon-sized bit of the Universe can grow large enough to encompass the entire observable Universe and much more. Because all that we can see today was once so extraordinarily small, it appears flat and smooth. This is unaffected by the expansion since then and so the Hubble radius today is much, much smaller than the curvature radius, implying 0 = 1. Lastly, the tremendous expansion stretches quantum fluctuations on truly microscopic scales ( 10-23 cm) to astrophysical scales ( Mpc).
The curvature perturbations created by inflation are characterized by two important features: 1) they are almost scale-invariant, which refers to the fluctuations in the gravitational potential being independent of scale - and not the density perturbations themselves; 2) because they arise from fluctuations in an essentially noninteracting quantum field, their statistical properties are that of a Gaussian random field.
Scale invariance specifies the dependence of the spectrum of density perturbations upon scale. The normalization (overall amplitude) depends upon the specific inflationary model (i.e., scalar-field potential). Ignoring numerical factors for the moment, the fluctuation amplitude is given by: ( / )HOR ~ V3/2 / mPL3 V'. (The amplitude of the density perturbation on a given scale at horizon crossing is equal to the fluctuation in the gravitational potential .) To be consistent with the COBE measurement of CBR anisotropy on the 10o scale, must be around 2 × 10-5. Not only did COBE produce the first evidence for the existence of the density perturbations that seeded all structure (Smoot et al, 1992), but also, for a theory like inflation that predicts the shape of the spectrum of density perturbations, it provides the overall normalization that fixes the amplitude of density perturbations on all scales. The COBE normalization began precision testing of inflation.