The value of the density parameter in the beginning is one of the most precisely determined numbers in all of science, and the natural inference is that the value is exactly 1. One important implication of this is that there must be a large amount of dark matter in the universe. Another is that the universe was made flat by inflation(cmn206)
Inflation can prevent the universe from becoming a singularity as soon as it is born while quantum physics allows the entire universe to appear out of nothing at all(cmn208)
Inflation seems to take place faster than the speed of light. This is possible because it is spacetime itself that is expanding, carrying matter along for the ride. Nothing is moving through spacetime faster than light(cmn211)
Inflation also predicts that the primordial perturbations may have left a trace in the form of gravitational radiation with particular characteristics, and it is hoped that detectors sensitive enough to identify this characteristic radiation may be developed within the next ten or twenty years(cmn216)
By turning inflation on twice, cosmologists have found a way to have all the benefits of the inflationary scenario, while still leaving the universe in an 'open' state, so that it will expand forever(cmn220)
When the universe had been homogenized by the original bout of inflation, a second burst of inflation could have occurred within one of the bubbles. As inflation begins (essentially at a point), the density is effectively 'reset' to zero, and rises towards the critical density as inflation proceeds and energy from the inflation process is turned into mass(cmn220b)