## Low density inflationary Universes

### How Much Matter Does the Universe Contain?

An important question today in cosmology is how much mass is contained in the Universe. If there were no matter filling the Universe, the Universe would expand forever and the recession velocity of objects at rest with respect to the expansion of the Universe would not change as the Universe expands.

We know, of course, that the Universe is not empty but filled with matter, and ordinary matter through gravity attracts other matter, causing the expansion of the Universe to slow down. If the density of the Universe exceeds a certain threshold known as the critical density, this gravitational attraction is strong enough to stop and later reverse the expansion of the Universe, causing it eventually to recollapse in what is known as the "Big Crunch." On the other hand, if the average density of the Universe falls short of the critical density, the Universe expands forever, and after a certain point the expansion proceeds much as if the Universe were empty. A critical Universe lies precariously balanced between these two possibilities.

### Why a Universe of Critical Density?

For quite some time it has been known that the mean density of our Universe agrees with the critical density to within better than a factor of ten. Even with such large margin of error this agreement is remarkable. Establishing initial conditions so that the mean density remains close to the critical density for more than a fleeting moment is much like trying to balance a pencil on its point. A Universe initially with slightly subcritical density rapidly becomes increasingly subcritical and soon virtually indistinguishable from an empty Universe. Similarly, an ever so slightly supercritical Universe rapidly collapses into a Big Crunch, never reaching the old age of our Universe - somewhere around twelve billion years. To obtain a Universe like ours seems to require fine tuning of the initial density to agree with the critical density to an accuracy around one part in 10^{60}!

For a long time it was regarded simplest and aesthetically most pleasing to postulate that our Universe is now of exactly critical density. The versions of inflation developed in the early 1980s provided a mechanism for setting the density of the Universe near the critical density with nearly unlimited precision. For many years an exactly critical Universe was touted as one of the few firm predictions of inflation.