The basic picture of Big Bang cosmology, a hot, uniform early universe expanding and cooling at late times, is very successful and has (so far) passed a battery of increasingly precise tests. It successfully explains the observed primordial abundances of elements, the observed redshifts of distant galaxies, and the presence of the cosmic microwave background. Observation of the CMB is a field that is currently progressing rapidly, allowing for extremely precise tests of cosmological models. The most striking feature of the CMB is its high degree of uniformity, with inhomogeneities of about one part in 105. Recent precision measurements of the tiny anisotropies in the CMB have allowed for constraints on a variety of cosmological parameters. Perhaps most importantly, observations of the first acoustic peak, first accomplished with precision by the Boomerang  and MAXIMA  experiments, indicate that the geometry of the universe is flat, with total = 1.02 ± 0.05 . However, this success of the standard Big Bang leaves us with a number of vexing puzzles. In particular, how did the universe get so big, so flat, and so uniform? We shall see that these observed characteristics of the universe are poorly explained by the standard Big Bang scenario, and we will need to add something to the standard picture to make it all make sense: inflation.
4.1. The flatness problem
We observe that the universe has a nearly flat geometry, tot 1. However, this is far from a natural expectation for an arbitrary FRW space. It is simple to see why. Take the defining expression for ,
Here the density of matter with equation of state p = w evolves with expansion as
Using this and the Friedmann equation (12) it is possible to derive a simple expression for how evolves with expansion:
This is most curious! Note the sign. For an equation of state with 1 + 3w > 0, which is the case for any kind of "ordinary" matter, a flat universe is an unstable equilibrium:
So if the universe at early times deviates even slightly from a flat geometry, that deviation will grow large at late times. If the universe today is flat to within 1 ± 0.05, then the time of recombination was = 1 ± 0.00004, and at nucleosynthesis = 1 ± 10-12. This leaves unexplained how the universe got so perfectly flat in the first place. This curious fine-tuning in cosmology is referred to as the flatness problem.