Another question concerning the Big Bang model concerns the geometry of the universe and has become known as the Flatness Problem.
Recall that a key prediction of general relativity is that matter distorts spacetime – i.e. the force of gravity is essentially a distortion of spacetime by mass. Hence the curvature of the spacetime of our universe will be determined by the density of matter in it. Assuming only that the universe is homogenous and isotropic, it can be shown from general relativity that three distinct types of universe are possible (first calculated by Alexander Friedmann).
If the universe has a high enough density of matter, gravity will triumph over the energy of expansion as time goes on and space will be pulled in on itself, much like a sphere (closed universe). On the other hand, if the universe has a low enough density of matter, gravity eventually loses the battle with the energy of expansion and space will curve outwards (open universe). A third but unlikely possibility is that the curvature of space caused by matter could be exactly balanced by the energy of expansion – in this case space would not be curved but have Euclidean geometry (flat universe).
Friedmann universes: 3 possibilities
We say the above mathematically by defining a flatness parameter Ω to be the ratio of the actual density of matter d to the critical density dc required for flatness i.e. Ω = d/dc . Hence we characterize an open,closed or flat universe as Ω < 1, Ω >1 and Ω = 1 respectively.
So what’s the problem? In the late 1960s, calculations by Bob Dicke showed that even the slightest deviation (1 in1015) from flatness in the early universe would quickly lead to either a runaway closed or a runaway open universe. As observation and mass calculations suggest that we live in neither of these, there is a clear implication that the geometry of the early universe must have been exactly flat. But why should the early universe have been so finely balanced between the energy of gravity and the energy of expansion? A curious example of fine tuning indeed…
This mystery has come back to the fore in recent years: measurements of the cosmic microwave background are strongly indicative of a universe that is exactly flat (at least to 1%) at the time of recombination. So now we also have experimental evidence of an exact balance between the density of matter in the universe and the energy of expansion. Again, why such a precise balancing act?