The local geometry of the universe is determined by whether Omega is less than, equal to or greater than 1. From top to bottom: a spherical universe, a hyperbolic universe, and a flat universe. The parameter Omega i will discuss in future, you can just take it for grant now.
According to general relativity, the flatness of the universe is related to its mass density, Omega=actual mass density / critical mass density. The critical mass density is determined by the expansion rate, and is given by p(c)=(3H^2)/(8*pi*G) where G is Newton’s gravitational constant. The critical density is defined to be that density which leads to a geometrically flat universe. (In the past cosmologists often said that a closed universe(p>p(c)) will recollapse and an open universe (p < p(c)) will expand forever, but these statements are invalidated by the possibility of a cosmological constant. A positive cosmological constant can allow a closed universe to expand forever, and a negative one can cause an open universe to collapse.)
The density of the universe also determines its geometry. If the density of the universe exceeds the critical density, then the geometry of space is closed and positively curved like the surface of a sphere. This implies that initially parallel photon paths converge slowly, eventually cross, and return back to their starting point (if the universe lasts long enough). If the density of the universe is less than the critical density, then the geometry of space is open, negatively curved like the surface of a saddle. If the density of the universe exactly equals the critical density, then the geometry of the universe is flat like a sheet of paper. Thus, there is a direct link between the geometry of the universe and its fate.
Flatness Problem
The large-scale uniformity can be seen in galaxy surveys, but the most dramatic evidence comes from the cosmic background radiation. Data from the COBE satellite, confirmed by subsequent ground-based observations, show that this radiation has the same temperature in all directions (after correcting for the motion of the Earth) to an accuracy of one part in 100,000.
According the Big Bang theory, we can logical think that the explosion can cause rapid change and thus the universe is birth , but from the what we see today,the large-scale uniformity can be seen in galaxy is not expected under the theory of Big Bang. "Under many circumstances such uniformity would be easy to understand, since anything will come to a uniform temperature if left undisturbed for a long enough time. In the traditional form of the big bang theory,however,the universe evolves so quickly that there is no time for the uniformity to be established. Calculations show that energy and information would have to be transported at about 100 times the speed of light in order to achieve uniformity by 300,000 years after the big bang. Thus, the traditional big bang theory requires us to postulate, without explanation, that the primordial fireball filled space from the beginning. The temperature was the same everywhere by assumption, but not as a consequence of any physical process. This shortcoming is known as the horizon problem, since cosmologists use the word “horizon” to indicate the largest distance that information or energy could have traversed, since the instant of the big bang, given the restriction of the speed of light.(Alan Guth)"
Inflationary Solution
"Inflation expands space, and like a balloon, as it expands its surface becomes less and less curved, regions of space become flatter and flatter, the enormous inflation of space world essentially eliminate any initial curvature.(Professor Sorbo)"
From Alan Guth, he relate the concept of false vacuum creates a repulsive gravitational forces, which is driving force of inflation. "Once a patch of the early universe is in the false vacuum state, the repulsive gravitational effect drives the patch into an inflationary period of exponential expansion. To produce a universe with the special features of the big bang discussed above, the universe must expand during the inflationary period by at least a factor of 1025. There is no upper limit to the amount of expansion. If the energy scale of the false vacuum is characteristic of the 1016 GeV scale of grand unified theories, then the time constant of the exponential expansion would be about 10–37 seconds. Eventually the false vacuum decays, and the energy that had been locked in the false vacuum is released. This energy produces a hot, uniform soup of particles, which is exactly the assumed starting point of the traditional big bang theory. At this point the inflationary theory joins onto the older theory, maintaining all of its successes.(Alan Guth)"
During the inflation period, the region undergo a tremendous spurt of expansion which has significant physical explanation of why the universe as it is today.
References
1. Guth, Alan Harvey. "Inflation and the New Era of High-Precision Cosmology." MIT Department of Physics. Web. June-July 2010.
2. Sorbo, Lorenzo. "Inflation Solves the Flatness Problems." From Big Bang to Black Holes. Amherst. 11 July 2010. Lecture.
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