There was a recent article in Scientific American about the topology of the universe. If the universe is flat, space does not need to be infinite if it has the right topology. Such a finite universe could have any number of possible overall shaped, with a 3-torus doughnut like shape being one of them. It seems like an easy enough idea to grasp. But wait. A flat universe would have an omega value of 1. For closed universes, I though it was the high amount of mass density that cause the universe to fold over, and get its hypersphere shape. If a flat universe has a much smaller value for omega, what cause it to have a hyperbolic shape?
i think that most people will think that Gravity will `bend` space-time. But perhaps the curvature of space time creates gravity... Gravity can be regarded as particles (gravitons) ,but these are linked directly with time and space. (i can imagine a universe without electrons, but i can`t imagine a universe without gravity ). This is of course `happens` in another dimension (the 4th ?). The universe may have any type of topography, what we can be sure of, after the map results, is that it is open. We have no way of knowing (for sure) if the density of our visible universe is typical of the universe in general. If the universe is a doh!nut shape then it will expand forever. if the universe is a sphere then it will expand forever... Generally there is nothing to suppose that the universe is not a sphere.
If the universe is open, omega is less than one, and space (on large scales) is negatively curved. Negatively curved spaces are sometimes called 'hyperbolic' spaces, because their curvature can look like that of a saddle. - Warren
Yes. and for those that dont know about omega... The flatness problem can be understood in terms of mass. The density parameter is a measure of the amount of gravitating material in the Universe, usually signified by the Greek letter omega (W). It is defined in such a way that if space-time is exactly flat then W = 1. One of the great difficulties in cosmology was the fact that the actual density of the Universe today is very close to this critical. This is extraordinary because as the Universe expands the density parameter should move away from the critical value. If the Universe starts out with the parameter less than one, W gets smaller as the Universe ages, while if it starts out bigger than one W gets bigger as the Universe ages. The fact that W is between 0.1 and 1 today means that in the first second of the Big Bang it was precisely 1 to within 1 part in 10^60. This makes the value of the density parameter in the beginning one of the most precisely determined numbers in all of science, and the instinctive deduction is that the value is exactly 1. One important feature of this is that there is a large amount of dark matter or energy in the Universe. Another is that the Universe was made flat by inflation. Another indication is the Cosmic Abundance of Helium and Hydrogen When the expanding universe had cooled to below about 10^9 K so that protons and neutrons could fuse to make stable deuterium nuclei (a hydrogen isotope with one proton and one neutron). Most of the helium in the universe was created from the primordial neutrons and protons (Although stars do produce some of the helium visible today), by the time the ucleosynthesis epoch ended. Stars to fuse hydrogen nuclei to make a helium nucleus use fusion. The fusion chain process in the early universe was slightly different than what occurs in stars because of the abundant free neutrons in the early universe. However, the general process is the same: protons react to produce deuterium (heavy hydrogen), deuterium nuclei react to make Helium-3 nuclei, and Helium-3 nuclei react to make the stable Helium-4 nucleus. The amount of the final Helium-4 product is not as sensitive to the density of the early universe ... The deuterium nucleus is the weak link of the chain process, so the fusion chain reactions could not take place until the universe had cooled enough. The exact temperature depends sensitively on the density at that time. Extremely small amounts of Lithium-7 were also produced during the early universe nucleosynthesis process. Lithium-7 and deuterium density depends sensitively upon the density of protons (2 up + 1 down quarks and neutrons during this time. If the universe were too dense, then most of the deuterium would have fused into helium. The more neutrons that decay before combining with protons, the smaller the abundances of heavier elements. Only in a low-density universe can the deuterium survive. A denser universe would have had more deuterium fused to form helium, so the amount of the remaining deuterium seen today is used as a probe of the early density because of the sensitivity of its production to the density of the protons and neutrons and temperature in the early universe. As another benefit of this inflation we can get rid of the great amounts of magnetic monopoles that would have been created during the symmetry breaking phase.
The simplest cosmological models have three possible outcomes: omega > 1 ... universe is closed, spherical geometry, will collapse omega < 1 ... universe is open, hyperbolic geometry, will expand forever omega = 1 ... universe is open, flat geometry, will expand forever The picture is complicated if we introduce a cosmological constant (thought to be caused by "dark energy"). In that case, it is possible, for example, to have a universe with omega < 1, which is closed and spherical but which will nevertheless expand forever.
SRY, to butt in again... omega < 1 ... universe is open, hyperbolic geometry, will expand forever (...and accelerate....)
But what causes the universe as a whole to fold over so that there is no edge? Specifically, in the case of a flat universe with the overall shape of a torus.
The term 'hyperbolic' doesn't imply that the universe is toroidal, or has any other more complex topology. 'Hyperbolic' just means the space is negatively curved. I don't know what might cause such a topology. - Warren
