From: helbig@astro.rug.nl (Phillip Helbig) Subject: Re: Shape of the universe -- questions Date: Thu, 15 Jun 2000 03:40:02 GMT Newsgroups: sci.physics.research Summary: [missing] In article <3943F1B9.EC57700A@home.com>, Alan Lewis writes: >My understanding is that the best experimental evidence >suggests the universe will expand forever. Yes. I would say this is "fairly certain". > Does this mean > the universe is flat now or just eventually? In itself, neither. Current evidence also suggests, though not quite as strongly as it suggests that it will expand forever, that the universe is close to being flat (and it can't be ruled out that it is exactly flat). > If the > universe is flat now, is it still finite in extent? No. Flat <--> infinite in spatial extent. (For the experts: assuming a trivial topology.) The geometry and expansion history is determined by two parameters, the density parameter Omega and the cosmological constant lambda, which can be measured through a variety of astronomical observations. Their SUM tells us about the geometry: more than 1, closed; 1, flat; less than 1, open. If Omega is less than 1 (for which there is much more evidence than for either of the above claims (expand forever and flatness), then the universe will expand forever as long as lambda is not negative, i.e. if it is 0 or positive. If lambda is negative, the universe is always decelrating and will collapse in the future. If it is positive, then after a finite time the universe will be accelerating. If it is zero, it will always decelerate, but nevertheless will expand forever. If Omega is greater than one, then there is a value of lambda > 0 such that the universe will expand forever if this value is exceeded. This value increases slowly with Omega. In the diagram below, imagine the vertical line turning into a curve bending slowly to the right as it is extended upward. Without a cosmological constant, if Omega > 1 one has a finite universe which collapses, if Omega < 1 an infinite universe which expands forever, if Omega = 1 a flat universe which expands forever but infinitely slowly after an infinitely long time. Thus, geometry and destiny were such that knowing one implied the other. (Similarly, if one sticks to flat universes, Omega > 1 implies collapse and Omega < 1 implies eternal expansion.) This is not true if a cosmological constant is allowed (and current evidence seems to indicate that it is there and positive): in this case, one can also have finite universes which expand forever (this is by no means ruled out by current data and is mathematically the most interesting case) and infinite universes which recollapse. Omega \| / 1 | / |\ / | \ / | \ / | X | / \ | / \ |/ \ 0 ---------------------- lambda 0 1 Knowing the values of lambda and Omega (0.7 and 0.3 are good choices based on current evidence), we can determine where we are in the above diagram. Universes to the left of the diagonal line extending from the upper left are infinite and spatially open (negative curvature), on it infinite and spatially open (0 curvature, flat) and to the right finite and spatially closed (positive curvature). Universes to the right of the other diagonal line are CURRENTLY accelerating, those to the left of it CURRENTLY decelerating. Universes to the left of the vertical (for Omega < 1) line will collapse in the future, those on it or to the right of it will expand forever. In general, the parameters lambda and Omega are time-dependent. Non-empty universes start out arbitrarily close to lambda=0, Omega=1. As they evolve, they move about in the diagram above. They cannot cross the first mentioned diagonal line, since that would imply moving from infinite to finite or vice versa. They cannot cross the vertical line, since that would change the sign of the cosmological constant. (It really is "constant" but lambda above refers to a value scaled with the square of the expansion rate, so its magnitude can change but not its sign.) They CAN cross the other diagonal line. -- Phillip Helbig Email .............. helbig@astro.rug.nl Kapteyn Instituut Email ................. helbig@man.ac.uk Rijksuniversiteit Groningen Tel. ................... +31 50 363 4067 Postbus 800 Fax .................... +31 50 363 6100 NL-9700 AV Groningen Web ... http://www.astro.rug.nl/~helbig/ My opinions are not necessarily those of my employer. HIRE ME!