You are asking the front line questions of the day in modern astrophysics.
There are now a number of observations that contribute to our present thoughts about the fate of the universe.
1. Cosmic microwave backgound. --> Shows that the curvature of the universe is very nearly "flat"
2. Supernovae dimming as a function of distance or ’red shift’. --> Shows acceleration of the Hubble expansion rate
3. Dynamics of galaxies --> Gives information on the presence of "dark matter"
There is plenty of it, a factor of 10 or so more that that of luminous (stars) matter.
(about a factor of 7 more than all the ordinary "baryonic" matter /mw)
Nevertheless, as best we can measure, it all adds up to the fact that there is a mysterious property of the ’vacuum’, sometimes referred to as ’Dark Energy’ that is forcing the ever increasing expansion of the universe. Stay tuned. More observations are being planned.
Professor Wayne Hu at the University of Chicago has a very nice web site that explains many of the details of moderen astrophyics. http://background.uchicago.edu/~whu/
p.s. I wonder if you may be questioning how we know how dark energy will behave in the future, since we don't really understand it. There used to be something like a type of dark energy driving very fast inflation of the universe, but it changed. Now dark energy has been driving a much slower inflation. It doesn't seem to have changed significantly in billions of years (the measurements are tough), but it's just a good guess that it will stay constant in the future.
(published on 10/22/2007)