First learn how to read the results. I assume you are looking at table 2 of
http://arxiv.org/pdf/0803.0547v2.pdf
Look in the WMAP+BAO+SN column which is where they combined ALL the data relevant to curvature. Not just the WMAP mission data.
Look in the row that says Positive curvature case. that would be the 3-sphere case.
Look where it gives the estimate for the RADIUS OF CURVATURE. That would be like the radius of the 3-sphere if you pictured it embedded in a higher dimensionality which we merely imagine, do not know exists.
The table tells you that with considerable confidence the radius R in that case has to be
> 22 h
-1 Gpc
OK you have to learn how to interpret that. G means billion. pc means parsec (i.e. 3.26 lightyear) so multiply 22 times 3.26 and call the answer billion lightyears.
And h is a little number cosmologists use which encodes what you believe the Hubble parameter is. A common value to assign to it is h = 0.71 because you think the present value of the Hubble parameter is 71 km/s per Mpc.
But you might also think h is 0.706 if you and your co-authors are using 70.6 km/s per Mpc in your work. Having this little order-of-magnitude unity (approx = 1) number in the formulas is helpful because it gives everybody the freedom to adjust, as estimates of the Hubble parameter are refined.
So multiply 22 times 3.26 and divide by 0.71 and call the lowerbound answer for the RADIUS that many billion lightyears.
Then if you want a lowerbound for the circumference you can multiply by 2 pi.
Space has to be AT LEAST that big. So if you could freeze expansion and head off straight in some direction at the speed of light it would take at least 600-some billion years before you completed circumnavigation and got back home. At least. It is only a 95% confidence lowerbound estimate.
Now the fascinating thing is how they arrive at the confidence interval for the curvature radius, or equivalently for the curvature parameter Omega-sub-k
Look at footnote g of table 2 to find out how you calculate the radius R from the parameter Ω
k.
One way might be to count the number of galaxies in balls of different radius. If we live in overall zero curvature then the number should increase with the cube of the radius. (after you adjust for expansion, and rates of galaxy formation). The idea is that if it weren't for distances expanding then in a nice flat (zero curvature) universe the volume would increase as the CUBE of how far out you look. And counting galaxies is a way of estimating the volume. that's oversimple but it gives an idea of how one might get a handle on curvature.
Another way is to study the SIZES of the TEMPERATURE FLUCTUATIONS in the ancient light.
this is a snapshot of the universe taken in year 380,000. All that light was released right about that time. Our region of space was 1000 times smaller and full of hot gas that had just then become transparent. We understand something about the PHYSICS of that hot gas. How its pressure and density would have been throbbing randomly in patches of lower/higher pressure. So astrophysicists can estimate the statistics of the sizes of the patches of lower/higher temperature. what the actual sizes were in year 380,000.
And then they can compare what the sizes were then with how big they look now. Admittedly it gets somewhat sophisticated and involved. But it gives another way to get a handle on the overall curvature. Several other people here (say e.g. B. Powell) can explain this more clearly.
There are several handles one can get on curvature. They are all "model-dependent". You have to set up a mathematical model of the geometry of the universe and make observational measurements that you put into the model and adjust the model to, and get curvature out. Where it says "WMAP+BAO+SN" that means using 3 different types of observation to get 3 different handles.
the model is called Friedman equation and it is derived from the 1915 Einstein equation, which is basically a law of geometry, how geometry (in conjunction with matter) evolves.
I wish I could do a better job of intuitively explaining how one infers estimates of curvature from various observations.
I think the confidence interval for Ω
k is one of the more beautiful achievements of cosmology. BTW there is a conventional sign reversal and negative Ω
k signifies positive curvature---i.e. finiteness.
As they continue to shrink the interval, it might continue to straddle zero (which would suggest flatness and possibly infiniteness) OR it might get a little off to one side, say the negative side and stop straddling zero. And that would signify spatial finiteness. To shrink the interval requires improved instruments. Currently the successor to WMAP is taking data. It is called Planck and it orbits the sun about a million miles farther out from the sun than the Earth is. The Planck data will be reported in 2013, I think. Not this year anyway.
