How to measure curvature of 3-sphere from inside its surface

[gulfcoastfella]

How is 3-sphere curvature measured? If a 2-D being living "in" the surface of a sphere tried to measure the 3-D curvature of the sphere, how would they go about it? They couldn't detect the curvature by looking for curvature in the paths of signals, because if the surface of their sphere was as transparent, isotropic, and homogeneous as our universe is, then wouldn't any signal they emit show no deviation from a geodesic when viewed on a cosmological scale, and we would see any 4-D geodesic in our universe as a straight line, the path followed by an "unperturbed" particle? The only way I can think of to detect higher-dimensional curvature would be to examine the effect on particles which have traveled over a cosmological distance. Since particles have non-zero dimension, then absent some kind of resistance by the particle or a field, they would tend to be stretched out more and more the further they traveled. (Two sides of the particle on opposite sides of it's geodesic path would tend to follow slightly different geodesics in the surface of a 3-sphere, leading to a "spreading out" of the particle as it traveled, assuming, again, that the particle or a field didn't counteract this somehow.) I remember from a quantum mechanics course I took in college that the wavefunctions of photons tend to spread out more and more the further they travel. I assumed, however, that this phenomenon was explained by quantum mechanics and didn't require curvature of space in higher dimensions.

Please don't respond with relativistic tests, as relativity doesn't deal with 4 spatial dimensions; it deals with 3 spatial dimensions and 1 time dimension.

 

[Bandersnatch]

You can measure the sum of angles in sufficiently large triangles.

Find out a faraway object of well-measured distance, whose size you can calculate from some well-established principles. Then measure the angular size on the 'sky' and compare with the predicted size for flat euclidean space.
If the surface is positvely curved, the angles in the triangle with vertices ABC where A is the observer and B and C are the opposite edges of the observed object, will add up to more than 180 degrees and the object will look larger. If negatively curved, it'll look smaller, and if it'll look about right it will mean that the space is flat within the error bars set by the accuracy of your measuring instruments.

 

[gulfcoastfella]

Thanks Bandersnatch! Has an experiment been conducted to uncover the answer to this question?

 

[Bandersnatch]

Yes. The recent WMAP and PLANCK missions have been doing just that.

They measure Baryon Acoustic Oscillations in the CMBR. These are regions of overdensities in the early universe that left a 'footprint' on the CMBR when the photons were released ('decoupling').
See here:
http://en.wikipedia.org/wiki/Baryon_acoustic_oscillations#Cosmic_sound
for a bit more thorough explanation.

Since the size the overdensities should have can be calculated, and the distance to the surface of last scattering (CMBR emission surface) is known, they can be compared with observations.

The steadily improving observations all appear to zero in on a flat universe. Practically every year the minimum curvature radius the universe would need to have to fit the data goes up as the error bars go down. The last estimate I remember seeing was at least 220 Gly.
The most recent PLANCK results have been just released (there's a featured thread on PF), so expect further refinement of the flatness estimates.

 

[gulfcoastfella]

Thanks for the link and reference to the PF thread Bandersnatch. I'll be interested to read it (understanding it is another matter altogether.) :)