On 1/17/2014 5:03 AM, Edgar L. Owen wrote:
Brent,

Of course you can calculate the radius of a sphere (in this case a 4-dimensional hypersphere) from the curvature of that sphere.

Just make the assumption the universe is a hypersphere and then what's the formula to calculate the radius from the curvature? And don't tell me it's not a hypersphere, just make the assumption then what's the formula?

There's no single formula. First, the spacial radius can be anything because it's flat. An FRW universe is a hypersphere because it's curved in the time direction. If the curvature is positive, then the universe recollapses and it's radius is half it's duration. But the curvature is a function of the energy density which in turn depends on the equation of state of matter in the universe. Taking just the cosmological constant (dark energy), matter, and radiation:

        rho = rho_0 + a^(1/3) + a^(1/4)

Then you have to plug this into the equation of motion

        2a" + (a')^2/a + k/a = rho*a

and integrate numerically. My friend Lawrence Crowell has already done this for the case neglecting the matter terms and got this result for different cosmological constant values. Notice that even with k=1 (positive curvature) the universe doesn't collapse for the CC>0, i.e. it's open and has no radius.


https://lh3.googleusercontent.com/-eD-j0h7GBN4/UcsHpX02_uI/AAAAAAAAAAw/o4mNGiglnmY/s1600/FLRW+integration.GIF


Brent

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