Tuesday, February 8, 2011
So, using standard trigonometry, and the fact that tan(θ) = opposite/ adjacent in flat space, you first calculate the size fluctuations should have grown in the universe when the CMB was emitted ~380,000 years after the big bang. (The opposite). Then you calculate the distance to those fluctuations from us. (The adjacent). And then, assuming all angles add up to 180 degrees and therefore tan(θ) = opposite/ adjacent applies, you solve for the angle θ that those fluctuations should make and then look at the CMB and ensure that the fluctuations are that size.
Then you make your measurement even more precise by turning again to our friend the power spectrum. The first peak in the power spectrum should be at the scale of the largest fluctuations in the CMB. The important formula is the peak at multiple l means the largest fluctuations are on the order of 180/l degrees.
The calculation: Anyways, using Ned Wright's Calculator I get the biggest fluctuation at the time of the CMB is about 0.64x10^6 light years across (opposite) and was emitted 41.5x10^6 light years away from us (adjacent) giving θ = 0.88 degrees which corresponds to a flat universe prediction is that the first peak should be at l ~ 205. (If you do this 100% correct and not just back of the envelop you should get l ~ 220)
Anyways, as you can see from the power spectrum this is exactly where we see the first peak showing triangles in the space-time of our universe add up to 180 degrees and thus demonstrating our universe is flat!