Evidence for a flat universe vs. circular logic

[kmarinas86]

Interpretation: The Cosmic Background Radiation is evidence of a flat universe.
Problem: How does a teacher show that all the knowledge necessary for this interpretation does not require the assumption of a flat universe?

 

[cesiumfrog]

I understand the flatness (or otherwise) of the universe is measured by comparing the size (eg. length) of the original background fluctuations with the angle they subtend now.

Perhaps someone could tell us how large were those original fluctuations are predicted to have been, and why?

 

[kmarinas86]

I understand the flatness (or otherwise) of the universe is measured by comparing the size (eg. length) of the original background fluctuations with the angle they subtend now.

Perhaps someone could tell us how large were those original fluctuations are predicted to have been, and why?

Well inflationary theory has made predictions of those parameters including the angular width of the hotter spots in the Cosmic Background Radiation. If those parameters were not observered in WMAP, then one would be more skeptical about inflationary theory. However, it was observed that the parameters matched those predicted by inflation. If inflationary theory is correct, this implies that the flatness problem of the Big Bang theory is taken care of.

But why are the values in Cosmic Background Radiation evidence of a flat universe? Is it because inflationary theory, which predicts these values, also predicts a flat universe?

Let ---
A: Inflationary Theory (theory)
B: Values in Cosmic Background Radiation (observation)
C: Flat universe (conclusion)
We know that A implies (B & C).

By assuming A and knowing B, how do we conclude that "B implies C"?
B is implied by A which implies C. But that alone does not mean B implies C.

What am I not getting?

 

[cesiumfrog]

Well inflationary theory has made predictions of [...] the angular width of the hotter spots in the Cosmic Background Radiation.

I was guessing it predicted the absolute width, and that the angular width depended on that and the flatness. Like.. if a triangle is drawn on the surface of a sphere then the angle at the top depends on: "how long it was" to the base, the actual width of the base, and the radius of curvature (or the flatness) of the surface.

 

[hellfire]

You can determine whether space is flat or curved measuring the angular size of the first peak. The impact of the curvature of space in the relation between actual size and measured angular size of the first peak is given through the definition of the angular diameter distance $d_A$.

In general, one has that the actual size $l$ relates to the angular size $\theta$:

$d_A = l / \theta$

This means that the angular diameter distance and therefore the curvature and the Hubble parameter $d_A = f(\Omega_{k, 0}, H_0)$, can be expressed as a relation between the measured angular size and the actual size.

The actual size of the first peak corresponds to the size of the horizon during recombination. This depends on the recombination redshift and on the curvature of space at recombination redshift.

The recombination redshift depends on the recombination temperature that in turn depends on the ionization energy of hydrogen and the photon to baryon ratio.

The curvature at recombination redshift can be assumed to be zero (flat space) because for the first peak to exist (or to be interpreted as such) the universe must have underwent an inflationary phase. This implies that the universe was very near to flatness after inflation. After inflation it deviates from flatness either to positive curvature or negative curvature, but for a high redshift such as the recombination epoch space was still nearly flat.

However, I guess you can make the calculations also without relying on this assumption.

 

[Mike2]

And how does anyone know that the curvature is the same as it was at recombination when the CMB was born? Could the curvature have changed since then?

 

[hellfire]

Well, I was trying to make clear how to make a simplified calculation, but I see I might have generated some confusion.

The curvature at recombination can be of course different from zero and you have to take this into account if you want to test inflation for example.

If inflation is correct, the curvature at recombination is nearly zero. After inflation the curvature evolves away from flatness. The more far away in time the more devitation, until the universe becomes dominated by the cosmological constant that will make the curvature tend to zero again.