Are rays from opposite sides of a hot spot of CMB parallel or not?

[alkmini]

hello
I have a question:
i am trying to understand how we find out that the curvature of the universe is zero using the angular size of the hot spots of the d microwave background radiation.
http://scienceblogs.com/startswithabang/2012/07/18/how-big-is-the-entire-universe/
there is a picture in this blog showing the light rays from the opposite sides of the hot spots.
My question is: doesn't the proof that the curvature is zero require that the rays are parallel to each other? If they are parallel in a positively curved universe they converge. But in a flat one they should remain parallel and in a negatively curved they should diverge. Then, in the last two cases, how do they meet in the eye?

 

[Lino]

Alkmini, The rays are not parallel in any of the examples.

Imagine being in a fairground hall of mirrors and looking at yourself. If you see yourself tall you know that the mirror is convex, if you see yourself small you know the mirror is concave, if you see yourself normal then you know that the mirror is straight. The blog is adopting the same approach (except it is a CMB hotspot that is being looked at and it is space that is curved instead of a mirror, and space is open / closed instead of concave / convex).

I hope that this helps.

Regards,

Noel.

 

[alkmini]

thank you so much Noel
I understand the similarity with the mirrors
but isn't it true that rays coming from a distant object are parallel?

 

[alkmini]

do you can consider the ENTIRE space between the spot and the eye as a mirror?

 

[torquil]

but isn't it true that rays coming from a distant object are parallel?

Rays from from the sources were sent out in all directions, just as our sun sends out rays in all directions. The two rays in question are the ones that happen to reach us, and are not parallel since they come from different positions, for the same reason that two sides of a nondegenerate triangle are necessarily not parallel.

Rays from a distant object are only approximately parallel if the size of the distant object is negligible compared to the distance to the object. As long as you are looking at pairs of different points in the CMB map this is no longer the case.

 

[Lino]

Alkmini,

The rays are a very close approximation to parallel ...but not quite and because of the huge distances, the very small deviation from parallel allows them to converge. As a result the measurements need to be very accurate, which is why it take science the effort and time that it does

In relation to space, it is better to consider the entire space between the spot and the eye as a lens (but that lens equates to a mirror in my simple example) - but it is the entire space.

Regards,

Noel.

 

[alkmini]

thank you both Noel and Torquil for your explanations. they helped