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Can someone please explain to me this image:

http://www.astro.ucla.edu/~wright/omega0.gif

from the page

http://www.astro.ucla.edu/~wright/cosmo_02.htm

What i do understand:

I understand the concept of light cones

I understand that the units used for the axes means the lines are a 45 degree angle indicate the particle/causaility horizon thing (i think)

I think i understand that non parallel worldlines implies relativistic effects taking place

I dont understand what the red line thing is.

So yeh i dont get much of this, any help would be appriciated =)

Thanks

-G