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according to An Introduction to Modern Astrophysics, 1263p, there is explanation about evolution of scale factor.

“ For the early universe($R<1$) there is little difference among the kinematic behaviors of a flat, a closed, an open universe because the early universe was essentially flat.”

what does it mean? i don’t know relationship between ‘kinematic behaviors’ and ‘early universe flat’

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Help me please

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This answer is based on the Figure 5 you added to your question. I believe that the concept of "kinetic behavior" in this context means the mathematics of the expanding universe based on the cosmological model related to Figure 5. The "early flat universe" refers to the similarity between the three assumed models (open, flat and closed) in Figure 5 on the left side where R approaches zero. Note that near R = 0, both the open and closed models look very similar to the flat model.

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  • $\begingroup$ the texture was written below the graph. so i think the relationship between graph and texture is obvious. $\endgroup$ – 빛나는밤 May 11 at 18:18
  • $\begingroup$ i post new picture. Can u check? $\endgroup$ – 빛나는밤 May 11 at 19:40
  • $\begingroup$ oh, you mean that R changes over time into the same functional form of the same function. I think I was mistaken because the origin of the graph didn't match. thanks. $\endgroup$ – 빛나는밤 May 13 at 2:11
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For the early universe(R<1) there is little difference among the kinematic behaviors of a flat, a closed, and open universe because the early universe was essentially flat.

Indeed somewhat confusing, because if the early universe were essentially flat, how can it develop in a closed, flat, or open universe? Won't it stay flat forever?

f you imagine the universe to be a balloon (reducing the number of space dimensions to two), it is said that for every model (closed, flat , or open) the developing balloons are identical in the early universe (around $T=1$). I'm not so sure what is meant by essentially flat means that the balloons locally don't differ from a flat spacetime (though their overall form is that of a balloon). This is also the case for the three types of universe in the distant future.
So while the balloons don't differ much from each other now, they will differ from each other in the future.

If you take accelerated expansion seriously you can nothing but conclude that we are living in an open universe (the balloon will not only expand forever, but it will do so in an accelerated way (though it remains to be seen if the accelerated expansion will continue forever). Before the accelerated expansion started the universe looked locally flat.

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  • $\begingroup$ thanks for your anser $\endgroup$ – 빛나는밤 May 13 at 2:13

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