Big Crunch and the Universe's shape

[LCSphysicist]

TL;DR

I was reading a book about cosmology, from Andrew Liddle, where in the chapter 5 he start to talk about implications of the differentes signs possible to k (k from the Friedmann equation). It is said, until this chapter, that the implication of k>0 inevitably implies a stop in the Universe expansion, and eventually a Universe's contraction. But, as far as i know, the Big Crunch was already discarded, so it is the same to say that k can not be greater than 0?

In another words, to discard the Big Crunch, we discard the spherical universe hypothesis too?

k>0 able us to find a solution for a dot = 0

 

[PeterDonis]

It is said, until this chapter, that the implication of k>0 inevitably implies a stop in the Universe expansion, and eventually a Universe's contraction. But, as far as i know, the Big Crunch was already discarded

When you say "the Big Crunch was already discarded", do you mean somewhere earlier in Liddle's textbook? If so, where?

If all you mean is that the Big Crunch is not currently predicted to happen for our actual universe, that's true, but it's irrelevant to what Liddle is saying in the chapter you refer to. He is talking about possible models in general, not about the specific model that we use to describe our actual universe.

to discard the Big Crunch, we discard the spherical universe hypothesis too?

If there is zero cosmological constant, then it is true that the universe having a hyperspherical spatial geometry and the universe ending in a Big Crunch must go together.

If there is a positive cosmological constant (as we believe is the case in our actual universe), that is no longer true. It is possible for a universe with hyperspherical spatial geometry to not end in a Big Crunch.