Flatness of space vs spacetime

[SkyHiker]

I had been reading about the universe being flat as opposed to negatively curved, which I thought was the consensus for at least a while. That made me wonder what is meant by that, exactly? I collected some easy-to-read links that explain things a bit but also show the confusion (negative energy and/or pressure, imaginary mass, cosmological constant, vacuum energy, dark this and that, the list goes on).

My standard reference is Weinberg's book from 1971 who assumes no cosmological constant. His FLRW metric and the available measurements lead to 3 possibilities for curvature and age of the universe: Positive (age=7.5B), zero (age=9B), or negative (age=13B). He does not differentiate the pressure part into baryons (regular matter), dark matter, microwave background or dark energy, but rather does the analysis first then starts to wonder where the missing quantities can be found. He concludes that the measurements support the 3rd option but that a lot remains to be investigated.

This educational link gives a nice overview of the four pressure terms in the first couple of paragraphs. About 2/3 of the way down is a time vs scale factor graph that corresponds well with the ages given by Weinberg.

The controversy about the various ways in which people discuss the matter is shown in this interesting physics stackexchange link. It clarifies that the references to flatness are just about the spatial part, which is not the same as a flat spacetime (Minkowski metric/special relativity). If you check the comments, you will see how confusing the terminology gets even among those who seem to know the matter well.

I recall Sabine Hossenfelder referencing the same issue, suggesting that we simply talk about negatively curved spacetime instead of a mysterious term like dark energy. Whatever the answer is, it can all turn out to be a hoax with the new JWST observations. Hope you enjoy the read, maybe it helps if you had the same impressions.

[Gmetric]

Thanks for the links, your thoughts and clarification, Henk. Much appreciated.