In the "Closed Universe" and in the "Opened universe" you have put the geodesic views from the plane, in the "Flat universe" following that logic (which I do not know if it is the correct one) would be wrong, it would be a "triangle".

In the last figure, notice that it begins as two "Opened universe" reflected, and that in the part that expands to infinity join.

Therefore I would say that the figure you are looking for would be the "Opened universe" rotated, like an hourglass

@Sergio Garcia Chimeno

Please note that the graph shows the size of the universe as a function of time, depending on the k and Ω parameters. While a closed universe reunites in one point (sometimes called Big Crunch), the open universe continues to grow. In the flat universe, however, growth is no longer recorded at some point due to the parameters, but there is an asymptotic approximation to a fixed size of the universe. This is at least the interpretation I was taught - and here the two-dimensional representation of a triangle - I think - would not be correct. But maybe I am wrong too.

Note that the fourth graphical representation says something strange if we accept that the x-axis represents time. So here we would have a universe that grows, then strikes backwards due to a too large exponential gradient - in this case into the temporal past (if x-axis is time). For me this is very difficult to understand and i wonder if there is anything mathematically to be read out of this graph, or if its just a nonsense plot.

Actually there is a common misunderstanding with the Friedmann-Lemaitre equations and the solutions for the Robertson-Walker metric. The point is that the parameters k and Ω define the metric, which is a local entity and does not imply the global structure of the space(time). Topologically, if k>0, the universe must be closed. But for k=0 or k

The other thing is: I do not understand the fourth picture. Where does the fourth option actually come from? I understand the graphics is meant to denote the timeline of the universe (bottom to top), but then I do not understand the fourth picture at all.

From what I said above, the parameter k defines the curvature of the RW metric. This by itself does not imply the global, topological properties of the universe. For this, additional input is needed, namely equations of state or in other words: models of matter. This then gives the "radius" as a function of a cosmological time parameter, and may lead to big bang, crunch, some periodicity maybe, ever expansion and so forth.

Oliver Tennert

Dear Oliver,

Thank you very much for the gainful answer.

To your question, where the fourth drawing did come from:

I just thought about how to graphically represent an entity that is infinitely extended in all directions, but is still a finite entity. The graphic thus has both a defined start and end point, but is nevertheless infinitely extended in all directions. As noted in the question, the derivation is purely graphical and I was just wondering if there is a mathematical interpretation of such a body.

I came up with the idea because the concepts of a closed universe as well as of an open universe were always coming towards me in my work, but they were then always closed "or" open. So the question for me was: Is there a description for a system that is open in all directions "and" closed with a defined start- and endpoint at the same time.

what about space-time curvature? there should more possibilities

Aleksandar, I think you're going about this the wrong way, you can't just draw a picture then look for an equation that fits, you need to first propose a density profile (Robertson and Walker assumed homogeneous and isotropic) then from that work out the metric and finally calculate a graphical representation if that is feasible. The first three on your poster cover all the three possibilities of the Robertson-Walker metric so you need a different distribution of energy to start from. To work that backwards which is what I think you are trying to do, you first need to turn the picture into the equations for the metric. I would advise you look at a variety of existing metrics to see the form of what would be needed.

George Dishman

Thank you for your comment. Basically I think you can draw a picture to see if there could turn out new thoughts or not. But of course there is no metric here. After the drawing I also did not find a corresponding formulation, so that I assume myself that it is a drawing that can be drawn, but nothing more.

Aleksandar wrote:

"... so that I assume myself that it is a drawing that can be drawn, but nothing more."

Not so fast. Recall Ringermacher's discussion of oscillating accelerations (2015, 2017), and his earlier (2012) speculations about combining negatively curved (hyperbolic) regions of space ("voids") within an FRW universe.

https://arxiv.org/abs/1201.5533 (2012)

https://arxiv.org/abs/1502.06140 (2015)

https://arxiv.org/abs/1706.00071 (2017)

https://astronomynow.com/2015/07/01/is-the-universe-ringing-like-a-crystal-glass/

Please keep going a little more!

Nigel

Nigel Nunn

A big thank you for these great articles I didn't know before!

Hi...

The FLRW metric is a solution of the Einstein eqns for a Constant-Curvature space. There are only 3 possibilities - already mentioned in your discussion. Anything else is not a solution of FLRW. To accommodate your 4th option requires a new metric, perhaps a time/space-dependent curvature solution. I have not yet found such a solution of the Einstein eqn.

I might add that my paper on a "Bipolar Model"...of hyperbolic space was rejected by Physics journals as being too mathematical and by Mathematics journals as being too physical. It primarily raises the question of what coordinates are "physical". This is not easy to answer. For example rotating coordinates are considered non-physical, but if you are in them, they are real and there is physics associated with them. As mentioned above, one needs to consider the matter distribution to make sense of them.