Thread: CMB, multiconnected space and isotropy.

Hi everyone, I was reading about how the universe might have a multi-connected topology, and how we might confirm that by finding opposing pairs of circles in the cosmic microwave background radiation, and a question came to my mind.

One of the postulates physics lay on is that there is no special direction in the universe, that is, physics laws are isotropic. But now, I get told that if the universe actually has a folded topology, I will be able to find matching pairs of circles in the CMB at specific places in the sky, these places depending on what exactly the large-scale topology of the universe is.

The way I understand it, these circles, if any, will be found in the same direction regardless of where you stand in space. So where has the symmetry gone?

Does the theory of a multi-connected space require the existence of an arbitrary reference set of directions that define the orientation of the base polyhedron that describes the universe’s topology? Its orientation would have at least one physically observable effect, the existence of pairs of circles in the CMB, but could there be more?

Or am I totally misguided?

It's supposed to be the same as how, on Earth, if you start traveling East, and go far enough, you'll arrive at the international date line. If you start traveling West, and go far enough, you'll arrive there as well.

Big Bang cosmologists believe that the universe is not infinite. It just starts over after you travel enough in any one direction.

That is almost correct, but how does that answer my question?

I wrote 'almost' because cosmologists do actually not know whether the universe is finite or not, they have theories but no definitive answer yet.

But my question is quite more specific than that. Thanks for answering anyway

Imagine I'm on the equator, and I send two robotic airplanes with video cameras in different directions. One east. One west. Sooner or later, they would meet, and be filming the same object. They'd criss cross and soon I'd be seeing the exact same objects on both sides of me.


If I understand correctly, Big Bang symmetry is based on that idea. You see similar objects in either direction because you're seeing the same object twice. Or maybe I have that notion confused?

I understand your question, but don't trust my answer. I'm no expert on this.

Anyway, I think the answer you're looking for here is spontaneous symmetry breaking. Of course, finding such a break would raise all kinds of new questions, but I think you're right about it creating a preferred direction.

If anyone is having trouble imagining this, the easiest way to think of it is to imagine a universe on a sheet of paper (topological paper, to be precise, but the difference won't matter in this example). You could make a paper universe out of a sheet of paper that's infinite in both directions, or you could roll the paper up so it's only infinite in one direction. You wouldn't be able to tell if the paper was rotated in the first case, but you could in the second.

Originally Posted by kojax

Imagine I'm on the equator, and I send two robotic airplanes with video cameras in different directions. One east. One west. Sooner or later, they would meet, and be filming the same object. They'd criss cross and soon I'd be seeing the exact same objects on both sides of me.


If I understand correctly, Big Bang symmetry is based on that idea. You see similar objects in either direction because you're seeing the same object twice. Or maybe I have that notion confused?

No you got it right.

If you were living on a sphere and were following a straight line on it, the exact same thing would happen regardless of which direction you picked. You would always be coming back to the same point and see the same objects again after having traveled the same distance, this distance being the sphere's perimeter.

But now here's the catch: from what I understood of cosmological topology (I'm far from being an expert), it seems that the universe's topology is not analogous to a sphere’s, but rather to a flat torus’. A flat torus’ topology is kind of similar to a sphere’s, in the sense that when you travel far enough you get back to where you first were. But when you ‘unfold’ the torus, it gives you a perfect rectangle. What this means is that living on a torus is a bit like living on a plane that contains infinitely repeating copies of the same rectangle put side to side. Just like in a video game, when get out on one side you enter on the other side. But now, the time it will take you to see the same object is not independent from the direction you’re moving in anymore.

Just picture that you’re living in a flat torus that has the same ‘size’ in both its dimensions. The unfolded torus is a square. Starting from any point, you can reach that point again by traveling from left to right or top to bottom, and in both cases you’ll get back having travelled the same distance. But now if you try and follow a top-right direction parallel to your square’s diagonal, it’ll take you square(2) more time to get back to where you started. Generally speaking, if you picked two directions it is very likely that the times it will take you to get back to your starting point will be very different.

So there are privileged, arbitrary directions in the flat torus’ topology, and it seems that there might be too in the universe’s. These would affect how much it takes you to get back to where you were, following a straight line, but that is quite devoid of pragmatic interest. Yet apparently, some effects could be seen, like pairs of circles that would have to appear in given directions in the CMB.
So my question was, is this compatible with the isotropy assumption, and could those arbitrary directions be responsible for other, maybe more tangible effects?

Originally Posted by MagiMaster

I understand your question, but don't trust my answer. I'm no expert on this.

Anyway, I think the answer you're looking for here is spontaneous symmetry breaking. Of course, finding such a break would raise all kinds of new questions, but I think you're right about it creating a preferred direction.

If anyone is having trouble imagining this, the easiest way to think of it is to imagine a universe on a sheet of paper (topological paper, to be precise, but the difference won't matter in this example). You could make a paper universe out of a sheet of paper that's infinite in both directions, or you could roll the paper up so it's only infinite in one direction. You wouldn't be able to tell if the paper was rotated in the first case, but you could in the second.

I don’t know much about spontaneous symmetry breaking, but the way I understand it it’s got to do with quite a different and unrelated thing, that is, symmetry breaking with respect to how the actions of a given unified field could yield very different effects at low temperature.

I’m not sure whether spontaneous symmetry breaking has much to say about topology, but hey I’m not to be trusted on this either.

Originally Posted by kojax

It's supposed to be the same as how, on Earth, if you start traveling East, and go far enough, you'll arrive at the international date line. If you start traveling West, and go far enough, you'll arrive there as well.

Big Bang cosmologists believe that the universe is not infinite. It just starts over after you travel enough in any one direction.

This is better said as being explained that the universes big bang happened everywhere, not just at some refined point in spacetime, nor does it answer the OP.

What do you mean by topology? In what sense are you using it?

Originally Posted by Jean-Pierre Liégeois

Originally Posted by MagiMaster

I understand your question, but don't trust my answer. I'm no expert on this.

Anyway, I think the answer you're looking for here is spontaneous symmetry breaking. Of course, finding such a break would raise all kinds of new questions, but I think you're right about it creating a preferred direction.

If anyone is having trouble imagining this, the easiest way to think of it is to imagine a universe on a sheet of paper (topological paper, to be precise, but the difference won't matter in this example). You could make a paper universe out of a sheet of paper that's infinite in both directions, or you could roll the paper up so it's only infinite in one direction. You wouldn't be able to tell if the paper was rotated in the first case, but you could in the second.

I don’t know much about spontaneous symmetry breaking, but the way I understand it it’s got to do with quite a different and unrelated thing, that is, symmetry breaking with respect to how the actions of a given unified field could yield very different effects at low temperature.

I’m not sure whether spontaneous symmetry breaking has much to say about topology, but hey I’m not to be trusted on this either.

You're thinking of a specific application of the idea of spontaneous symmetry breaking. The basic idea though just means any situation where it's perfectly symmetrical until something has to give. All the possible outcomes are equally likely, but since only one can be chosen, the result is a broken symmetry.

The classic example is the ball on top of a round hill. You could rotate the hill and ball around the center axis and you couldn't tell the difference (which is the basic definition of symmetry). Assuming the ball is just sitting there, it's unstable, and will eventually roll to the bottom of the hill. After that happens, the system's rotational symmetry is broken.

As I (poorly) understand it, what is at stake in measurements of the CMB (Cosmic Microwave Background) is the topology of the Universe, and so also the Copernican principle.

The Copernican principle is that The Earth does not inhabit any special place in the Universe. It has been a good guideline for ruling out suspect cosmologies such as the Ptolemaic one.

If CMB studies show patterns that support a topology something like a foam of bubbles, then the Earth could be said to have a special place at the centre of one of these bubbles.

It seems that interest in has arisen in tossing The Copernican Principle among some cosmologists as a way of side-stepping the Dark Energy problem.

I think the reasoning goes like this:
To explain the accelleratng expansion of the Universe, as is observed, one needs something like Dark Energy to drive this expansion.
If, however, Earth just happens to be at the centre of a little bubble that just happens to be expanding, then Dark Energy would not be needed.

The expansion seen from Earth would just be a local phenomenon, and not a Universe wide one as usually thought when applying The Copernican Principle.

I'm sure the cosmologists proposing this idea would explain it much better, but that's my limited understanding of the matter.