# Eternal inflation and topology of the universe

Hi,

I have a question regarding the idea of eternal inflation happening in a multiverse and the topology of our universe.

Looking at the current data it seems plausible that our universe is flat or slightly negatively curved, i.e. has a non-compact topology. But the idea of eternal inflation tells us that our universe (every universe) was created from a small quantum fluctuation of a "mother-universe; therefore I would say that our universe created from a small region of cannot have a non-compact topology.

Is there an idea whether and how during eternal inflation something like a topology change can happen? Or is a local deformation of geometry sufficient to explain the shape of the universe we observe?

## Answers and Replies

Chalnoth
Is there an idea whether and how during eternal inflation something like a topology change can happen? Or is a local deformation of geometry sufficient to explain the shape of the universe we observe?
Well, first of all, it is relatively easy to write down universes that wrap back on themselves, but nonetheless have zero or negative curvature.

But even if the universe starts off with positive curvature, it is very easy within an inflationary framework to see how it we might measure a flat or negatively-curved universe: if inflation lasts long enough, then the overall curvature goes to zero, and whatever curvature we measure will just be an artifact of any curvature fluctuations that were generated during inflation.

Well, first of all, it is relatively easy to write down universes that wrap back on themselves, but nonetheless have zero or negative curvature.
Do you have a reference? Or an equation that shows how the metric looks like?

Chalnoth
Do you have a reference? Or an equation that shows how the metric looks like?
Well, you just identify the coordinates over some surface. For a flat universe that wraps back on itself, you can, for instance, make the x,y,z coordinates periodic on (-1,1]. This would be a three-torus.

For a negatively-curved universe in spherical coordinates, you can identify the coordinate $(1,\theta,\phi)$ with $(1,\pi - \theta,-\phi)$. I'm not sure what that would be, but if you travel in any direction from the origin, you'll end up returning to the origin after some time.

for the torus is trivial, I agree; but how do you get a negative curvature on the sphere?

Chalnoth
for the torus is trivial, I agree; but how do you get a negative curvature on the sphere?
Well, take the normal metric for negatively-curved space in spherical coordinates, and then identify opposite points on a sphere centered at the origin.

Does that mean that in eternal inflation one would change the topology of the universe such that it consists of a) a large (expanding) but finite region of negative curvature (which is "our" universe) + a small piece that connects this universe to its mother?

Chalnoth
Does that mean that in eternal inflation one would change the topology of the universe such that it consists of a) a large (expanding) but finite region of negative curvature (which is "our" universe) + a small piece that connects this universe to its mother?
I don't understand this.

Let me explain: in the "mother universe" there is a small piece from which - due to a quantum fluctuation - a new "baby universe" is spawned. The original manifold is (close to this quantum fluctuation) free of singularities and boundaries. So if we want to avoid topology change the "mother universe" + the "baby universe" must remain connected. As the "baby universe" starts as a finite topological manifold it cannot become infinite during the spawning; but as we observe zero (or even negative) curvature we have to find a topology for the "baby" which allowes for a finite universe with constant or negative curvature.

Chalnoth
Let me explain: in the "mother universe" there is a small piece from which - due to a quantum fluctuation - a new "baby universe" is spawned. The original manifold is (close to this quantum fluctuation) free of singularities and boundaries. So if we want to avoid topology change the "mother universe" + the "baby universe" must remain connected. As the "baby universe" starts as a finite topological manifold it cannot become infinite during the spawning; but as we observe zero (or even negative) curvature we have to find a topology for the "baby" which allowes for a finite universe with constant or negative curvature.
Typically "baby" universes start off connected, but quickly "pinch off" and become disconnected. This is because the baby universe, from the outside, looks like a black hole, which rapidly evaporates.

But then there IS a topology change and the topology of the baby universe is unconstrained; if you allow a "pinch off" everything else could happen, too.

Is there a theory to understand this "pinch off" mathematically?

(we could speculate about quantum gravity which would allow for a discrete structure w/o classical manifold near the "pinch off")

Chalnoth
But then there IS a topology change and the topology of the baby universe is unconstrained; if you allow a "pinch off" everything else could happen, too.

Is there a theory to understand this "pinch off" mathematically?

(we could speculate about quantum gravity which would allow for a discrete structure w/o classical manifold near the "pinch off")
There may be. Unfortunately I haven't looked into it.

let me ask again: is there a theory to understand this "pinch off" mathematically?

... For a flat universe that wraps back on itself...

I am not a physicist nor a mathematician and can't understand the above sentence. How can a flat Universe wrap to itself? For a trivial speaking "flat" mean "with no curvature" like a sheet of paper.

What is the meaning of "flat" on the GTR (or on a FLRW space)?

That is difficult to explain b/c it escapes your intuition.

Think about your sheet of paper or a Pac Man game where Pac Man leaves the screen on the right and re-enters the screen from the left. That means that the x-coordinate has been compactified to a circle. The same applies to the y-coordinate. Topologically that means that the screen is no longer a square but a 2-torus. It has no boundaries as it has been compactified; the left and the right edge have been glued together which results in a cylinder. Now gluing together the two circles at the ends of the cylinder results in a 2-torus.

If you look at the curvature of the screen it is obviously flat. If you look the the 2-torus is seems that it's no longer flat. So what created the curvature?

One can show that the curvature is only due to the embedding of the 2-torus in 3-space, not due to the gluing itself! There are embeddings in higher dimensional spaces with flat geometry w/o curvature. Mathematicaly such embeddings are not required at all! That means one can still work with a flat 2-torus w/o ever referring to any embedding into 3-space, 4-space, ... Of course you cannot visualize it, but you can describe it mathematically.

That is difficult to explain b/c it escapes your intuition.
... Of course you cannot visualize it, but you can describe it mathematically.

Thanks Tom. I understand now mathematically concept.

But in the PCMan analogy some specific coordinates (the edges of the screen) are translated to other coordinates. In Universe I think such special locations does not exist. Or we can consider that all coordinates of Universe can be translated to other values?

In today cosmology what is considered as flat (or very close to), the 3-D space or the space-time? I suppose that when one cosmologist speak about of curvature of Universe he speak about curvature of space-time of Universe.

Chalnoth
But in the PCMan analogy some specific coordinates (the edges of the screen) are translated to other coordinates. In Universe I think such special locations does not exist.
Consider that you can shift the whole screen to the right by any number of pixels and the behavior will be the same. So there is no special location.

Ich
I suppose that when one cosmologist speak about of curvature of Universe he speak about curvature of space-time of Universe.
They speak always of the curvature of space, not spacetime. The concept of space is not trivial in GR, so be warned that cosmological "space" is not defined the same way as in special relativity.

In many cases the topology of the Universe is M4 = M3 * Time.

That means (using your sheet-of-paper analogy) the universe is a book of sheets of paper. Now if you compactify each sheet to a 2-torus, spacetime consists of infinitly many 2-tori. That means that the time-dimension adds no further element to the topology.

There are exceptions where instead of compactifying space you compactify time, i.e. the last page of the book is identical to the first page - which means you can travel into your own past - but I think we should exclude these strange scenarios gere.

From what I know the observation of Universe show us that it is very close to flat with 98-99% accuracy. This 1-2% can be an error of measurement method, a local fluctuation of curvature of Universe or even a real constant curvature of it.

If this is not just an observational error, what is curved? An additional spatial dimension? Just one extra curved dimension exist or many?

Your answers make me to think that my GTR understanding is wrong. I know that in GTR mass curve the space-time not the normal 3D space. Am I wrong with this?

<<I'm sorry for so many questions but it seems like many people (including myself) had a lot of misconceptions about this notions and it is hard to find accurate answers expressed with natural language (not mathematical one).>>

If there is a curvature then 4-dim. spacetime is curved.

The slicing I introduced above means that the topology in the time direction is trivial and that non-trivial topology is restricted to 3 space dimensions. But that does not mean that the geometrical property "curvature" is restricted to 3-space; unfortunatly we cannot visualize curved spacetime, only curved space.

So curved 4-dim. spacetime means that exactly 4 dimensions are there and that 4 dimensions (3 space + 1 time) are curved; but there is no additional dimension "in" which these 4 dimensions are curved; this is not necessary mathematically.

PhilKravitz
What is eternal inflation? Does it mean only once here but it can happen only once at other locations? Can it happen more than once here? Are the different location always far enough apart that they do not interact? If not, how do they interact? Do we have any data to back any of this up? Can we ever have data to back any of this up?

To tom.stoer, you ask: Is there a theory to understand this "pinch off" mathematically? I can refer you to "The Birth and Death of the Sun" by George Gamow, read Chapter 10, re tear drops. As for mathematial calculations, see "The Creation of the Universe" by George Gamow, Chapter II. The man had quite a lot to say.

For a lesser read with no math, try Dwyer's Sun Creation Theory. There are no pinch offs, only creation of suns which of course leads to galaxies. The math is waiting to be done.

Chalnoth
What is eternal inflation?
The idea with internal inflation is this:

During inflation, you have quantum zero-point fluctuations which randomly perturb the energy density. These density fluctuations are very small, but can, in some cases, be large enough to change how inflation behaves.

You see, the rate of inflation is given by the energy density. A region with higher energy density inflates faster than a region with lower energy density. Furthermore, the overall trend of any given region is for its energy density to drop.

But if you have a rather special situation where the small probability for a region to increase in energy density is compensated for by the increased expansion rate, then there will always be regions that continue on inflating. Most of the volume will decrease in energy, and become a normal universe like our own, but if the probability of the energy density to jump up is high enough, then inflation just keeps on going forever, forever spitting out new regions of the universe like our own.

I can refer you to "The Birth and Death of the Sun" by George Gamow
This is about 70 years old, whereas the idea of inflation is about 30 years old.

I would like to ask again whether there is - in the context of inflation - a theory trying to model the "baby universe pinch off" mathematically?