favoured topology of Universe candidates

Boud Roukema boud w astro.uni.torun.pl
Nie, 13 Sty 2002, 01:40:15 CET

Dear Ken & Alison,
   Glad you're getting back to me for feedback. This is definitely
a good way to check we understand each other. :-)

> Date: Sat, 12 Jan 2002 14:28:52 EST
> Subject: Re: favoured topology of Universe candidates
> To: boud w astro.uni.torun.pl
> Dear Boud,
> Many thanks for your prompt reply. We have read through your e-mail and BASI 
> review, and would like to check that we have understood correctly (not being 
> experts in this field).
> We understand that you are researching possible shapes of the universe which 
> are finite, unbounded, and multiply connected (such as a hypersphere, 
> 3-torus, 3-Klein Bottle or other 4D polyhedra). This would imply that we are 

OK, two comments.
(a) science
 Correct, except that I'm also researching possible *infinite*,
unbounded, and multiply connected models. A finite model would be more
aesthetically pleasing, but it is best to do observational work with as
few prior hypotheses as possible. The Universe is how ever it is, not how
I would like it to be.

(b) pedagogical, terminology
 The "shapes" such as the hypersphere, the 3-torus, the 3-Klein Bottle
are *not* "4D polyhedra".  Technically, they are called "3-manifolds",
but since that is rather scary to non-specialists (including many
astronomers!), a more friendly term would be

"3-spaces" or "3-dimensional spaces"
- which can be thought of with the *help* of a 4th dimension 
- or which can be thought of as 3D polyhedra where certain faces are identified
in some way.

It is *very* important to explain that when a 4th dimension is used to
help think about a 3-space, this is purely a psychological tool, a mental 
crutch for human beings having difficulties thinking in a multiply connected
and/or curved 3-space, simply because we are biased from our childhood 
experiences and school education to think in terms of Euclidean, simply
connected 3-space.

It might sound like gestalt psychotherapy ;-) to say this, but as far
as a mathematician or physicist is concerned, a 3-dimensional space
exists in and of itself, it has no need for the existence of extra
dimensions to be itself, even if an extra dimension or too helps it
(or beings living inside it!) to understand what it is.

In "brane theory", it is true that 4th & 5th dimensions are thought of
as true physical dimensions, but the field of observational studies
of the topology of the Universe is separate from brane theory. 

But then again, the 4-dimensional or 5-dimensional models could again
be thought of in spaces with an extra "psychological" dimension of no
physical meaning, i.e. 5D or 6D spaces, again with the extra dimension
just as a thinking aid. So I think the issue of a dimension as a
thinking aid with no physical meaning cannot be avoided.

I would recommend that in your article you explain something of this
issue - the fact that as far as most observational cosmologists
interested in either curvature and/or topology are concerned, a 4th
dimension is a useful psychological tool for thinking, but is used
without any physical meaning.

If we go one dimension down, to make things easier, I can refer to
Figure 1 of my BASI article http://uk.arXiv.org/abs/astro-ph/0010185 .

Here I discuss a 2-space, the "2-torus". There are three different ways
(i)-(iii) of thinking about the same space. It's a good exercise to try
switching between the three ways and checking that you can understand
them as the same thing.

In (i) (lower figure), I use ordinary 3-space to help the reader imagine
what the 2-space is.

In (ii) and (iii), the reader does not need to think at all in 3-space.
2-space is enough, even though it may seem a little weird.

Going back up a dimension, it is possible to imagine various 3-spaces, 
either like (i) within a 4-space or like (ii) and (iii) just within 
ordinary 3-space, but a little weird.

> seeing repeated copies of a small universe (rather than a single very large 
> or infinite one). Your research centres around studying the sky to identify 
> repeated patterns of objects (celestial bodies and/or temperature signatures 
> in the microwave sky) to infer a size and shape for this "tiled" universe.

Correct. For more clarity, you might want to put "...studying the sky in
three dimensions to identify..."

BTW, when you say "tiled" Universe, you're referring to mode (iii) of
> Presumably, in order to decide in which directions, and at which distances, 
> to look for these repetitions, you must be working from a hypothesis about 
> the possible shape. For example, according to John Gribbin's New Scientist 
> article from that date, in 1997 you seemed to be looking for quasar patterns 
> to support a "twisted torus" or "3-Klein bottle" hypothesis. 

Nope. I'm a skeptical observer who works the other way around!  There
is no serious theory for the extension of general relativity which
would give a theory of global geometry, and even if there was, the
best bet would still be to make observations with as few preconceived
ideas as possible. The standard Big Bang model, by the way, is a
preconceived idea which I and my colleagues *do* assume, since it (and
the theory of general relativity) is (are) extremely well established

Rather than working from a specific hypothesis about the shape, my
general approach is on finding the best ways of using existing (or
near future) observational data catalogues. It happens that during
some of these projects, serendipitous candidate topologies have turned
up.  I think all of (1), (2) and (3) that I gave you in the previous
message are best described as serendipitous, though (2) comes from a
(slightly) more systematic approach than (1) and (3).

The candidate from my John Gribbin article was found using a very
systematic approach, but at the moment it is just sitting in my
"list-of-things-to-do". Although it was found systematically, I would
probably class its "subjective probability" as 5%. But my subjective
probabilities are... subjective. The only serious way to check
candidates is by further observational work.
BTW, the implicit 3-Klein-bottle-like model was what the observations
gave, not what I was looking for. I would have preferred to find a
3-torus than a 3-Klein-bottle-like model!

> Your e-mail suggests that your current research focuses on the use of galaxy 
> clusters and microwave patterns to identify a "model class: 2-torus". Are we 

Hmmm. I don't know if the following is too subtle for a general audience.
See how you like it.

I would prefer to say:

 My current research focuses on the use of galaxy clusters and
microwave patterns in which serendipitously discovered 2-torus models
offer good prospects of observational tests. Later on, if more
complicated models are offered as candidates, the experience in
testing 2-torus models will be important for testing the more
complicated models. After all, if it is not possible to
observationally test a 2-torus model, which is relatively simple, how
could we possibly hope to test a more complicated model?

Candidate (3) is even less ambitious: it involves just one generator,
or path between two images of (hypothetically) the same object.
[plus a modification of the term "2-torus" as per the comment below]

> right in assuming that this refers to an actual universe in the shape of a 
> (non-twisted?) 4D 3-torus, which is modelled as a 3D 2-torus?

OK, I used an abbreviation without explaining it, sorry! 

Firstly, as I explained above, the 3-torus is really 3D, and the 2-torus
is really 2D, even if an extra dimension is used as a psychological crutch.

The model I've called "2-torus" for my observational work on the real
Universe (as opposed to pedagogical explanations with one dimension
subtracted) is really the 3-torus, but one side length is considered
bigger than the horizon diameter, so big that observations made that
far away would require the light to have been emitted before the Universe
was born. 

If you called it a "3-torus with one very long side" or a "3-torus
with one presumably infinite side length" that would be correct, would
probably help avoid confusing readers, and would avoid experts in the
field getting confused about what I'm doing.
> We would also be interested in your views on other possible tessellated 
> polyhedra models. In particular, we are interested in any topologies which 

Well, again, I would prefer to say "other possible 3-spaces" or 
"other possible, multiply connected 3-spaces, which can be thought of
as tesselated polyhedra models".

> may be consistent with the recently emerging field of braneworlds. For 
> example, Michael R. Feltz's website - http://www.cyburban.com/~mrf/ - 
> identifies brane theory with the Riemannian hypersphere topology.
Well, I quite like the idea of non-experts writing web pages (if I
hadn't been job-hunting for the past few years I probably would have
made more effort to interact with non-specialists, and maybe 
sometime in the future I will be able to...), but I'm afraid there is
some confusion in Michael Feltz's page:

: The discussion here attempts to answer this question as it relates to
: the often hypothesized, but little explored, "finite but unbounded"
: universe. The formal name for this model in topology and cosmology is
: a "closed cosmic hypersphere".
There are many finite but unbounded models other than the hypersphere,
e.g. the 3-torus.

: Otherwise the days of an expanding universe are numbered because this
: high density model will expand only to certain maximum size and then
: contract into a "big crunch" ("big bang" in reverse) at some future
: date at least tens of billions of years from today.
Correct. And the geometry of *this* model is a 3-sphere, also known as
a hypersphere.

: The alternate model explored in this series of essays is the long
: suspected and often hypothesized "closed cosmic hypersphere" which
: incorporates a fourth spatial dimension.

The hypersphere provides one of the three possible curvatures, plus
the assumption of trivial topology. Calling it the "closed cosmic
hypersphere" is OK.

But it does *not* incorporate a fourth spatial dimension. A fourth,
psychological, dimension is just one possible way to think about it.

And it is not an "alternate" model. It is the model which (for reasonable
values of the local cosmological parameters, which show that the Universe
is "approximately" flat, like any continent on the Earth is "approximately" 
flat) will expand to a maximum size and then contract into a "big crunch".

Back to your question! My views on other possible "tesselated
polyhedra" models (and now you know they're physically only 3D, not
4D), are that I'm totally open to any of these multiply connected
models, whether for a flat Universe (where the angles of a triangle
add up to 180°, and this includes the "3-torus with a very long side"
models - this is a flat model!), for a spherical Universe
("hypersphere", where the angles of a triangle add up to more than
180°) or for a "saddle-shape" Universe ("hyperbolic", where than
angles of a triangle add up to *less* than 180°). Of course, in either
the spherical or the saddle-shape cases, the observable part of the
Universe would be approximately flat, again like any continent on the
Earth is approximately flat.

> Given that the readers of Astronomy are mainly non-scientists, we would be 
> very grateful if you could provide as simple an explanation as possible.

I know very little about brane-theory, but I think it is still much
too theoretical to have any serious links with observational work.  I
think the best link you could make between observational cosmic
topology work and brane theory could be something like the following:

 Scientists trying to measure the 3-dimensional shape of space (with a
type of non-expanding map of the Universe called "comoving" - although
in reality the Universe is expanding, in this special map, the
Universe can be thought of as static) sometimes use a psychological
4th dimension to explain or think about different possibilities for
the shape of space, e.g. a 3-space which seems to be tesselated by
many copies of the Universe.

Other, more theoretical scientists, working on "brane theory", think
that 4th and 5th dimensions might have real physical meaning (of
course, there's also time, making a 6-dimensional world). 

However, the scientists trying to measure the 3-dimensional shape of
space (also called the topology and curvature of space) prefer a more
conservative, purely empirical approach and would like to measure the
shape of 3-space without any preconceived notions, apart from the
standard Big Bang model. They hope to detect any of many possible
shapes of 3-space, whether it's the 3-torus with one very long side or
any other.

For more on brane theory, you might want to ask
Nathalie.Deruelle w obspm.fr, who gave a very nice explanation during a
workshop in Paris last year, but unfortunately I was too tired and
stressed to really listen properly :-(, though she did seem to give a
good explanation. She's definitely an expert in brane theory, and is
very supportive of work in cosmic topology. And she's probably one of
the best people around who can make an intelligent comment on whether
there's any link between the two.
> Many thanks once again for your help,

My pleasure :-) - feel free to keep passing draft texts to me for comment
and/or other questions. 


boud w astro.uni.torun.pl
Torun Centre for Astrophysics, University of Nicolas Copernicus, Torun
(affiliation by the time the article is published!)


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