Tim Thompson

FYI, I have updated my own webpage on the cosmic microwave background, to include the WMAP results and links to all of the preprints submitted thus far. They will probably appear together, eventually, in a special issue of the Astrophysical Journal, devoted to the WMAP results (this is what happened in 1991 when the COBE results came out).

COBE was the first all sky CMB mapper, and it was not known until COBE that the CMB was in fact thermal (though by then it was generally assumed that it would be). WMAP is only the 2nd all sky CMB mapper, a decade after COBE. There have been several other high resolution CMB experiments, all either ground based or balloon based, and all observing only very small portions of the sky. The real significance of WMAP is that it has mapped the entire sky, and, like COBE, will continue to build overlay maps, increasing its sensitivity.

I think that the real significance of the results from WMAP is that they support what has come to be called the "concordance cosmology". The CMB observations from all of the high-res experiments, including WMAP, clearly indicate a Hubble constant around 71, the presence of a considerable amount of non-baryonic dark matter, and the presence of a non-zero cosmological constant which implies that the expansion of the universe should be accelerating. But the same results are also derived from the high redshift supernova data, and from the HST Key Project data. All of the cosmological indicators now point towards the same model for the universe, and that I think is a first.

The WMAP derived age of the universe is of particular interest (13.7 +/- 0.2 billion years, according to the cosmological parameters paper). But the best fit age for the universe, based on the median age of old globular cluster stars (about 12.6 billion years) is 13.4 billion years, which nicely matches the WMAP results (see the January 3, 2003 issue of Science, and the cover story features on globular clusters).

I think it is quite interesting how our understanding of the CMB has changed dramatically, since 1989, when COBE launched. Before that, it was almost ephemeral, subject to wide ranging interpretation. But now, after COBE, WMAP, and other experiments, we see the CMB in exacting detail, not as something "ephemeral", but as a concrete, well measured quantity in the universe.

With that in mind, I look forward to the same thing happening, over say the next 20 years, for gravitational wave astronomy. We understand gravitational waves in the same "ephemeral" sense as we did the CMB. But once the new gravitational wave observatories (like LIGO or LISA) come online, it won't take too long for some kind of definitve indication to come out (though it may take the next generation of instruments to be sure about it). Then we well see gravitational waves become concrete & well measured.

Shadowy Man

Hehe... actually to most astronomers those experiments observed large portions of the sky!

Shake

Shadowy Man, Demosthenes, Tim Thompson, yes that makes sense and helps a lot. Very informative. I'm looking over the many links you all have provided now.

wade-w

So this means that the geometry of spacetime is Euclidean?

Answerer

Guys, I have a similar question too. Since the universe is flat, does it mean that we have a total curvature of zero (or close to zero) or not?

Demosthenes

I think it's close to the value of zero, but I'm not sure. Why don't you look it up yourself?

Friar Bellows

A flat universe has by definition a curvature (everywhere, on the large scale) of zero. But it's important to note that curvature is a local concept. One can have (mathematical) spaces where the curvature varies from point to point. To talk about the "shape of the universe" is really to talk about topology, which is a global concept. Topology doesn't care too much about curvature. As an often-used example, a coffee mug is topologically equivalent to a dougnut. Topology is concerned with those properties of space which are insensitive to continuous transformations like stretching, squeezing and "kneading" -- but not tearing, cutting or making holes. Sort of like a frigid pacifist.

For example, take a flat piece of paper and draw a triangle on it. Measure the interior angles of such a triangle, then add them up and find the result to be 180°. No big surprises. Now join one edge of the paper to its opposite edge to form a cylinder. What about our triangle now? Do the interior angles still add up to 180°? If you answered yes, then you are right. The curvature has remained unchanged (i.e. zero everywhere), but the topology has changed: from a flat piece of paper to a cylinder (from R^2 to S^1 x R, I think the mathematicians would say).

Now take the cylinder and join the ends together. What you have now is a torus (i.e. a doughnut). Embedded in 3D Euclidean space, the surface's curvature is no longer zero everywhere. In some places it has positive curvature, and in other places it has negative curvature. But there are ways to create a torus which preserve a zero curvature over the entire surface. This flat torus can't be embedded in 3D Euclidean space, but it can in other spaces (like 4D Euclidean space, I think, but don't trust me on that, my memory is hazy).

The torus is an example of a topology which is compact (i.e. finite area or volume) and multiply connected (i.e. there exist closed loops which cannot be continually contracted to a point). Other examples of multiply connected 2D topologies are the Möbius strip and the Klein bottle. Look them up on Google if you don't know about them: you'll be impressed, I think.

But enough of 2D! Let's go up a dimension. In 3D, the possibilities multiply. The simplest example of a multiply connected flat 3D space is the following. Take a cube. Join the front face to the back face. Then join the right face to the left face. Finally, join the top face to the bottom face. What you've created is a hypertorus. You can't embed it in 3D Euclidean space, so stop trying to visualise it! Actually you can sort of visualise it with this pictorial metaphor: think of an infinite cubic lattice where each cubic cell is an exact physical copy of every other cubic cell. That's it. But don't get confused by or too hung up on this metaphor: the hypertorus has a finite volume. And it's flat, everywhere.

If the (spatial) universe was a hypertorus, what would an observer see? If the universe were old enough, then he would see multiple copies of astronomical objects in different areas of the sky and at different redshifts (distances). In fact, WMAP can be used to detect the topology of our universe. It may be, though, that our topology is on too large a scale to be detected, now or even forever. That would be sad. But observations are our only guide at the moment, since the general theory of relativity has nothing to say about the topology of our universe. Maybe a more complete theory will have something to say.

(Warning: The preceding paragraphs probably contain a few factual and/or conceptual errors, possibly many. The author is not a scientist or a mathematician and now gladly cedes the floor to those who are. You learn a lot by saying stupid things or asking stupid questions. )

fando

Well, the concept of curvature of spacetime is different from curvature of just space. By closed universe, we mean that spacetime expands then converges. All that means is that two photons travelling diametrically opposite to each other will again meet at the end of the universe when spacetime converges. In a flat or open universe, these photons will never see each other again. It's as simple as that. There is no contorted geometry to think about in open vs. closed vs. flat if all you want is the Ultimate Fate Of The Universe. Moreover, I don't think there is any good solution for being able to see the back of your head in General Relativity (no hypertorus solution for space in finite time), but I haven't done the math or checked my notes.

Jesse

Don't the versions of string theory which postulate "rolled-up" extra dimensions (apparently not all of them do anymore--see this article) suggest that the universe has the topology of a type of hypertorus?

fando

Now that you mention it, maybe there are solutions for seeing the back of your head. It's true in string theory. I haven't thought too much about that in the typical 4D spacetime of General Relativity. Rolled-up or not, the regular dimensions are still stretching and the heat death seems inevitable according to the recent CMB measurements.