Measuring the curvature of the universe

Factual Questions
1

I was reading the book “Poetry of the Universe” by Robert Osserman last night, and a discussion on non-euclidean geometry suggested to me a way to experimentally measure the curvature of the universe.

Imagine the universe to be a two-dimensional sheet of paper, with the earth as a point in the center. If the surface has zero curvature, it will be flat like a normal piece of paper. If it has positive curvature, it will be dome-shaped, like the sliced-off end of a sphere. If it has negative curvature, it will be saddle-shaped.

Now for a few assumptions: First, we’ll assume that stars (or, more likely on the scale the thought experiment will require, galaxies) are roughly evenly distributed through the cosmos, if we pick a large enough scale. Second, I’m assuming a static universe. I’m not sure how finite age and expansion would affect the calculations. Third, I’m assuming that we can make a small-scale survey of the density of stars (or galaxies) where the curvature of the universe is negligible, and use that density over much larger scales where the curvature becomes important.

So, on our paper universes, we’ll begin by drawing circles of a fixed radius centered on the earth. The circumference of the circle on the surface with positive curvature will be smaller than that of the circle on the flat surface, which will be smaller than that of the circle on the surface of negative curvature. If stars (or galaxies) are located, on average, every x units throughout space, there will be the smallest number of stars along the circumference of the circle on the surface with positive curvature, and the larges number of stars on the circumference of the circle on the surface with negative curvature. In essence, there is more “space” for the stars to be in on surfaces with negative curvature.

So the universe should be brighter than one would expect if it is negatively curved – there would be more stars a given distance away than in a flat universe – and a positively curved universe should be dimmer.

Could this experiment work to measure the curvature of the universe? It seems like such an obvious property of a curved universe that it must have been conceived and tried if it is possible. Has it been proposed or carried out and I am just not aware of it? If not, is it because of the assumptions? Is the matter in the universe not sufficiently evenly spread? Is it not possible to compensate in the calculations for an expanding universe of finite age? Is it not possible to make a small-scale survey to determine what the large-scale brightness should be, and then apply it? Chronos, Bad Astronomer, Dr. Matrix, I’m looking to you for this one.

-b

2

I can’t answer your question but if I might I think there is a simpler way to do this. Just get a few laser beams and setup a nice equilateral triangle (or right triangle or any triangle for that matter). If the universe is curved the angles should not add-up quite right.

Why hasn’t this been done? Not sure but I think the issue is that the universe is so big that you’d need to separate your lasers by a HUGE amount of distance (say light years but I’m just guessing) to get a measurable effect.

3

I think your reason is probably correct: the experiment hasn’t been done because it is not currently technologically feasible to separate the lasers by a large enough distance. Travelling on the Voyager probes, they’d probably take about 10,000 years to get far enough away. It’s hard enough to get funding for five years. :wink:

-b

4

Well, we think that matter in the Universe is relatively uniformly distributed, but only on very large scales. By the time that you get to those scales, curvature is significant, so we can’t get a good baseline for the number of galaxies without first knowing the curvature. We could use a method somewhat like this if we knew the distances to all of the objects we observe, but again, all of the methods we have for measuring distance depend on the curvature of the Universe. If you combine different methods of measuring the distance to something, then you can in principle solve for the true distance and the curvature, but errors in your measurements usually swamp out the results.

Of course, it’s not completely impossible, and what can be done, is done. Cosmologists are currently looking forward to the results of the MAP mission, which should give us a very good handle (relatively speaking) on the various cosmological parameters, including curvature. The basic idea there, is that the satellite will make a map of the cosmic background radiation, and measure the apparent size of fluctuations in the background. We think we have a pretty good idea of what the true sizes of those structures should be, and by comparing the true sizes to the apparent sizes, we should be able to learn a lot.

By the way, it’d take a lot longer than a measly few thousand years for the Voyagers to get out far enough. The curvature scale is at least billions of light years, and the Voyagers aren’t travelling at anywhere near the speed of light.

5

First, cosmology always makes my head hurt. Part of the problem is that the universe seems to be very nearly flat. So any curvature, either positive or negative, will be very hard to measure. Figuring out exactly how many super clusters should be observed per unit radian will depend closely on the exact age of the universe and its rate of expansion, numbers which aren’t perfectly known.

Second, and more fundamentally, I think you’ve got it backwards. In your hypotheses, you’re looking “out” along the ray towards the “horizon.” In fact, you’re looking backward in time towards the big bang. In other words, you’re standing on the “horizon” looking back to the “center” when the universe was much smaller.

I’m also not certain that the curvature of the universe will make any difference to the number of super clusters per unit radian. Think of it this way. You want to count the number of galaxies in a one degree cone reasoning that if the universe is open, there will be a greater volume of space in the cone and, therefore, a greater number of galaxies. However, what’s happening is that the galaxies are expanding into that space. So, though it is true that at specific distance from the observer there would be more space in an open universe than a closed universe, there would be the same amount of matter in that space – not per unit volume, but per unit radian. There would be more space between galaxies in an open universe than in a closed one. However, you wouldn’t able to tell that because your sight-line would be be curved, just like the path of the galaxies that expanded into that space. To put it another way, the increase in volume at a given distance would be exactly cancelled by the decrease in galactic density. It would be a wash . . . I think.

6

Truth Seeker: I think cosmology makes everybody’s head hurt. :slight_smile:

Chronos: Thanks for the info. Thought I might’ve found a research subject for grad school for a minute there. :slight_smile: Also, the 10,000 years I mentioned was just for the Voyagers to get the few light-years away Whack-a-Mole suggested (I’m assuming his experiment uses uber-sensative lasers). I completely agree that actually performing the experiment would require much greater distances, and therefore a much longer time.