A friend asked about the geometry of the universe, and Rob Sheldon replied, saying News could quote him:
On a positive curvature surface, the sum of the interior angles of a triangle > 180.
On a negative curvature surface, the sum of the interior angles of a triangle < 180.
On a flat surface, e.g., a sheet of paper, the sum of the interior angles of a triangle = 180.
Now a small triangle drawn with ball point pen on a sufficiently large balloon will look flat, the angles add up to 180. But if the triangle is made bigger and bigger, the angle sums start to get more and more obviously different, so if you draw from the north pole to the equator, and then 90 degrees of latitude along the equator, and then back to the north pole, you will have drawn a triangle with 3 right angles = 270 degrees. So in order to determine the curvature of the universe, one needs to use distant objects—quasars and galaxies and suchlike, to determine the angles.
When one does this, the universe appears remarkably flat. It takes a lot of effort to find any curvature at all, and certainly it is difficult to get good agreement between different types of measurement.
What curvature implies
Philosophically, of course, a “closed” universe that collapses back down to itself solves the problem of infinite time, and so one would like it to have positive curvature to avoid infinities. Negative curvature suggests an “open” universe that will expand forever, ending “not with a bang, but a whimper”, and gives a feeling of the infinite emptiness of existence. I’m not really sure what a “flat” universe portends, perhaps a feeling of driving the speed limit across Kansas.
Most astronomers (e.g. experimentalists) find that the universe is flat, though there are some arguments as to why it should have a very, very slight negative curvature. The Nobel prize was awarded 2 years ago for a claim that some observed supernovae are further away than the flat universe predicts, and therefore the expansion rate is accelerating. Of course, this claim requires a model to predict what the expansion rate was at various epochs in the past, and that model has all sorts of assumptions. Personally, I think the Nobel Prize was a big mistake. But it shows that experimentally it is really difficult to find absolute distances between galaxies and quasars, so that most of our models are only weakly supported by data.
But even if we all agree that the universe is flat, we still don’t know what is causing it to be so flat. That is, gravity is always attractive, so it will make a positive curvature universe, collapsing down to a point, unless something else were counterbalancing it. The Big Bang is outward kinetic energy, and that gives a negative curvature. So in computing the sum of gravitational and kinetic energy, which one wins? Well, we can measure the expansion rate—the red shift of distant galaxies—and we can measure the number of stars in the volume from here to a distant galaxy to compare the gravitational attraction to the Big Bang explosion rate. According to these calculations, only about 10% of the mass needed to balance the kinetic energy is visible, is in stars. We can add in red dwarfs, dust and molecular clouds, and we get up to about 30%. But roughly 70% of the necessary mass to “flatten” the universe is invisible, is unaccounted for.
a) In the present epoch, as in right now, the universe is experimentally as flat as can be measured—the “fine-tuned Big Bang” problem.
b) About 70% of the matter that is needed to make it flat is unobserved—the “Dark Matter” problem.
c) There is some controversial data that the universe is becoming negatively curved in the next epoch. The unknown cause of this “anti-gravity” is called the “Dark Energy” problem.
d) “Inflation” is not occurring now, but is a strong “negative curvature” term from a previous epoch that presumably solves problem (a).
But it introduces a whole new set of “Fine-Tuning” parameters into the model which are even more contingent than the ones it was meant to replace! Inflation also interacts with (c) in ways that cause trouble. Nor does it explain (b) at all. So by and large, I would ignore “inflation” as a solution for anything, since it merely “solves” one problem by introducing a dozen more.
BTW, the hope that the search for the Higgs boson would reveal novel physics needed by the “inflaton”-field were dashed, making inflation a fast-receding threshold of confirmation.
Advice on books
If you stick to observational astronomy books, they will all talk about a flat universe.
Cosmologists tend to be faddish thrill-seekers, and will tell you anything. The past 30 years of publishing has been particularly brutal, with nearly every cosmological model having a shelf-life in single digits. The “flat” cosmology is almost apophatically defined—as one cosmology after another is denied. Perhaps there is something religious about cosmology that prevents the “flat” universe from being acceptable theology.
Big Bang exterminator wanted, will train
Copernicus, you are not going to believe who is using your name. Or how.
As if the multiverse wasn’t bizarre enough …meet Many Worlds
Note: Yes, that’s Mars up there beside “Now a small triangle …” Useful as any sphere for picturing the cosmology.
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