Statements about cosmology should be based on science. All ontological arguments aside, it's becoming more and more apparent that the universe is indeed infinite, with an inifnite amount of matter and energy. Scientists are led to this by current theories of physics and astronomical observations.

After I originally posted this, I argued this subject back and forth via /msg with various noders for several weeks. I was forced to go back and make sure that I remembered the cosmology I learned so long ago correctly. It turns out I had, with a big *however* at the end.

In the early 20th century, Albert Einstein's theory of General Relativity showed that matter curves the space around it. Our idea of space in the Universe as a whole is a 3-manifold embedded in a larger spacetime. This is really a matter for another node^{1}, but we will touch on it later.

Meanwhile, one interpretation of Einstein's theory was that the universe was continually expanding (which Einstein disliked so much that he threw a fudge factor called the "cosmological constant" into the theory to get rid of it). The universe's expansion became widely accepted after Edwin Hubble's observation of red shift in almost every other galaxy in the late 1920's, and confirmed by the observation of cosmic microwave background radiation of Arno Penzias and Robert Wilson in the early 1960's.

So, physicists were led to one of two possibilities:

- Either matter in the Universe is dense enough to slow down the expansion, make it reverse itself, and contract and eventually cause a Big Crunch (a closed universe);
- or it is not, in which case the Universe will keep expanding forever (an open universe as well as a flat universe).

The density of matter in the Universe determines not only its fate, but also its size. Strangely enough, it is an *inverse* relation.

- A closed universe has a finite volume, like the finite surface area of a balloon. Consequently, it can hold only a finite amount of matter. Not only that, the denser matter is in a closed Universe, the sooner the Big Crunch will come. In other words, the denser the matter is, the less matter there is overall! An analogy may help you understand this better: We can observe a closed universe from the outside, as a black hole in another universe that contains it. Now, Stephen Hawking showed that black holes evaporate due to a strange process of quantum physics now called Hawking radiation. The less matter in a black hole, the smaller it is, and the sooner it will evaporate.
- An open universe is
*infinite* in extent, and was infinite as soon as the Big Bang happened. Now, matter has to be (more or less) evenly distributed throughout the universe. Otherwise, the Universe contains a finite "speck" of matter with an infinite void around it (if there is more than one, each is an infinite distance from the other, effectively putting each in its own infinite open universe). Spacetime isn't an object in and of itself, it is a structure built from the interactions between its contents. An infinite void with nothing in it simply can't exist. Put another way, the laws of physics^{2} would have had to be different in the vicinity of the "speck" for it to be created, violating the concept of Universality, one of the basic assumptions underlying the theory of relativity. We must conclude that an open universe contains an inifite amount of matter.

Unexpectedly, it has been recently observed that the expansion of the Universe is speeding up. The cosmological constant has been resurrected as an idea, but we are also forced to discard the possibility of a closed universe. Without some new physics, we will have to accept the idea of a forever-expanding, open, infinite universe.

There are two flies in our ointment that may lead to "new physics":

- The theory of the Big Bang has actually been replaced with one of cosmic inflation, which has led some to the notion that the Universe we observe arose from a quantum fluctuation in some other primoridal outer universe that got out of hand. So, we may not understand the structure of our Universe completely. I'd like to believe that the topology of spacetime is conserved, and our Universe is connected somehow to the one that spawned it. This would definitely lead to a localized area with "different laws". We are then left with the conundrum of what the "outer universe" is like, and whether there is an infinite regress or not.
- More recent astronomical observations have postulated that the fine-structure constant had a slightly different value billions of years ago, soon after the Big Bang. This variance of the laws of physics through time does not mean they vary through space at a particular point in time, but it opens the door to possibility.

There is an important valid objection to the picture I've painted above, which has to be addressed: A universe that expands forever doesn't necessarily have to be infinite in extent to start with (if it wasn't to start with, it never will be). If we delve into the subject matter of Which 3-manifold do we live in? and Which 4-manifold do we live in?, 3-D slices of the Universe for particular points in time could possibly be compact (= finite for our purposes) even if the slices extend forever. It is possible that we are embedded in some sort of 4-dimensional hypercone or hypertrumpet.

All of that said, the standard model of cosmology still portrays an open universe as spatially infinite at any given point in time. According to General Relativity, the curvature of space (determined by the matter embedded in it) controls its shape. A closed universe has negative curvature; an open universe has zero or negative curvature.

While our hypertrumpet and hypercone have negative curvature in timelike directions, they both have positive curvature in spacelike directions. An infinite volume is still the only 3-manifold with zero curvature (and an infinite hyperbolic paraboloid is still the only manifold with negative curvature) explained by General Relativity alone.

**HOWEVER**

It turns out that compact hyperbolic manifolds (such as the Weeks Space) exist. A 3-torus would work in a flat universe. But then, we would have to live in a universe with a really funky topology (I should have read Which 4-manifold do we live in? first). However, results from the Wilkinson Microwave Anisotropy Probe seem to indicate that the universe is *flat*.

Another objection: An infinite amount of matter will result in an infinite gravitational pull. There are a couple of possible ways out of this, all hinging on the observed fact that the Universe is expanding:

- If gravity behaves in a quantum manner similar to the electromagnetic force, with gravitons that behave like photons, then some analogy of redshift applies to gravity. In a way, the "infinite gravity" problem is similar to Olbers' paradox, which of course is resolved by redshift.
- If, as a few physicists think, gravity is transmitted instantaneously, it still has to overcome the initial "force" of the Big Bang, not to mention whatever is making the expansion speed up. Since an infinite universe is an open or flat Universe, we already know it's flinging itself apart too fast for gravity to overcome.

^{1}Please read Which 3-manifold do we live in? and Which 4-manifold do we live in?

^{2}By the "laws of physics", cosmologists usually mean the values of certain "magical" fundamental constants, rather than its underlying mathematical basis.