Geometry of the Universe:

Can the Universe be finite in size? If so, what is ``outside'' the Universe? And is there a center of the Universe? The answer to these questions involves a discussion of the intrinsic geometry of the Universe.

General Relativity describes gravity as a warping or distortion of space and time near a massive object. In General Relativity, four-dimensional spacetime is curved. There are basically three possible shapes to the Universe; a flat Universe (Euclidean or zero curvature), a spherical or closed Universe (positive curvature) or a hyperbolic or open Universe (negative curvature). A high mass density Universe has positive curvature, a low mass density Universe has negative curvature.

To help you understand what curved spacetime means, let's use the analogy of a two-dimensional world curving into the third dimension. Pretend you are confined to the surface of a balloon and you only know about ``front'', ``back'', ``left'', and ``right'', but not ``up'' and ``down''. In your 2D universe you cannot see the third dimension. Your universe appears flat. Yet you know that your 2D universe must be curved because if you walk in a straight line, you eventually arrive back at where you started! The balloon universe has a finite size but no edge. You also know that the angles of large triangles add up to a number larger than 180°! For example, on the balloon the lines of longitude running north-south intercept the equator at a 90° angle and converge at the poles. So a triangle made of one point on the equator + the north pole + another point on the equator will have the angles add up to more than 180°. In a truly flat universe, the angles would add up to exactly 180°. You would be able to deduce that your universe is positively curved.

No Center to the Expansion in 3-D Space:

The idea of a curved surface also explains why astronomers in every galaxy will see the other galaxies moving away from it and, therefore, derive the same Hubble Law. Go back to the balloon analogy, imagine that there are flat houses on it. As the balloon expands, the elastic material moves the houses apart from each other. A person sitting on their front porch see everybody else moving away from her and she appears to be the center of the expansion.

Now add another dimension and you have our situation. Just like there is not new balloon material being created in the 2D analogy, new three-dimensional space is not being created in the expansion. Like any analogy, though, the balloon analogy has its limits. In the analogy, the balloon expands into the region around it---there is space beyond the balloon. However, with the expanding universe, space itself is expanding in three dimensions---the whole coordinate system is expanding. Our universe is NOT expanding ``into'' anything ``beyond''.

Mass Density and the Geometry of the Universe:

The description of the various geometries of the Universe (open, closed, flat) also relate to their futures. There are two possible futures for our Universe, continual expansion (open and flat), turn-around and collapse (closed). Note that flat is the specific case of expansion to zero velocity (recall the newtonian orbits case as a function of escape velocity).

The key factor that determines which history is correct is the amount of mass/gravity for the Universe as a whole. If there is sufficient mass, then the expansion of the Universe will be slowed to the point of stopping, then retraction to collapse. If there is not a sufficient amount of mass, then the Universe will expand forever without stopping. The flat Universe is one where there is exactly the balance of mass to slow the expansion to zero, but not for collapse.

The parameter that is used to measure the mass of the Universe is the critical density, Omega. Omega is usually expressed as the ratio of the mean density observed to that of the density in a flat Universe.

Given all the range of values for the mean density of the Universe, it is strangely close to the density of a flat Universe. And our theories of the early Universe (see inflation) strongly suggest the value of Omega should be exactly equal to one. If so our measurements of the density by galaxy counts or dynamics are grossly in error and remains one of the key problems for modern astrophysics.

Cosmological Constants:

The size, age and fate of the Universe are determined by two constants:

The measurement of these constants consumes major amounts of telescope time over all wavelengths. Both constants remain uncertain to about 30%; however, within this decade we can expect to measure highly accurate values for both due to the Hubble Space Telescope and the Keck twins.