The Geometry of the Universe

The most profound insight of General Relativity was the conclusion that the effect of gravitation could be reduced to a statement about the geometry of spacetime. In particular, Einstein showed that in General Relativity mass caused space to curve, and objects travelling in that curved space have their paths deflected, exactly as if a force had acted on them.

### Curvature of Space in Two Dimensions

The idea of a curved surface is not an unfamiliar one since we live on the surface of a sphere. More generally, mathematicians distinguish 3 qualitatively different classes of curvature, as illustrated in the following image (Source):

These are examples of surfaces that have two dimensions. For example, the left surface can be described by a coordinate system having two variables (x and y, say); likewise, the other two surfaces are each described by two independent coordinates. The flat surface at the left is said to have zero curvature, the spherical surface is said to have positive curvature, and the saddle-shaped surface is said to have negative curvature.

### Curvature of 4-Dimensional Spacetime

The preceding is not too difficult to visualize, but General Relativity asserts that space itself (not just an object in space) can be curved, and furthermore, the space of General Relativity has 3 space-like dimensions and one time dimension, not just two as in our example above. This IS difficult to visualize! Nevertheless, it can be described mathematically by the same methods that mathematicians use to describe the 2-dimensional surfaces that we can visualize easily.

### The Large-Scale Geometry of the Universe

Since space itself is curved, there are three general possibilities for the geometry of the Universe. Each of these possibilites is tied intimately to the amount of mass (and thus to the total strength of gravitation) in the Universe, and each implies a different past and future for the Universe:
• If space has negative curvature, there is insufficient mass to cause the expansion of the Universe to stop. The Universe in that case has no bounds, and will expand forever. This is termed an open universe.
• If space has no curvature (it is flat), there is exactly enough mass to cause the expansion to stop, but only after an infinite amount of time. Thus, the Universe has no bounds in that case and will also expand forever, but with the rate of expansion gradually approaching zero after an infinite amount of time. This is termed a flat universe or a Euclidian universe (because the usual geometry of non-curved surfaces that we learn in high school is called Euclidian geometry).
• If space has positive curvature, there is more than enough mass to stop the present expansion of the Universe. The Universe in this case is not infinite, but it has no end (just as the area on the surface of a sphere is not infinite but there is no point on the sphere that could be called the "end"). The expansion will eventually stop and turn into a contraction. Thus, at some point in the future the galaxies will stop receding from each other and begin approaching each other as the Universe collapses on itself. This is called a closed universe.
Which of these scenarios is correct is still unknown because we have been unable to determine exactly how much mass is in the Universe.

### Is the Universe Open, Flat, or Closed?

The Density Parameter of the Universe
Source Value
Baryons (BB nucleosynthesis) (0.013 +/- 0.005) h-2
Stars in Galaxies 0.004
Intergalactic Stars <0.04
Rich Clusters 0.01
Dynamics (r < 10 h-1 Mpc) ~0.05 - 0.2
Dynamics (r > 30 h-1 Mpc) ~0.05 - 1
Source: P. J. E. Peebles, Principles of Physical Cosmology

The geometry of the Universe is often expressed in terms of the density parameter, which is defined to the the ratio of the actual density of the Universe to the critical density that would just be required to cause the expansion to stop. Thus, if the Universe is flat (contains just the amount of mass to close it) the density parameter is exactly 1, if the Universe is open with negative curvature the density parameter lies between 0 and 1, and if the Universe is closed with positive curvature the density parameter is greater than 1.

The density parameter determined from various methods is summarized in the adjacent table. In this table, BB nucleosynthesis refers to constraints coming from the synthesis of the light elements in the big bang, +/- denotes an experimental uncertainty in a quantity, and the parameter h lies in the range 0.5 to 0.85 and measures the uncertainty in the value of the Hubble parameter.

Although most of these methods (which we will not discuss in detail) yield values of the density parameter far below the critical value of 1, we must remember that they have likely not detected all matter in the Universe yet. The current theoretical prejudice (because it is predicted by the theory of cosmic inflation) is that the Universe is flat, with exactly the amount of mass required to stop the expansion (the corresponding average critical density that would just stop the is called the closure density), but this is not yet confirmed. Therefore, the value of the density parameter and thus the ultimate fate of the Universe remains one of the major unsolved problems in modern cosmology.

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