

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 saddleshaped surface
is said to have negative curvature.
Curvature of 4Dimensional 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 spacelike 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 2dimensional surfaces
that we can visualize easily.
The LargeScale 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 noncurved 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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