``Only two things are infinite, the universe and human stupidity - and I'm not sure about the former.'' - Albert Einstein

- On large scales, the universe is apparently homogeneous and isotropic.
- The universe could be positively or negatively curved, but is probably flat.
- The large scale curvature of the universe is determined by its density.

First assumption of cosmology: the universe is **homogeneous**
on large scales. ``Homogeneous'' merely means that every region
of the universe is pretty much the same as every other region;
there are no special locations. On small scales, the universe
is obviously very **inhomogeneous**; it's full of lumps.
Stars are much denser than the interstellar medium. Galaxies are
much denser than the intergalactic medium. Even superclusters are
denser than the voids between them. HOWEVER, on scales larger
than superclusters and voids (> 100 Mpc), the universe is finally
homogeneous. The average density of stuff within a sphere of
radius 100 Mpc is the same as the average density of any other
sphere of the same size.

Second assumption of cosmology: the universe is **isotropic**
on large scales. ``Isotropic'' merely means that every direction
in the universe is pretty much the same as every other direction;
no matter which way you look, you see the same view. On small
scales, the universe is obviously anisotropic; there exist preferred
directions. Look down & you see rock; look up & you see sky. Look
toward the Virgo cluster & you see lots of galaxies; look away from
the the Virgo cluster & you see fewer galaxies. HOWEVER,
on scales larger than superclusters and voids (> 100 Mpc), the
universe is finally isotropic.

Another assumption of cosmology (and of all fields of science) is that
the laws of physics are **universal**; that is, they
are the same everywhere in the universe. For instance, we assume
that Kepler's Third Law applies to a binary galaxies millions of
light years away just as well as it applies to planets within
the Solar System.

Combine the assumptions of homogeneity, isotropy, and universality, and you have the

Consider the analogy of an ant wandering
over the surface of an orange. The ant will encounter small
local dimples (the pores of the orange), but if the ant
wanders far enough, it will discover that the orange is
spherical on average.

On large scales, there are three
possibilities for the average curvature of space.

First possibility: Space is

The two-dimensional analog for flat space is a plane (illustrated below).

On a plane, and in flat space, the standard laws of plane geometry apply: for instance, the sum of the vertices of a triangle equals 180 degrees. A plane has infinite area; similarly, flat space has infinite volume.

Second possibility: Space has

The two-dimensional analog for positively curved space is a sphere, illustrated below.

On a sphere, and in positively curved space, the laws of plane geometry no longer apply: the sum of the vertices of a triangle, for instance, is greater than 180 degrees. A sphere has a FINITE area; similarly, positively curved space has FINITE volume (but no edge).

Third possibility: Space has

The two-dimensional analog for negatively curved space is a saddle shape (called a hyperboloid by mathematicians), illustrated below.

On a hyperboloid, and in negatively curved space, the laws of plane geometry don't apply: the sum of the vertices of a triangle, for instance, is less than 180 degrees. A hyperboloid has an INFINITE area; similarly, a negatively curved space has an INFINITE volume.

So what IS the curvature on large scale? It must be one of the three possibilities, but which?

It's hard to tell, since we see only a limited volume within our cosmic particle horizon. It's comparable to the difficulty that early cultures had in determining that the Earth was spherical -- positively curved -- rather than flat. Actually, it's even worse than you might think, since the local curvature due to stars, galaxies, clusters, and superclusters tends to mask the global positive or negative curvature. (Imagine trying to determine the curvature of the Earth if you were confined to Switzerland. The local curvature, due to the Alps, would totally swamp the global curvature due to the Earth's spherical shape.)

The most promising technique for determining the curvature
of the Earth involves looking at the angular size of
very distant objects, such as ``hot spots'' in the
Cosmic Microwave Background. In flat space, light
from the hot spots travels along straight lines.
In positively curved space, though, light travels
along converging lines. This has the effect of making
the hot spots look **larger** than they
would in flat space. Conversely, in a negatively
curved space, the hot spots would look **smaller**
than they would in flat space. As it turns out,
the actual size of hot spots (about one degree across)
is just what cosmologists would have expected in
a flat universe. Precise measurements lead cosmologists
to conclude that the universe is **flat**,
and thus has infinite volume. (We can't rule out,
however, the possibility that the universe has
a tiny amount of positive curvature, leading to a universe
whose volume is finite, although very very much
larger than the volume within our cosmic particle
horizon.)

The mass of clusters of galaxies can be determined
by the application of Kepler's Third Law to galaxies
within the cluster. A census of all the clusters of
galaxies within a few hundred Mpc of us leads to
the conclusion that the mass in clusters only amounts
to 30 percent or so of the critical density. What
provides the rest of the mass (or energy) required
to flatten the universe? The Cosmic Microwave Background,
although it has no mass, has energy. Every cubic meter
of space contains about 400 million CMB photons. The
total energy of all those photons contributes 4 x 10^{-14}
joules per cubic meter. If you divide this energy
density by c^{2} to find an equivalent mass
density, it comes to merely 5 x 10^{-31} kilograms
per cubic meter, only 0.005 percent of the critical
density. The Cosmic Microwave Background doesn't contribute
significantly toward flattening the universe. (And
starlight doesn't help, either; all the light emitted
by stars during the past 14 billion years has an average
energy density less than the Cosmic Microwave Background.)

Where is the rest of the mass (or energy) hidden? [Come back next week for a further discussion of the mysterious hidden energy (or mass)...]

Prof. Barbara Ryden (ryden@astronomy.ohio-state.edu)

Updated: 2003 Mar 6

Copyright © 2003, Barbara Ryden