Astronomy
162: Professor Barbara Ryden
``Only two
things are infinite, the universe and human stupidity - and I'm not sure about
the former.'' - Albert Einstein
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 cosmological
principle. The cosmological principle states that on large scales
(> 100 Mpc), the universe looks the same from every vantage point. In
particular, we here on Earth are not in a special position in the universe
(shades of Copernicus!) The cosmological principle, if true, implies that the
universe cannot have an edge or a center. An observer at the edge of the
universe would see a very different view from an observer at the center of the universe,
thus violating the cosmological principle.
Einstein
told us, in his theory of General Relativity, that on small scales, space is
``dimpled'' by massive objects such as stars, galaxies, or clusters of
galaxies. On large scales, however, where the assumptions of homogeneity and
isotropy apply, space must have the same average curvature
everywhere.
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 FLAT
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 POSITIVE curvature.
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 NEGATIVE curvature.
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.)
General
relativity relates the curvature of space (and of time) to the amount of mass
(and energy) in the universe. Space is flat if the density of mass (plus energy
divided by c2) is equal to a value known as the critical
density. (It's negatively curved if the density is lower, and
positively curved if the density is higher.) In an expanding universe like our
own, the critical density depends on the expansion rate. The mathematical
relations of general relativity reveal that the critical density is
proportional to the square of the Hubble constant H0. For the
measured value, H0 = 70 km/sec/Mpc, the critical density required
for the universe to be perfectly flat turns out to be only 9 x 10-27
kg/meter3. The critical density is equivalent to one hydrogen atom
for every 180 liters of volume. This doesn't seem like much, but remember that
most of the universe consists of empty intergalactic voids. Dense regions like
our galaxy take up a small fraction of the total volume.
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 c2 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)...]
Updated: 2003 Mar 6
Copyright © 2003, Barbara Ryden