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Section

Part 1: Observations
of Global Properties
Part 2: Homogeneity and
Isotropy; Many Distances; Scale Factor
Part 3: Spatial Curvature;
Flatness-Oldness; Horizon
Part 4: Inflation;
Anisotropy and Inhomogeneity

FAQ | Tutorial : Part 1 | Part 2 | Part 3 | Part 4 | Age | Distances | Bibliography | Relativity

Spatial Curvature

One consequence of general relativity is that the curvature of space depends
on the ratio of rho to rho(crit). We call
this ratio Omega = rho/rho(crit). For Omega less than 1, the Universe has
negatively curved or hyperbolic geometry. For Omega = 1, the Universe has
Euclidean or flat geometry. For Omega greater than 1, the Universe has
positively curved or spherical geometry. We have already seen that the zero
density case has hyperbolic geometry, since the cosmic time slices in the
special relativistic coordinates were hyperboloids in this model.

The
figure above shows the three curvature cases plotted along side of the
corresponding a(t)'s. These a(t) curves assume that the cosmological
constant is zero, which is not the current standard model. Omega > 1
still corresponds to a spherical shape, but could expand forever even though the
density is greater than the critical density because of the repulsive
gravitational effect of the cosmological constant.

The age of the
Universe depends on Omegao as well as Ho. For Omega=1,
the critical density case, the scale factor is
a(t) = (t/to)2/3

and the age of the Universe is
to = (2/3)/Ho

while in the zero density case, Omega=0, and
a(t) = t/to with to = 1/Ho

If Omegao is greater than 1 the age of the Universe is even
smaller than (2/3)/Ho.

The
figure above shows the scale factor vs time measured from the present for
Ho = 65 km/sec/Mpc and for Omegao = 0 (green),
Omegao = 1 (black), and Omegao = 2(red). The age of the
Universe is 15, 10 and 8.6 Gyr in these
three models. The recollapse of the Omegao = 2 model occurs when the
Universe is 11 times older than it is now, and all observations indicate
Omegao < 2, so we have at least 80 billion more years before any
Big Crunch.

The value of Ho*to is a dimensionless number that
should be 1 if the Universe is almost empty and 2/3 if the Universe has the
critical density. In 1994 Freedman et al. (Nature, 371, 757) found
Ho = 80 +/- 17 and when combined with to = 14.6 +/- 1.7
Gyr, we find that Ho*to = 1.19 +/- 0.29. At face value
this favored the empty Universe case, but a 2 standard deviation error in the
downward direction would take us to the critical density case. Since both the
age of globular clusters used above and the value of Ho depend on the
distance scale in the same way, an underlying error in the distance scale could
make a large change in Ho*to. In fact, recent data from
the HIPPARCOS satellite
suggest that the Cepheid distance scale must be increased by 10%, and also that
the age of globular
clusters must be reduced by 20%. If we take the latest HST value for
Ho = 72 +/- 8 (Freedman et al. 2001, ApJ, 553, 47) and the
latest globular cluster ages giving to = 13.5 +/- 0.7 Gyr, we find
that Ho*to = 0.99 +/- 0.12 which is consistent with an
empty Universe, but also consistent with the accelerating Universe that is the
current standard model.

Flatness-Oldness Problem

However, if Omegao is sufficiently greater than 1, the Universe
will eventually stop expanding, and then Omega will become infinite. If
Omegao is less than 1, the Universe will expand forever and the
density goes down faster than the critical density so Omega gets smaller and
smaller. Thus Omega = 1 is an unstable stationary point unless the expansion of
the universe is accelerating, and it is quite remarkable that Omega is anywhere
close to 1 now.

The
figure above shows a(t) for three models with three different densities at a
time 1 nanosecond after the Big Bang. The black curve shows the critical density
case with density = 447,225,917,218,507,401,284,016 gm/cc. Adding only 1 gm/cc
to this 447 sextillion gm/cc causes the Big Crunch to be right now! Taking away
1 gm/cc gives a model with Omega that is too low for our observations. Thus the
density 1 ns after the Big Bang was set to an accuracy of better than 1 part in
447 sextillion. Even earlier it was set to an accuracy better than 1 part in
1059! Since if the density is slightly high, the Universe will die in
an early Big Crunch, this is called the "oldness" problem in cosmology. And
since the critical density Universe has flat spatial geometry, it is also called
the "flatness" problem -- or the "flatness-oldness" problem. Whatever the
mechanism for setting the density to equal the critical density, it works
extremely well, and it would be a remarkable coincidence if Omegao
were close to 1 but not exactly 1.

Manipulating Space-Time Diagrams

The critical density model is shown in the space-time diagram below.

Note that the worldlines for galaxies are now curved due to the
force of gravity causing the expansion to decelerate. In fact, each worldline is
a constant factor times a(t) which is (t/to)2/3 for this
Omegao = 1 model. The red pearshaped object is our past lightcone.
While this diagram is drawn from our point-of-view, the Universe is homogeneous
so the diagram drawn from the point-of-view of any of the galaxies on the
diagram would be identical.

The
diagram above shows the space-time diagram drawn on a deck of cards, and the
diagram below shows the deck pushed over to put it into A's point-of-view.

Note that this is not a Lorentz transformation, and that these
coordinates are not the special relativistic coordinates for which a Lorentz
transformation applies. The Galilean transformation which
could be done by skewing cards in this way required that the edge of the deck
remain straight, and in any case the Lorentz transformation can not be done on
cards in this way because there is no absolute time. But in cosmological models
we do have cosmic time, which is the proper time since the Big Bang measured by
comoving observers, and it can be used to set up a deck of cards. The presence
of gravity in this model leads to a curved spacetime that can not be plotted on
a flat space-time diagram without distortion. If every coordinate system is a
distorted representation of the Universe, we may as well use a convenient
coordinate system and just keep track of the distortion by following the
lightcones.

Sometimes it is convenient to "divide out" the expansion of the Universe, and
the space-time diagram shows the result of dividing the spatial coordinate by
a(t). Now the worldlines of galaxies are all vertical lines.

This division has expanded our past line cone so much that we have
to replot to show it all:

If
we now "stretch" the time axis near the Big Bang we get the following space-time
diagram which has straight line past lightcones:

This kind of space-time diagram is called a "conformal" space-time
diagram, and while it is highly distorted it makes it easy to see where the
light goes. This transformation we have done is analogous to the transformation
from the side view of the Earth on the left below and the Mercator chart on the
right.

Note that a constant SouthEast course is a straight line on the
Mercator chart which is analogous to having straight line past lightcones on the
conformal space-time diagram.

Also remember that the Omegao = 1 spacetime is infinite in extent
so the conformal space-time diagram can go on far beyond our past lightcone,

as
shown above.

Other coordinates can be used as well. Plotting the spatial coordinate as
angle on polar graph paper makes the translation to a different point-of-view
easy. On the diagram below,

an
Omegao = 2 model (which really is "round") is plotted this way with
a(t) used as the radial coordinate. The past lightcone of an observer reachs
halfway around the Universe in this model.

Horizon Problem

The conformal space-time diagram is a good tool use for describing the
meaning of CMB anisotropy observations. The Universe was opaque before protons
and electrons combined to form hydrogen atoms when the temperature fell to about
3,000 K at a redshift of 1+z = 1090. After this time the photons of the CMB have
traveled freely through the transparent Universe we see today. Thus the
temperature of the CMB at a given spot on the sky had to be determined by the
time the hydrogen atoms formed, usually called "recombination" even though it
was the first time so "combination" would be a better name. Since the
wavelengths in the CMB scale the same way that intergalaxy distances do during
the expansion of the Universe, we know that a(t) had to be 0.0009 at
recombination. For the Omegao = 1 model this implies that
t/to = 0.00003 so for to about 14 Gyr the time is about
380,000 years after the Big Bang. This is such a small fraction of the current
age that the "stretching" of the time axis when making a conformal space-time
diagram is very useful to magnify this part of the history of the Universe.

The
conformal space-time diagram above has exaggerated this part even further by
taking the redshift of recombination to be 1+z = 144, which occurs at the blue
horizontal line. The yellow regions are the past lightcones of the events which
are on our past lightcone at recombination. Any event that influences the
temperature of the CMB that we see on the left side of the sky must be within
the left-hand yellow region. Any event that affects the temperature of the CMB
on the right side of the sky must be within the right-hand yellow region. These
regions have no events in common, but the two temperatures are equal to better
than 1 part in 10,000. How is this possible? This is known as the "horizon"
problem in cosmology.

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