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

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.

The age of the Universe
depends on Omega_{o} as well as H_{o}. For Omega=1,
the critical density case, the scale factor is

a(t) = (t/tand the age of the Universe is_{o})^{2/3}

twhile in the zero density case, Omega=0, and_{o}= (2/3)/H_{o}

a(t) = t/tIf Omega_{o}with t_{o}= 1/H_{o}

The figure above shows the scale factor vs time measured from the present for H

The value of H_{o}*t_{o}
is a dimensionless number that should be 1 if
the Universe is almost empty and 2/3 if the Universe has the critical
density. Taking H_{o} = 65 +/- 8 and
t_{o} = 14.6 +/- 1.7 Gyr, we find that
H_{o}*t_{o} = 0.97 +/- 0.17.
At face value this favors 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 H_{o}
depend on the distance scale in
the same way, an underlying error in the distance scale could make a
large change
in H_{o}*t_{o}.
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 H_{o} = 60 +/- 7
and t_{o} = 11.7 +/- 1.4 Gyr, we find that
H_{o}*t_{o} = 0.72 +/- 0.12 which is perfectly
consistent with a critical density Universe.
It is best to reserve judgement until better data is obtained.

However, if Omega_{o} is greater than 1, the Universe will eventually
stop expanding, and then Omega will become infinite. If Omega_{o} 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, 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 10

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/t

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 Omega_{o} = 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 Omega

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 = 1000. 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 0.001 at recombination.
For the Omega_{o} = 1 model this implies that
t/t_{o} = 0.00003 so for t_{o} about 10 Gyr the time is about
300,000 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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© 1996-1998 Edward L. Wright. Last modified 7-Nov-1998