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

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One consequence of general relativity is that the curvature of space
depends on the ratio of
rho to
rho(crit). We call this ratio
Ω = rho/rho(crit). For Ω less than 1, the Universe has
negatively curved or
hyperbolic geometry. For Ω = 1, the Universe has Euclidean
or flat geometry. For Ω 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. Ω > 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 Ω_{o} as well as H_{o}. For Ω=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, Ω=0, and_{o}= (2/3)/H_{o}

a(t) = t/tIf Ω_{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. In 1994 Freedman *et al.* (Nature, 371, 757) found
H_{o} = 80 +/- 17 and
when combined with
t_{o} = 14.6 +/- 1.7 Gyr, we find that
H_{o}*t_{o} = 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 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 the latest
HST value for H_{o} = 72 +/- 8 (Freedman *et al.* 2001,
ApJ, 553, 47) and the latest globular cluster ages giving
t_{o} = 13.5 +/- 0.7 Gyr, we find that
H_{o}*t_{o} = 0.99 +/- 0.12 which is
consistent with an empty Universe, but also consistent with the
accelerating Universe that is the current standard model.

However, if Ω_{o} is sufficiently greater than 1,
the Universe will eventually
stop expanding, and then Ω will become infinite. If Ω_{o} is
less than 1, the Universe will expand forever and the density goes
down faster than the critical density so Ω gets smaller and smaller.
Thus Ω = 1 is an unstable stationary point unless the expansion of the
universe is accelerating, and it is quite
remarkable that Ω 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 a critical density case that matches the WMAP-based concordance model, which has density = 447,225,917,218,507,401,284,016 gm/cc at 1 ns after the Big Bang. Adding only 0.2 gm/cc to this 447 sextillion gm/cc causes the Big Crunch to be right now! Taking away 0.2 gm/cc gives a model with a matter density Ω

Note that the old version of this figure was based on a model with higher current matter density, and also rounded the true Δρ of 0.4 gm/cc to 1 based on rounding the logarithm.

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 Ω_{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 Ω

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 Ω_{o} = 1 model this implies that
t/t_{o} = 0.00003 so for t_{o} 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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© 1996-2009 Edward L. Wright. Last modified 03 July 2009