Fate of Universe (2) ...
The geometrical issues that determine the fate of the Universe also determine how old it is.
Age of the Universe
In Chapter 24 we
identified
the inverse of the Hubble constant with an approximate age of the Universe.
We noted there that generally the true age of the Universe
was expected to be less than 1/H because we expect that the
Hubble constant changes with time as the expansion is slowed by gravity.
By that reasoning, the inverse of the
Hubble constant should generally
overestimate the age of the Universe. We illustrate in the top right
figure where the size of the Universe (more precisely, the scale factor) is plotted versus time for
several different geometrical assumptions. As seen in the inset magnified view,
the curves have been adjusted so that they all have the same slope
at the present time (that is, they all correspond to the same
current value of the Hubble constant),
but they diverge from each other in the distant past
and distant future.
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Age of the Universe and Geometry
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| Geometry |
Lifetime
|
| Massless Universe |
1/H
|
| Closed |
Less than 2/(3H)
|
| Flat |
2/(3H)
|
| Open |
Between 2/(3H) and 1/H |
|
|
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Age and Geometry
The age of the Universe corresponds to the difference between the time marked "Now" and the
intersection of the relevant curve with the bottom axis (corresponding to zero radius for the Universe)
in the above figure.
These ages are indicated by the green arrows at the bottom. All curves give an age that is less than
1/H.
It may be shown that a flat Universe has
an age 2/(3H), a closed Universe has an
age less than 2/(3H), and an open Universe has an age
lying between 2/(3H) and 1/H, where H is the
current value of the Hubble constant. These results are summarized in the adjacent
table.
Effect of a Finite Cosmological Constant
The discussion above concerning the fate and the age of the Universe assumed that the cosmological
constant was zero. As the adjacent left
figure illustrates, the behavior of the Universe can be modified
substantially if there is a finite cosmological constant. The purple curve corresponds to the pure
Hubble law (constant expansion rate) and the orange curve corresponds to the previous flat Universe
example (deceleration parameter of 0.5), assuming in both cases a zero cosmological constant.
The blue curve corresponds to the inclusion of a positive cosmological constant with a flat
geometry. We see that the introduction of the cosmological constant
leads to two effects of fundamental significance.
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At large times the cosmological constant causes the expansion of
the Universe to accelerate.
A cosmological constant (vacuum energy) implies that the Universe can be open and expand
forever, even if it contains more than a
critical density of matter.
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The age of the Universe that we would infer from the present value of the Hubble constant is increased.
If the cosmological constant is sufficiently large (as in this example), the age of the
Universe could even be
larger than 1/H.
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The existence of a finite cosmological constant
would have revolutionary implications both for the age
of the Universe (it would be older than we
would estimate otherwise) and for the fate of
the Universe (it would probably expand forever, no matter
whether the matter density were critical or not). As we have already
noted, there is growing evidence from Type Ia supernovae
that the Universe has
a non-zero value of the cosmological constant. Therefore, let us
now let us
look explicitly at how the preceding discussion for a Universe with
zero cosmological constant is modified for a finite cosmological constant.
