Fate of the Universe
The Universe is currently expanding.
One extremely important cosmological question
is whether this expansion will continue forever.
Ultimately, this
hinges on how much mass is contained in the
Universe (that is, its average density, and where by mass we include the
contributions of mass, energy, and pressure of all matter and radiation),
and on whether the
cosmological constant has a finite value.
To simplify the discussion, we shall first assume that the
cosmological constant is zero and then return to the consequences that are implied if this is
not so.
In this case of vanishing cosmological constant, if the density is below a
critical amount the Universe will expand forever. If it is above the critical
amount, the expansion will eventually reverse and the Universe will collapse on
itself, leading to what has been termed the big crunch. If it is exactly
equal to the critical amount, the expansion will slow, but will stop only after an
infinite amount of time. Thus, in this case the Universe will expand forever too.
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Value of the Critical Density
The value of the critical density for our Universe is a remarkably small number.
In mass and equivalent energy units,
ρc
= 7.9 x 10-27
kg/m3
= 4.5
GeV/m3
This is an average density of only about 5 hydrogen atoms for every cubic meter
of space in the Universe. However, we can express the critical density in yet another set of
units that is also instructive.
ρc
= 6.6 x 1011
solar masses/Mpc3
This is close (within a factor of 10) to the actual density of the Universe, since the
mass is close to that of a galaxy and the average spacings between galaxies are near
1 Mpc. This tells us
immediately that our Universe cannot be very far from the critical density.
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Is the Universe Open, Flat, or Closed?
The geometry of the Universe is
often expressed in terms of the density parameter, which is
defined to be the ratio of the actual density of the Universe to the critical density
that would just be required to cause the expansion to stop. This critical density is
also called
closure density. Its value is given in the adjacent left box.
In terms of the actual
density ρ and
the critical density ρc
Density Parameter = Ωo =
ρ/ρc
Thus, if
the Universe is flat (contains just the amount of mass to close it) the density
parameter is exactly 1, if the Universe is open with negative curvature the density
parameter lies between 0 and 1, and if the Universe is closed with positive curvature the
density parameter is greater than 1. These three possible categories for the
large-scale geometry of the Universe are summarized in the
following table and in the top right figure
in terms of the density parameter Ωo =
ρ/ρc.
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Density and the Fate of the Universe
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| Ωo =
ρ/ρc |
Geometry |
Fate of the Universe
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| < 1 |
Open |
Expansion forever
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| >1 |
Closed |
Expansion, then contraction
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| 0 |
Flat (Euclidean) |
Expansion stops only after
infinite time |
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Remember that
these considerations assume that the cosmological constant is zero. As we shall discuss
further below, if the cosmological constant is not zero the fate of the Universe is more
complex. In that case,
the future behavior of the cosmos
depends not only on the density of matter and radiation, but also on the vacuum
energy density. The condition for a flat geometry also becomes more complex because it must
account for both the influence of matter and radiation, and the vacuum energy density, on the
curvature of the Universe.
The Observed Value of the Mass Density Parameter
The mass density parameter can be estimated from various
observational methods. For example, big bang nucleosynthesis (to be
discussed in Chapter 27), coupled with observations of
light-element abundances, places a constraint that the density of
baryons (the stuff like neutrons and protons of ordinary matter)
can be no more than 3-4 percent of the closure density.
Detailed observations also indicate
that the stars, which we tend to think of as the
most obvious building blocks of the Universe, contribute less than
1 percent of the closure density.
Although most methods for estimating the mass in the Universe
yield values of the
density parameter far below the critical value of 1, we must remember that they
likely have not detected all matter in the Universe yet.
A typical modern number for the total
amount of mass in the Universe (both luminous and dark) based on extrapolating
from what is observed to a conjectured total mass
gives no more than 30-40 percent
of the closure density.
