224
R. Barrett and P. P. Delsanto
A solution to this puzzle has already been hinted at in Sect. 10.5: when
calculating the distance to a supernova by measuring the red shift and using
the Hubble constant, the assumption is made implicitly that the Hubble
constant is a constant. Its very name is an indication of our confidence in
this assumption, which is an application of Occam’s Razor. The data from
the two sets of measurements suggest that the expansion rate of the universe
at the time of emission of the light (measured by the red shift) was less than it
is today. The higher expansion rate in the universe today means that the path
length travelled by the light to reach us (on which the apparent luminosity of
the supernova observed through our telescopes depends) is longer than one
would expect from the initial expansion rate. In other words: the expansion of
the universe appears to be accelerating.
So how do these new results accord with Einstein’s Theory of General
Relativity, which is the theory underlying most of our understanding of
cosmology? Not very well at all, as it turns out. In fact, depending on the
actual value of the average energy/matter density in the universe, General
Relativity admits three different solutions, but in all cases the gravitational
attraction between galaxies slows down the initial expansion.
In the first, high density solution, the expansion reaches a maximum and
the universe begins to contract back, ending in a final Big Crunch. The Big
Crunch can be considered to be the inverse of the initial Big Bang singularity.
This solution constitutes a closed universe. (See Fig. 11.3.)
The second solution occurs when the energy/matter density has a value of
approximately five hydrogen masses per cubic meter. This value is critical, and
even a small deviation from it will not lead to this solution of Einstein’s equations. In this solution, the expansion slows down and stops asymptotically at
an infinite cosmic time. This would result in a flat and open universe.
In the third solution, which occurs for smaller densities, the expansion
rate also slows down, but tends to an asymptotic value different from zero,
so that the expansion never ceases. This would result in an open, but not flat
universe. As we can see, under no circumstances do the equations of General
Relativity yield a solution where there is an acceleration of the expansion rate.
The conclusion is clear: either General Relativity is wrong, or we are missing
something in our understanding of the universe.
In the preceding paragraphs we have used the terms “open” and “closed”
to describe the universe, and “flat” and “curved” to describe space–time.
What we mean by these can be illustrated by Fig. 11.3. The upper figure
corresponds to a closed universe with curved space–time. The parameter 0
describes the energy/mass density in arbitrary units. In this space, Euclidean
geometry does not apply, and the angles in a triangle sum to more than 180°.
R. Barrett and P. P. Delsanto
A solution to this puzzle has already been hinted at in Sect. 10.5: when
calculating the distance to a supernova by measuring the red shift and using
the Hubble constant, the assumption is made implicitly that the Hubble
constant is a constant. Its very name is an indication of our confidence in
this assumption, which is an application of Occam’s Razor. The data from
the two sets of measurements suggest that the expansion rate of the universe
at the time of emission of the light (measured by the red shift) was less than it
is today. The higher expansion rate in the universe today means that the path
length travelled by the light to reach us (on which the apparent luminosity of
the supernova observed through our telescopes depends) is longer than one
would expect from the initial expansion rate. In other words: the expansion of
the universe appears to be accelerating.
So how do these new results accord with Einstein’s Theory of General
Relativity, which is the theory underlying most of our understanding of
cosmology? Not very well at all, as it turns out. In fact, depending on the
actual value of the average energy/matter density in the universe, General
Relativity admits three different solutions, but in all cases the gravitational
attraction between galaxies slows down the initial expansion.
In the first, high density solution, the expansion reaches a maximum and
the universe begins to contract back, ending in a final Big Crunch. The Big
Crunch can be considered to be the inverse of the initial Big Bang singularity.
This solution constitutes a closed universe. (See Fig. 11.3.)
The second solution occurs when the energy/matter density has a value of
approximately five hydrogen masses per cubic meter. This value is critical, and
even a small deviation from it will not lead to this solution of Einstein’s equations. In this solution, the expansion slows down and stops asymptotically at
an infinite cosmic time. This would result in a flat and open universe.
In the third solution, which occurs for smaller densities, the expansion
rate also slows down, but tends to an asymptotic value different from zero,
so that the expansion never ceases. This would result in an open, but not flat
universe. As we can see, under no circumstances do the equations of General
Relativity yield a solution where there is an acceleration of the expansion rate.
The conclusion is clear: either General Relativity is wrong, or we are missing
something in our understanding of the universe.
In the preceding paragraphs we have used the terms “open” and “closed”
to describe the universe, and “flat” and “curved” to describe space–time.
What we mean by these can be illustrated by Fig. 11.3. The upper figure
corresponds to a closed universe with curved space–time. The parameter 0
describes the energy/mass density in arbitrary units. In this space, Euclidean
geometry does not apply, and the angles in a triangle sum to more than 180°.