of phi. This factor will depend on the slope of the potential, but will be at least 10 for
reasonable potentials. This means the fluctuations in the microwave background that the
density perturbations produce will be at least 10 times bigger than from the gravitational
waves. Thus the upper limit on the energy density at the time of wave function freezing
is only 10 − 12 of the Planck density. This is well within the range of the validity of the approximations I have been using. Thus it seems we don’t need string theory even for the
beginning of the universe.
The spectrum of the fluctuations with angular scale agrees within the accuracy of the
present observations with the prediction that it should be almost scale free. And the size
of the density perturbations is just that required to explain the formation of galaxies and
stars. Thus it seems the no boundary proposal can explain all the structure of the universe
including little inhomogeneities like ourselves.
One can think of the perturbations in the microwave background as arising from
thermal fluctuations in the scalar field φ.
The inflationary period has a temperature
of the expansion rate over 2 π because it is approximately periodic in imaginary time.
Thus, in a sense, we don’t need to find a little primordial black hole: we have already
observed an intrinsic gravitational temperature of about 1026 degrees, or 10 − 6 of the Planck temperature.
COBE predictions plus
⇒ upper limit on energy density
gravitational wave perturbations
10 − 10 Planck density
plus density perturbations
⇒ upper limit on energy density
10 − 12 Planck density
intrinsic gravitational
≈ 10 − 6 Planck temperature
temperature of early universe
= 1026 degrees
What about the intrinsic entropy associated with the cosmological event horizon. Can
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we observe this. I think we can and that it corresponds to the fact that objects like galaxies and stars are classical objects even though they are formed by quantum fluctuations. If one
looks at the universe on a space like surface Σ that spans the whole universe at one time,
then it is in a single quantum state described by the wave function Ψ. However, we can
never see more than half of Σ and we are completely ignorant of what the universe is like
beyond our past light cone. This means that in calculating the probability for observations,
we have to sum over all possibilities for the part of Σ we don’t observe. The effect of
the summation is to change the part of the universe we observe from a single quantum
state to what is called a mixed state, a statistical ensemble of different possibilities. Such decoherence, as it is called, is necessary if a system is to behave in a classical manner rather than a quantum one. People normally try to account for decoherence by interactions with
an external system, such as a heat bath, that is not measured. In the case of the universe
there is no external system, but I would suggest that the reason we observe classical
behavior is that we can see only part of the universe. One might think that at late times
one would be able to see all the universe and the event horizon would disappear. But this
is not the case. The no boundary proposal implies that the universe is spatially closed. A
closed universe will collapse again before an observer has time to see all the universe. I
have tried to show the entropy of such a universe would be a quarter of the area of the
event horizon at the time of maximum expansion. However, at the moment, I seem to be
getting a factor of 3 rather than a 1 . Obviously I’m either on the wrong track or I’m
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4
missing something.
I will end this lecture on a topic on which Roger and I have very different views,
the arrow of time. There is a very clear distinction between the forward and backward
directions of time in our region of the universe. One only has to watch a film being run
backwards to see the difference. Instead of cups falling off tables and getting broken, they