b
south pole, like distance to the third. This means the south pole is actually a singularity of the four dimensional geometry. However, it is a very mild singularity, with a nite value of the trace surface K
term, on a boundary around the singularity at the south pole. This means the actions of perturbations 7
of the four dimensional geometry are well dened, despite the singularity. One can therefore calculate the uctuations in the microwave background, as I shall describe later.
The deep reason behind this good behavior of the singularity was rst seen by Garriga. He dimensionally reduced ve dimensional Euclidean Schwarzschild, along the direction, to get a four
dimensional geometry, and a scalar eld.
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These were singular at the horizon, in the same manner as at the south pole of the instanton. In other words, the singularity at the south pole, can be just an artefact of dimensional reduction, and the higher dimensional space, can be non singular. This is true quite generally. The scale factor, ,b will go like distance to the third, when the internal space, collapses to zero size in one direction.
When one analytically continues the deformed sphere to a Lorentzian metric, one obtains an open universe, which is inating initially.
Hawking-Turok Instanton
Region I: Open Universe
Singularity
Null surface
t
Region II
Surfaces of homogeneity
Instanton
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One can think of this as a bubble in a closed, de Sitter like universe. In this way, it is similar to the single bubble inationary universes, that one obtains from Coleman De Lucia instantons. The di erence
is, the Coleman De Lucia instantons, required carefully adjusted potentials, with false vacuum local minima. But the singular Hawking-Turok instanton will work for any reasonable potential. The price 8
one pays for a general potential, is a singularity at the south pole. In the analytically continued Lorentzian spacetime, this singularity would be time like, and naked. One might think that anything could come out of this naked singularity, and propagate through the big bang light cone, into the open inating region. Thus one would not be able to predict what would happen. However, as I already said, the singularity, at the south pole of the four sphere, is so mild that the actions of the instanton, and of perturbations around it, are well dened.
This behavior of the singularity, means one can determine the relative probabilities of the instanton, and of perturbations around it. The action of the instanton itself is negative, but the e ect of
perturbations around the instanton is to increase the action. That is, to make the action less negative.
According to the no boundary proposal, the probability of a eld conguration is to minus its action.
e
Thus perturbations around the instanton, have a lower probability, than the unperturbed background.
This means that the more quantum uctuations are suppressed, the bigger the uctuation, as one would hope. This is not the case with some versions of the tunneling boundary condition.
How well do these singular instantons account for the universe we live in? The hot big bang model seems to describe the universe very well, but it leaves unexplained a number of features.
Problems of a Hot Big Bang
1. Isotropy
2. Amplitude of uctuations
3. Density of matter
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4. Vacuum energy
There is the overall isotropy of the universe, and the origin and spectrum of small departures from isotropy. Then there’s the fact that the density was suciently low to let the universe expand to its present size, but not so low that the universe is empty now. And the fact that despite symmetry breaking, the energy of the vacuum is either exactly zero, or at least, very small.
Ination was supposed to solve the problems of the hot big bang model. It does a good job with the rst problem, the isotropy of the universe. If the ination continues for long enough, the universe would now be spatially at, which would imply that the sum of the matter and vacuum energies had the critical value.
But ination, by itself, places no limits on the other linear combination of matter and vacuum energies, and does not give an answer to problem two, the amplitude of the uctuations. These have to be fed in, as ne tunings of the scalar potential, . Also, without a theory of initial conditions, it V