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This lecture is the intellectual
property of Professor S.W. Hawking. You may not reproduce, edit or distribute
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Cosmology used to be regarded as a pseudo science, an area where wild speculation, was unconstrained by any reliable observations. We now have lots and lots of observational data, and a generally agreed picture of how the universe is evolving. But cosmology is still not a proper science, in the sense that as usually practiced, it has no predictive power. Our observations tell us the present state of the universe, and we can run the equations backward, to calculate what the universe was like at earlier times. But all that tells us is that the universe is as it is now, because it was as it was then. To go further, and be a real science, cosmology would have to predict how the universe should be. We could then test its predictions against observation, like in any other science.
The task of making predictions in cosmology is made more difficult by the singularity theorems, that Roger Penrose and I proved.
A theory of quantum cosmology has three aspects.
I think we should interpret these loop divergences, not as a break down of the supergravity theories, but as a break down of naive perturbation theory. In gauge theories, we know that perturbation theory breaks down at strong coupling. In quantum gravity, the role of the gauge coupling, is played by the energy of a particle. In a quantum loop one integrates over… So one would expect perturbation theory, to break down.
In gauge theories, one can often use duality, to relate a strongly coupled theory, where perturbation theory is bad, to a weakly coupled one, in which it is good. The situation seems to be similar in gravity, with the relation between ultra violet and infra red cut-offs, in the anti de Sitter, conformal field theory, correspondence. I shall therefore not worry about the higher loop divergences, and use eleven-dimensional supergravity, as the local description of the universe. This also goes under the name of M theory, for those that rubbished supergravity in the 80s, and don't want to admit it was basically correct. In fact, as I shall show, it seems the origin of the universe, is in a regime in which first order perturbation theory, is a good approximation.
The second pillar of quantum cosmology, are boundary conditions for the local theory. There are three candidates, the pre big bang scenario, the tunneling hypothesis, and the no boundary proposal.
The quantum-tunneling hypothesis, is not actually a boundary condition on the space-time fields, but on the Wheeler Dewitt equation. However, the Wheeler Dewitt equation, acts on the infinite dimensional space of all fields on a hyper surface, and is not well defined. Also, the 3+1, or 10+1 split, is putting apart that which God, or Einstein, has joined together. In my opinion therefore, neither the pre bang scenario, nor the quantum-tunneling hypothesis, are viable.
To determine what happens in the universe, we need to specify the boundary conditions, on the field configurations, that are summed over in the path integral. One natural choice, would be metrics that are asymptotically Euclidean, or asymptotically anti de Sitter. These would be the relevant boundary conditions for scattering calculations, where one sends particles in from infinity, and measures what comes back out. However, they are not the appropriate boundary conditions for cosmology.
We have no reason to believe the universe is asymptotically Euclidean, or anti de Sitter. Even if it were, we are not concerned about measurements at infinity, but in a finite region in the interior. For such measurements, there will be a contribution from metrics that are compact, without boundary. The action of a compact metric is given by integrating the Lagrangian. Thus its contribution to the path integral is well defined. By contrast, the action of a non-compact or singular metric involves a surface term at infinity, or at the singularity. One can add an arbitrary quantity to this surface term. It therefore seems more natural to adopt what Jim Hartle and I called the no boundary proposal. The quantum state of the universe is defined by a Euclidean path integral over compact metrics. In other words, the boundary condition of the universe is that it has no boundary.
The Anthropic Principle is usually said to have weak and strong versions. According to the strong Anthropic Principle, there are millions of different universes, each with different values of the physical constants. Only those universes with suitable physical constants will contain intelligent life. With the weak Anthropic Principle, there is only a single universe. But the effective couplings are supposed to vary with position, and intelligent life occurs only in those regions, in which the couplings have the right values. However, quantum cosmolog, and the no boundary proposal remove the distinction between the weak and strong Anthropic Principles. The different physical constants are just different modulie of the internal space, in the compactification of M theory, or eleven-dimensional supergravity. All possible modulie will occur in the path integral over compact metrics. By contrast, if the path integral were over non compact metrics, one would have to specify the values of the modulie at infinity. But why should the modulie at infinity, have those particular values, like four uncompactified dimensions, that allow intelligent life. In fact, the Anthropic Principle, really requires the no boundary proposal, and vice-versa.
One can make the Anthropic Principle precise, by using Bayes statistics.
There are several ways of doing this.
If the Euclidean four-sphere were perfectly round, both the closed and open analytical continuations, would inflate for ever. This would mean they would never form galaxies. A perfect round four sphere has a lower action, and hence a higher a-priori probability than any other four metric of the same volume. However, one has to weight this probability, with the probability of intelligent life, which is zero. Thus we can forget about round 4 spheres.
On the other hand, if the four sphere is not perfectly round, the analytical continuation will start out expanding exponentially, but it can change over later to radiation or matter dominated, and can become very large and flat. This provides a mechanism whereby all eleven dimensions can have similar curvatures, in the compact Euclidean metric, but four dimensions can be much flatter than the other seven, in the Lorentzian analytical continuation. But the mechanism doesn't seem specific to four large dimensions. So we will still need the Anthropic Principle, to explain why the world is four-dimensional.
In the semi classical approximation, which turns out to be very good, the dominant contribution, comes from metrics near solutions of the Euclidean field equations. So we need to study deformed four spheres, in the effective theory obtained by dimensional reduction of eleven dimensional supergravity, to four dimensions. These Kaluza Klein theories, contain various scalar fields, that come from the three index field, and the modulie of the internal space. For simplicity, I will describe only the single scalar field case.
However, if the field phi is not at a stationary point of V, it can not have zero gradient everywhere. This means that the solution can not have O5 symmetry, like the round four sphere. The most it can have, is O4 symmetry. In other words, the solution is a deformed four sphere.
However, for general potentials without a false vacuum, the behavior is different. The scalar field will be almost constant over most of the four-sphere, but will diverge near the south pole. This behavior is independent of the precise shape of the potential, and holds for any polynomial potential, and for any exponential potential, with an exponent, a, less then 2. The scale factor, b, will go to zero at the 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 finite value of the trace K surface term, on a boundary around the singularity at the south pole. This means the actions of perturbations of the four dimensional geometry, are well defined, despite the singularity. One can therefore calculate the fluctuations in the microwave background, as I shall describe later.
The deep reason, behind this good behavior of the singularity, was first seen by Garriga. He pointed out that if one dimensionally reduced five dimensional Euclidean Schwarzschild, along the tau direction, one would get a four-dimensional geometry, and a scalar field. 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 artifact 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 inflating initially.
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 effect 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 field configuration, is e to minus its action. Thus perturbations around the instanton have a lower probability, than the unperturbed background. This means that quantum fluctuation are suppressed, the bigger the fluctuation, 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.
In fact, the present matter and vacuum energy densities can be regarded as two axes in a plane of possibilities. For some purposes, it is better to deal with the linear combinations, matter plus vacuum energy, which is related to the curvature of space. And matter minus twice vacuum energy, which gives the deceleration of the universe.
Inflation was supposed to solve the problems of the hot big bang model. It does a good job with problem one, the isotropy of the universe. If the inflation continues for long enough, the universe would now be spatially flat, which would imply that the sum of the matter and vacuum energies had the critical value. But inflation 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 fluctuations. These have to be fed in, as fine tunings of the scalar potential, V. Also, without a theory of initial conditions, it is not clear why the universe should start out inflating in the first place.
The instantons I have described predict that the universe starts out in an inflating, de Sitter like state. Thus they solve the first problem, the fact that the universe is isotropic. However, there are difficulties with the other three problems. According to the no boundary proposal, the a-priori probability of an instanton, is e to the minus the Euclidean action. But if the Reechi scalar is positive, as is likely for a compact instanton with an isometry group, the Euclidean action will be negative.
The larger the instanton, the more negative will be the action, and so the higher the a-priori probability. Thus the no boundary proposal, favors large instantons. In a way, this is a good thing, because it means that the instantons are likely to be in the regime, where the semi classical approximation is good. However, a larger instanton, means starting at the north pole, with a lower value of the scalar potential, V. If the form of V is given, this in turn means a shorter period of inflation. Thus the universe may not achieve the number of e-foldings, needed to ensure omega matter, plus omega lambda, is near to one now. In the case of the open Lorentzian analytical continuation considered here, the no boundary a-priori probabilities, would be heavily weighted towards omega matter, plus omega lambda, equals zero. Obviously, in such an empty universe, galaxies would not form, and intelligent life would not develop. So one has to invoke the anthropic principle.
If one is going to have to appeal to the anthropic principle, one may as well use it also for the other fine tuning problems of the hot big bang. These are the amplitude of the fluctuations, and the fact that the vacuum energy now, is incredibly near zero. The amplitude of the scalar perturbations depends on both the potential, and its derivative. But in most potentials, the scalar perturbations are of the same form as the tensor perturbations, but are larger by a factor of about ten. For simplicity, I shall consider just the tensor perturbations. They arise from quantum fluctuations of the metric, which freeze in amplitude when their co-moving wavelength, leaves the horizon during inflation.
Thus amplitude of the tensor perturbation, will thus be roughly one over the horizon size, in Planck units. Longer co-moving wavelengths, leave the horizon first during inflation. Thus the spectrum of the tensor perturbations, at the time they re-enter the horizon, will slowly increase with wavelength, up to a maximum of one over the size of the instanton.
We haven't yet taken into account the anthropic requirement, that the cosmological constant is very small now. Eleven dimensional supergravity contains a three-form gauge field, with a four-form field strength. When reduced to four dimensions, this acts as a cosmological constant. For real components in the Lorentzian four-dimensional space, this cosmological constant is negative. Thus it can cancel the positive cosmological constant, that arises from super symmetry breaking. Super symmetry breaking is an anthropic requirement. One could not build intelligent beings from mass less particles. They would fly apart.
Unless the positive contribution from symmetry breaking cancels almost exactly with the negative four form, galaxies wouldn't form, and again, intelligent life wouldn't develop. I very much doubt we will find a non anthropic explanation for the cosmological constant.
In the eleven dimensional geometry, the integral of the four-form over any four cycle, or its dual over any seven cycle, have to be integers. This means that the four-form is quantized, and can not be adjusted to cancel the symmetry breaking exactly. In fact, for reasonable sizes of the internal dimensions, the quantum steps in the cosmological constant, would be much larger than the observational limits. At first, I thought this was a set back for the idea there was an anthropically controlled cancellation of the cosmological constant. But then, I realized that it was positively in favor. The fact that we exist shows that there must be a solution to the anthropic constraints.
But, the fact that the quantum steps in the cosmological constant are so large means that this solution is probably unique. This helps with the problem of low omega I described earlier. If there were several discrete solutions, or a continuous family of them, the strong dependence of the Euclidean action on the size of the instanton, would bias the probability to the lowest omega and fluctuation amplitude possible. This would give a single galaxy in an otherwise empty universe, not the billions we observe. But if there is only one instanton in the anthropically allowed range, the biasing towards large instantons, has no effect. Thus omega matter and omega lambda, could be somewhere in the anthropically allowed region, though it would be below the omega matter plus omega lambda =1 line, if the universe is one of these open analytical continuations. This is consistent with the observations.
As I said, quantum fluctuations around the instanton are well defined, despite the singularity. Perturbations of the Euclidean instanton, have finite action if and only, they obey a Dirichelet boundary condition at the singularity. Perturbation modes that don't obey this boundary condition, will have infinite action, and will be suppressed. The Dirichelet boundary condition also arises, if the singularity is resolved in higher dimensions.
When one analytically continues to Lorentzian space-time, the Dirichelet boundary condition implies that perturbations reflect at the time like singularity.
I will finish on that note.