Lectures physucs colloquiums
Quantum Cosmology, M-theory and the Anthropic Principle (January '99)
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This talk will be based on work with Neil Turok and Harvey Reall. I
will describe what I see as the framework for quantum cosmology, on
the basis of M theory. I shall adopt the no boundary proposal, and
shall argue that the Anthropic Principle is essentia l, if one is to
pick out a solution to represent our universe, from the whole zoo of
solutions allowed by M theory.
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 cosmolo gy 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 tim es. 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 i n any other science. The task of making
predictions in cosmology is made more difficult by the singularity
theorems, that Roger Penrose and I proved. These showed that if
General Relativity were correct, the universe would have begun with a
singularity. Of course, we would expect classical General Relativity
to break down near a singularity, when quantum gravitational effects
have to be taken into acco unt. So what the singularity theorems are
really telling us, is that the universe had a quantum origin, and that
we need a theory of quantum cosmology, if we are to predict the
present state of the universe.
A theory of quantum cosmology has three aspects. The first, is the
local theory that the fields in space-time obey. The second, is the
boundary conditions for the fields. And I shall argue that the
anthropic principle, is an essential third element. As far as the
local theory is concerned, the best, and indeed the only consistent
way we know, to describe gravitational forces, is curved space-time.
And the theory has to incorporate super symmetry, because otherwise
the uncancelled vacuum energies of all the modes would curl space-time
into a tiny ball. T hese two requirements, seemed to point to
supergravity theories, at least until 1985. But then the fashion
changed suddenly. People declared that supergravity was only a low
energy effective theory, because the higher loops probably diverged,
though no on e was brave, or fool hardy enough to calculate an
eight-loop diagram. Instead, the fundamental theory was claimed to be
super strings, which were thought to be finite to all loops. But it
was discovered that strings were just one member, of a wider class of
extended objects, called p-branes. It seems natural to adopt the
principle of p-brane democracy. All p-branes are created equal. Yet
for p greater than one, the quantum theory of p-branes, diverges for
higher loops.
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 inf ra 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 approximati on.
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 pre big bang scenario claims that the boundary condition, is some
vacuum state in the infinite past. But if this vacuum state develops
into the universe we have now, it must be unstable. And if it is
unstable, it wouldn't be a vacuum state, and it wou ldn't have lasted
an infinite time before becoming unstable.
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 d e 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 fro m 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
p ath integral over compact metrics. In other words, the boundary
condition of the universe is that it has no boundary.
There are compact Reechi flat metrics of any dimension, many with high
dimensional modulie spaces. Thus eleven-dimensional supergravity, or M
theory, admits a very large number of solutions and compactifications.
There may be some principle that we haven' t yet thought of, that
restricts the possible models to a small sub class, but it seems
unlikely. Thus I believe that we have to invoke the Anthropic
Principle. Many physicists dislike the Anthropic Principle. They feel
it is messy and vague, it can be us ed to explain almost anything, and
it has little predictive power. I sympathize with these feelings, but
the Anthropic Principle seems essential in quantum cosmology.
Otherwise, why should we live in a four dimensional world, and not
eleven, or some other number of dimensions. The anthropic answer is
that two spatial dimensions, are not enough for complicated
structures, like intelligent beings. On the other hand, four or more
spatial dimensions would mean that gravitational and electric forces
would fall off faster than the inverse square law. In this situation,
planets would not have stable orbits around their star, nor electrons
have stable orbits around the nucleus of an atom. Thus intelligent
life, at least as we know it, could exist only in four dim ensions. I
very much doubt we will find a non anthropic explanation.
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 phys ical
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 hav e the right
values. However, quantum cosmology, 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.
One takes the a-priori probability of a class of histories, to be the
e to the minus the Euclidean action, given by the no boundary
proposal. One then weights this a-priori probability, with the
probability that the class of histories contain intelligent life. As
physicists, we don't want to be drawn into to the fine details of
chemistry and biology, but we can reckon certain features, as
essential prerequisites of life as we know it. Among these are the
existence of galaxies and stars, and physical const ants near what we
observe. There may be some other region of modulie space, that allows
some different form of intelligent life, but it is likely to be an
isolated island. I shall therefore ignore this possibility, and just
weight the a-priori probability , with the probability to contain
galaxies.
The simplest compact metric that could represent a four dimensional
universe, would be the product of a four sphere, with a compact
internal space. But the world we live in has a metric with Lorentzian
signature, rather than a positive definite Euclidean one. So one has
to analytically continue the four-sphere metric, to complex values of
the coordinates.
There are several ways of doing this.
One can analytically continue the coordinate, sigma, as sigma equator,
plus i t. One obtains a Lorentzian metric, which is a closed Friedmann
solution, with a scale factor that goes like cosh Ht. So this is a
closed universe that collapses to a minimum si ze, and then expands
exponentially again.
However, one can analytically continue the four-sphere in another way.
Define t = i sigma, and chi = i psi. This gives an open Friedmann
universe, with a scale factor like sinh Ht.
Thus one can get an apparently spatially infinite universe, from the
no boundary proposal. The reason is that one is using as a time
coordinate, the hyperboloids of constant distance, inside the light
cone of a point in de Sitter space. The point itself, and its light
cone, are the big bang of the Friedmann model, where the scale factor
goes to zero. But they are not singular. Instead, the spacetime
continues through the light cone to a region beyond. It is this region
that deserves the name, the pre big bang scenario, rather than the
misguided model that commonly bears that title.
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 proba bility 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 dime nsions. 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 sca lar field case.
The scalar field, phi, will have a potential, V of phi. In regions
where the gradients of phi are small, the energy momentum tensor will
act like a cosmological constant, Lambda =8 pi G V, where G is
Newton's constant in four dimensions. Thus it will curv e the
Euclidean metric, like a four-sphere.
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.
One can write the metric of an O4 instanton, in terms of a function, b
of sigma. Here b is the radius of a three sphere of constant distance,
sigma, from the north pole of the instanton. If the instanton were a
perfectly round four-sphere, b would be a si ne function of sigma. It
would have one zero at the north pole, and a second at the south pole,
which would also be a regular point of the geometry. However, if the
scalar field at the north pole, is not at a stationary point of the
potential, it will var y over the four sphere. If the potential is
carefully adjusted, and has a false vacuum local minimum, it is
possible to obtain a solution that is non-singular over the whole
four-sphere. This is known as the Coleman De Lucia instanton.
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 potent ial, 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 th e 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 wel l 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 s ingular. 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.
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 inflationary universes
that one obtains from Coleman De Lucia instantons. The difference is
that the Coleman De Lucia instantons require d carefully adjusted
potentials, with false vacuum local minima. But the singular
Hawking-Turok instanton, will work for any reasonable potential. The
price one pays for a general potential, is a singularity at the south
pole. In the analytically continue d Lorentzian space-time, 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 inflating region. Thus
one would not be able to p redict 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 defined.
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.
First is the isotropy. Why are different regions of the microwave sky,
at very nearly the same temperature, if those regions have not
communicated in the past? Second, despite this overall isotropy, why
are there fluctuations of order one part in 10 to th e minus 5, with a
fairly flat spectrum? Third, why is the density of matter, still so
near the critical value, when any departure would grow rapidly with
time? Fourth, why is the vacuum energy, or effective cosmological
constant, so small, when symmetry b reaking might lead one to expect a
value ten to the 80 higher?
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. A nd 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 t o 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. Accordin g 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 th e 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 u niverse 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 we ighted 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 sh all 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 th e horizon, will
slowly increase with wavelength, up to a maximum of one over the size
of the instanton.
The time, at which the maximum amplitude re-enters the horizon, is
also the time at which omega begins to drop below one. One has two
competing effects. The a-priori probability from the no boundary
proposal wants to make the instantons large, and probabi lity of the
formation of galaxies, which requires that both omega, and the
amplitude of the fluctuations, not be too small. This would give a
sharp peak in the probability distribution for omega, of about ten to
the minus three. The probability for the te nsor perturbations will
peak at order ten to the minus eight. Both these values, are much less
than what is observed. So what went wrong.
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 a s 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 cosmologic al 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 f act, 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 cancellati on 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 t hem, 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 ma tter plus omega lambda =1 line, if the
universe is one of these open analytical continuations. This is
consistent with the observations.
The red eliptic region, is the three sigma limits of the supernova
observations. The blue region is from clustering observations, and the
purple is from the Doppler peak in the microwave. They seem to have a
common intersection, on or below the omega tota l =1 line.
Assuming that one can find a model that predicts a reasonable omega,
how can we test it by observation? The best way is by observing the
spectrum of fluctuations, in the microwave background. This is a very
clean measurement of the quantum fluctuations, a bout the initial
instanton. However, there is an important difference between the
non-singular Coleman De Lucia instantons, and the singular instantons
I have described. 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.
This has an effect on the two-point correlation function of the
perturbations, but it seems to be quite small. The present
observations of the microwave fluctuations are certainly not sensitive
enough to detect this effect. But it may be possible with the new
observations that will be coming in, from the map satellite in two
thousand and one, and the Planck satellite in two thousand and six.
Thus the no boundary proposal, and the pea instanton, are real
science. They can be falsified by observation. I will finish on that
note.
