Lectures a brief history of mine
  Inflation: An Open and Shut Case (April '98)
  Slides and audio for this talk can be downloaded from the ITP at UCSB.
  This talk will be based on joint work with Neil Turok, at Cambridge.
  Neil has always been interested in what might be called, alternative
  cosmology. He pushed the idea that topological defects like cosmic
  strings or textures, were the origin of the large scale structure of
  the universe. And he was a proponent of what is called, open
  inflation. This is the idea that the universe is infinitely large, and
  of low density, despite having been through a period of exponential
  expansion, in the very early stages. My opinion was, that these were
  all nice ideas, but that nature probably hadn't chosen the use any of
  them. I included open inflation in that list, because I believed
  strongly that the universe came into being, at a finite size, and I
  felt that implied that the universe now, was still of finite size, or
  closed. However, after Neil gave a seminar on open inflation in
  Cambridge, we got talking. We realized it was possible for the
  universe to come into existence, at a finite size, but nevertheless,
  be either a finite, or an infinitely large universe now. My talk will
  be about this idea, and new developments that have occur since then.
  One of these, which is so recent that it is not yet fully worked out,
  is that it seems there is an observational signature, of the kind of
  inflation Neil and I are proposing. The present measurements are not
  sensitive enough to see this effect, but it should be possible to test
  for it, in the observations the Planck satellite will make.
  Another recent development, is that observations of supernovas, have
  suggested that the universe may have a small cosmological constant, at
  the present time. Even before these observations, Neil and I had
  realized that if there was a four form gauge field, one could invoke
  anthropic arguments to make it cancel the cosmological constant, that
  one would expect from symmetry breaking. But the anthropic argument
  would not require it to cancel exactly. So there could be a small
  residual cosmological constant. This is exciting. I shall describe the
  observational evidence later.
  As you probably know, the universe is remarkably isotropic on a large
  scale. That is to say, it looks the same in all directions, if one
  goes beyond such local irregularities as the Milky Way, and the Local
  Group of galaxies. By far the most accurate measurement of the
  isotropy of the universe, is the faint background of microwave
  radiation, first discovered in 1965. At the present time at least, the
  universe is transparant to microwaves, in directions out of the plane
  of our galaxy. Thus the microwave background, must have propagated to
  us, from distances of the order of the Hubble radius, or greater. It
  should therefore give a sensitive measurement, of any anisotropy in
  the universe.
  The remarkable fact is, that the microwave background is the same in
  every direction, to a high degree of accuracy. It wasn't until 1982,
  that differences between different directions were detected, at the
  level of one part in a thousand, with a dipole pattern on the sky.
  However, this could be interpreted as a consequence of our galaxy's
  motion through the universe, which blue shifted the microwave
  radiation in one direction, and red shifted it in the opposite
  direction. It need not represent any intrinsic anisotropy in the
  universe.
  It was not until 1992, that tiny fluctuations on angular scales of 10
  degrees, were detected by the Cosmic Background Explorer satellite,
  cobee. Since then, similar fluctuations have been found on smaller
  angular scales. The shape of the spectrum of fluctuations against
  angular scale, is still rather uncertain, but it is clear that the
  general size of the departures from uniformity, is only one part in
  ten to the five.
  This uniformity of the microwave background in different directions,
  was very difficult to understand. It seems the microwave background,
  is the last remnant of the radiation, that filled the hot early
  universe. What we observe, would have propagated freely to us, from a
  time of last scattering, when the universe was a thousandth of the
  size now. But according to the accepted hot big bang theory, the
  radiation coming from directions on the sky more than a degree apart,
  would be coming from regions of the early universe, that hadn't been
  in communication since the big bang. It was therefore truely
  remarkable, that the microwaves we observe in different directions,
  are the same to one part in ten to the five. How did different regions
  in the early universe, know to be at almost exactly the same
  temperature. It is a bit like handing in home work. If the whole class
  produce exactly the same, you can be sure they have communicated with
  each other. But according to the hot big bang model, there wasn't time
  since the big bang, for signals to get from one region to another. So
  how did all the regions, come up with the same temperature for the
  microwaves.
  If we assume that the universe is roughly homogeneous and isotropic,
  it can be described by one of the Friedmann Robertson Walker models.
  These are characterized by a scale factor S, which gives the distance
  between two neighbouring points, in the expanding universe. There are
  three kinds of Friedmann model, according to the sign of k, the
  curvature of the surfaces of constant time. If k =+1, the surfaces of
  constant time, are three spheres, and the universe is closed and
  finite in space. If k = minus 1, the surfaces have negative curvature,
  like a saddle, and the universe is infinite in spatial extent. The
  third possibility, k=0, a spatially flat universe, is of measure zero,
  but it is an important limiting case.
  Because the universe is expanding, the scale factor, S, is increasing
  with time.
  The second derivative of S, is given by the Einstein equation, in
  terms of the energy density and pressure, of matter in the universe,
  and the cosmological constant, lambda. For the moment, I will take
  lambda to be zero. For normal matter, both the energy density and
  pressure, will be positive. Thus the expansion of the universe, will
  be slowing down. In particular, for a universe dominated by radiation,
  like the early stages of the hot big bang model, the scale factor will
  go like t to the half.
  In such a model, one can ask how far one can see, before one sees
  right back to the big bang. It is easy to work out, that this is just
  the integral of one over the scale factor. For the hot big bang model,
  this integral converges. This means that a point in an early hot big
  bang universe, could have communicated only with a small region round
  it. Why then did it have almost exactly the same temperature, as
  regions far away.
  A possible explanation, was provided by the theory of inflation, which
  was put forward independently in the Soviet union, and the west,
  around 1980. The idea was to make regions able to communicate, by
  changing the expansion of the early universe, so that S double dot was
  positive, rather than negative. In other words, so that the expansion
  of the universe was being accelerated, rather than slowed down by
  gravity.
  As you can see from the Einstein equations, such accelerating
  expansion, or inflation, as it was called, required either negative
  energy, or negative pressure. One gets in a lot of trouble, if one
  allows negative energy. One would get runaway creation of particle
  pairs, one with positive energy, and the other with negative. But
  there is no reason to rule out negative pressure. That is just
  tension, which is a very common condition in the modern world.
  The original idea for inflation, was that in some way, the universe
  got trapped in what was called, a false vacuum state. A false vacuum
  state, is a Lorentz invariant meta-stable state, that has more energy
  than the true vacuum, which is taken to have zero energy density.
  Because a false vacuum is Lorentz invariant, its energy momentum
  tensor must be proportional to the metric. Since the false vacuum has
  positive energy density, the coefficient of proportionality must be
  negative. This means that the pressure in the false vacuum, is minus
  the energy density. The Einstein equations, then imply that the scale
  factor, increases exponentially with time. In such a universe, the
  integral of one over the scale factor, diverges as one goes back in
  time. This means that different regions in the early universe, could
  have communicated with each other, and come to equilibrium at a common
  state, explaining why the microwaves, look the same in different
  directions.
  The original model of inflation, which came to be known as old
  inflation, had various problems. How did the universe get into a false
  vacuum state in the first place, and how did it get out again. Various
  modifications were proposed, that went under the names of new
  inflation, or extended inflation. I won't describe them, because I
  have got into trouble in the past, about who should have credit for
  what, and because I now consider them irrelevant. As Lindeh first
  pointed out, it is not necessary for the universe to be in a false
  vacuum, to get inflation.
  A scalar field with a potential V, will have the energy momentum
  tensor shown on the screen. If the field is nearly constant in a
  region, the gradient terms will be small, and the energy momentum
  tensor, will be minus half V, times the metric. This is just what one
  needs for inflation.
  In the false vacuum case, the scalar field sits in a local minimum of
  the potential, V. In that case, the field equation allows the scalar
  field, to remain constant in space and time. If the scalar field is
  not at a local minimum, it can not remain constant in time, even if it
  is initially constant in space. However, Lindeh pointed out that if
  the potential is not too steep, the expansion of the universe, will
  slow down the rate at which the field rolls down the potential, to the
  minimum. The gradient terms in the energy momentum tensor, will remain
  small, and the scale factor will increase almost exponentially. One
  can get inflation with any reasonable potential V, even if it didn't
  have local minima, corresponding to false vacua. The work that Neil
  and I have done, is a logical extension of Andrei's idea. But I'm not
  sure if Andrei agrees with it, though I think he's coming round.
  Andrei's idea removed the need to believe that the universe began in a
  false vacuum. However, one still needed to explain, why the field
  should have been nearly constant over a region, with a value that was
  not at the minimum of the potential.
  To do this, one has to have a theory of the initial conditions of the
  universe. There are three main candidates. They are, the so called
  pre-big bang scenario, the tunneling hypothesis, and the no boundary
  proposal. In my opinion, the pre-big bang scenario is misguided, and
  without predictive power. And I feel the tunneling hypothesis, is
  either not well defined, or gives the wrong answers. But then I'm
  biased, for it was Jim Hartle and I, that were responsible for the no
  boundary proposal.
  This says that the quantum state of the universe, is defined by a
  Euclidean path integral over compact metrics, without boundary.. One
  can picture these metrics, as being like the surface of the Earth,
  with degrees of latitude, playing the role of imaginary time. One
  starts at the north pole, with the universe as a single point. As one
  goes south, the spatial size of the universe, increases like the
  lengths of the circles of latitude. The spatial size of the universe,
  reaches a maximum size at the equator, and then shrinks again to a
  point at the south pole.
  Of course, spacetime is four dimensional, not two dimensional, like
  the surface of the Earth, but the idea is much the same. I shall go
  through it in detail, because it is basic to the work I'm going to
  describe. The simplest compact four dimensional metric that might
  represent the universe, is the four sphere.
  One can give its metric in terms of coordinates, sigma, chi, theta and
  phi. One can think of sigma, as an imaginary time coordinate, and chi,
  theta and phi, as coordinates on a three sphere, that represents the
  spatial size of the universe. Again, one starts at the north pole,
  sigma =0, with a universe of zero spatial size, and expands up to a
  maximum size at the equator, sigma = pi, over 2H. But we live in a
  universe with a Lorentzian metric, like Minkowski space, not a
  Euclidean, positive definite metric.
  One therefore has to analytically continue, the Euclidean metrics used
  in the path integral, for the no boundary proposal. There are several
  ways one can analytically continue, the metric of the four sphere, to
  a Lorentzian spacetime metric. The most obvious is to follow the
  Euclidean time variable, sigma, from the north pole to the equator,
  and then go in the imaginary sigma direction, and call that real
  Lorentzian time, t. Instead of the size of the three spheres going as
  the sine of H sigma, they now go as the cosh of H t. This gives a
  closed universe, that expands exponentially with real time. At late
  times, the expansion will change from being exponential, to being
  slowed down by matter in the normal way. This departure of the scale
  factor from a cosh behavior, will occur because the original Euclidean
  four sphere, was not perfectly round. But the universe would still be
  closed, however deformed the four sphere.
  For nearly 15 years, I believed that the no boundary proposal,
  predicted that the universe was spatially closed. I also believed that
  the cosmological constant was zero, because it seemed unreasonable to
  suppose that it was less than the observational limit of 10 to the
  minus 120 Planck units, unless it were exactly zero. But the Einstein
  equations, relate the energy density in the universe, plus lambda, to
  the rate of expansion, and the curvature, k, of the surfaces of
  constant time.
  Define omega matter and omega lambda, to be the density and lambda,
  divided by the critical value. If the universe is closed, that is,
  k=+1, omega matter plus omega lambda, must be greater than one.
  Observations of luminous matter, like stars and gas clouds give an
  omega matter of about 0 point 0 2. We know that galaxies and clusters
  of galaxies, must contain non luminous, or dark matter, but the best
  estimate of this, is that it contributes at most 0 point 2 of the
  critical density. Still, Eddington once said, if your theory doesn't
  agree with the observations, don't worry. The observations are
  probably wrong. But if your theory doesn't agree with the second law
  of thermodynamics, forget it. I firmly believed in the no boundary
  proposal, and I thought it implied that the universe had to be closed.
  Since a closed universe, is not incompatible with the second law of
  thermodynamics, I was sure the observers had missed something, and
  there really was enough matter to close the universe. At that time, I
  didn't take the seriously the possibility of a small cosmological
  constant.
  The observations do not yet indicate that the universe is definitely
  open, or that lambda is non zero, but it is begining to look like one
  or the other, if not both. I won't go through all the observations,
  but shall just show what I consider to be the most significant pieces
  of evidence.
  The first is the distribution of large scale inhomogeneities in the
  universe. On the very largest scales, this can be measured by
  fluctuations in the microwave background, and on smaller scales by the
  galaxy galaxy correlation function. One can then try and fit these
  observations, with the predictions of inflationary theory. If one
  assumes the universe is filled with cold dark matter, the predicted
  spectrum of irregularities, depends on a quantity gamma. This is the
  product of omega matter, with the Hubble constant, or rate of
  expansion, in units of a hundred kilometers per second, per
  Megaparsec. (Astronomers use funny units). It is generally believed
  that the Hubble constant, is somewhere between 50 and 100 of those
  funny units. Thus if omega matter is one, gamma must be at least 0
  point 5. As you can see, a gamma of 0 point 5, would predict much less
  irregularity on large angles, than is observed. One can get a
  reasonable fit to the observations, with a gamma of 0 point 2. If
  omega matter were one, this would imply a Hubble constant of only 20.
  As a theorist, I would be happy with such a figure, because it would
  make the universe older, and remove a possible conflict with the ages
  of some stars. But the observers claim the Hubble constant, has to be
  in the range, 50 to 100. This would imply that omega matter is at most
  0 point 4.
  Thus dynamical measurements, give us a vertical strip in the omega
  matter, omega lambda plane.
  One can obtain further limits in this plane, from observations of
  supernova. Type 1 supernova, are standard candles. That is, the total
  energy in the explosion, is always the same, within a factor close to
  one. One can thus use their observed brightness, as a distance
  measurement, and compare it with their red shifts. This gives the
  limits shown on the diagram, for which I'm grateful to Ned Wright and
  Shawn Carol. The yellow, red and green areas represent the formal
  errors, and the large pink area, other possible errors. Also shown in
  blue, are the limits set by the position of a peak in the angular
  spectrum, of the variations of the microwave background. As you can
  see, the observations suggest that the universe is close to the open
  closed divide, but with a non zero lambda.
  Despite these indications of a low density lambda universe, I
  continued to believe that the cosmological constant was zero, and the
  no boundary proposal, implied that the universe must be closed. Then
  in conversations with Neil Turok, I realized there was another way of
  looking at the no boundary universe, that made it appear open. One
  starts with the point that Andrei Lindeh made, that inflation doesn't
  need a false vacuum, a local minimum of the potential. But if the
  scalar field is not at a stationary point of the potential, then it
  can not be constant on an instanton, a Euclidean solution of the field
  equations. In turn, this implies that the instanton can't be a
  perfectly round four sphere. A perfectly round four sphere, would have
  the symmetry group, O5.
  But with a non constant scalar field, the largest symmetry group that
  an instanton can have, is O4. In other words, the instanton 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 sine 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 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. The non constant scalar field,
  will cause the instanton not to be a perfectly round four sphere, and
  in fact there will be a singularity at the south pole. But it will be
  a very mild singularity, and the Euclidean action of the instanton
  will be finite.
  This Euclidean instanton, has been described as the universe begining
  as a pea. In fact, a pea is quite a good image for a deformed sphere.
  Its size of a few thousand Planck lengths, makes it a very petty pea.
  But the mass of the matter it contains, is about half a gram, which is
  about right for a pea.
  I actually discovered this pea instanton in 1983, but I thought it
  could describe the birth of close universes only.
  To get a closed universe, one starts with sigma =0 at the north pole,
  and proceeds to the equator, or rather the value of sigma at which the
  radius, b, of the three sphere is maximum. One then analytically
  continues sigma in the imaginary direction, as Lorentzian time. As I
  described earlier, this gives a closed universe with a scale factor
  that initially goes like cosh t. The scalar field, will have a small
  imaginary part, but that can be corrected by giving the initial value
  of the scalar field at the north pole, a small imaginary part.
  According to the no boundary proposal, the relative probability of
  such a closed universe, is e to minus twice the action of the part of
  the pea instanton, between the north pole, and the equator. Notice
  that as this part, doesn't contain the singularity at the south pole,
  there is no ambiguity about the action of a singular metric. The
  action of this part of the instanton, is negative, and is more
  negative, the bigger the pea. Thus the probability of the pea, is
  bigger, the bigger the pea. The negative sign of the action, may look
  counter intuitive, but it leads to physically reasonable consequences.
  As I said, I thought the no boundary proposal, implied that the
  universe had to be spatially closed, and finite in size. But Neil
  Turok and I, realized his ideas on open inflation, could be fitted in
  with the no boundary proposal. The universe would still be closed and
  finite, in one way of looking at it. But in another, it would appear
  open and infinite.
  Let's go back to the metric for the pea instanton, and analytically
  continue it in a different way. As before, one analytically continues
  the Euclidean latitude coordinate, in the imaginary direction, to
  become a Lorentzian time, t. The difference is that one goes in the
  imaginary sigma direction at the north pole, rather than the equator.
  One also continues the coordinate, chi, in the imaginary direction, as
  a coordinate, psi. This changes the three sphere, into a hyperbolic
  space. One therefore gets an exponentially expanding open universe.
  One can think of this open universe, as a bubble in a closed, de
  Sitter like universe. In this way, it is similar to the single bubble
  inflationary universes, that have been proposed by a number of
  authors. The difference is, the previous models all required carefully
  adjusted potentials, with false vacuum local minima. But the pea
  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 continued Lorentzian spacetime, this singularity would be
  time like, and naked. One might think that this naked singularity,
  would mean one couldn't evaluate the action of the instanton, or of
  perturbations about it. This would mean that one couldn't predict the
  quantum fluctuations, or what would happen in the universe. However,
  the singularity at the south pole, the stalk of the pea, is so mild,
  that the actions of the instanton, and of perturbations around it, are
  well defined. This 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. On the other hand,
  according to the tunneling hypothesis, favored by Vilenkin and Lindeh
  , probabilities are proportional to e to the ~plus action. This would
  mean that quantum fluctuation would be ~enhanced, the bigger the
  fluctuation. There is no way this could lead to a sensible description
  of the universe. Lindeh therefore proposes to take e to the ~plus
  action, for the probability of the background universe, but e to the
  ~minus action, for the perturbations. However, there is no invariant
  way, in which one can divide the action, into a background part, and a
  part due to fluctuations. So Lindeh's proposal, does not seem well
  defined in general. By contrast, the no boundary proposal, is well
  defined. Its predictions may be surprising, but they are not obviously
  wrong.
  To recapitulate. A general potential, without false vacuums, or local
  minimums, leads to the pea instanton. This can be analytically
  continued, to either an open, or a closed universe. The no boundary
  proposal, then allows one to calculate the relative probabilities of
  different backgrounds, and the quantum fluctuations about them.
  There isn't just a single pea instanton, but a whole family of them,
  labeled by different values of the scalar field at the north pole. The
  higher the value of the potential at the north pole, the smaller the
  instanton, and the less negative the value of the action. Thus the no
  boundary proposal, predicts that large instantons, are more probable
  than small ones. This is a problem, because large instantons, will
  lead to a shorter period of exponential expansion or inflation, than
  small ones. In the closed universe case, a short period of inflation,
  would mean the universe would recollapse before it reached the present
  size and density. On the other hand, an open universe with a short
  period of inflation, would become almost empty early on.
  Clearly, the universe we live in, didn't collapse early on, or become
  almost empty. So we have to take account of the anthropic principle,
  that if the universe hadn't been suitable for our existence, we
  wouldn't be asking why it is, the way it is. Many physicists don't
  like the anthropic principle, but I think some version of it is
  essential, in quantum cosmology. M theory, or whatever the ultimate
  theory is, seems to allow a very large number of possible solutions,
  and compactifications. One has to have some criterion, for discarding
  most of them. Otherwise, why isn't the universe, eleven dimensional
  Minkowski space.
  The approach Neil Turok and I took, was to invoke the weakest version
  of the anthropic principle. We adopted Bayes statistics. In this, one
  starts with an a-priori probability distribution, and then modifies it
  in light of ones knowledge of the system. In this case, we took the
  a-priori distribution, to be the e to the minus action, predicted by
  the no boundary proposal. We then modified it, by the probability that
  the model contained galaxies, which are presumably a necessary
  condition, for the existence of intelligent observers. An open
  universe, has an infinite spatial volume. Thus the total number of
  galaxies in an open universe, would always be infinite, no matter how
  low the probability of finding a galaxy, in a given comoving volume.
  One therefore can not weight the a-priori probability, given by the no
  boundary proposal, by the total number of galaxies in the universe.
  Instead, we weighted by the comoving density of galaxies, predicted
  from the growth of quantum fluctuations, about the pea instanton. This
  gives a modified probability distribution for omega, the present
  density, divided by the critical density. For the open models, this
  probability distribution, is sharply peaked at an omega of about zero
  point zero one. This is lower than is compatible with the
  observations, but it is not such a bad miss. As far as I'm aware, this
  is the first attempt to ~predict a value of omega for an open
  universe, rather than fine tune a false vacuum potential, to obtain a
  value in the range indicated by observation.
  The anthropic arguments we have used, are fairly crude, and could be
  refined. But the best hope of getting a more realistic omega, seems to
  be to include other fields. Eleven dimensional super gravity, which is
  the best candidate we have for a theory of everything, has a three
  form potential, with a four form field strength. When dimensionally
  reduced to four dimensions, this can act as a cosmological constant.
  For a real four form in dimensions, the contribution to the
  cosmological constant is negative. It can therefore cancel the
  positive contribution to the cosmological constant, that must arise
  because super symmetry is broken, in the universe we live in. Indeed,
  super symmetry breaking, is a necessary condition for life. But
  galaxies will not form, unless the total cosmological constant, is
  almost zero. Thus the anthropic principle fixes the value of the four
  form field strength, which is a free parameter of the theory, so it
  almost cancels the positive contribution from symmetry breaking. But
  it need not cancel it exactly. The anthropic requirement, can probably
  be satisfied by any omega lambda between about minus one point five,
  and plus one point five, with a fairly flat probability distribution.
  This is consistent with the observations. My student, Harvey Reall and
  I, are now working on an eleven dimensional supergravity version of
  the pea instanton. One gets a reduced action in four dimensions with a
  four form, and two scalar fields, which describe the size, and the
  squashing of a seven sphere.
  The squashing scalar field, phi, has a potential with a minimum at the
  round metric, and a maximum at the squashed sphere with an Einstein
  metric. One can get a pea instanton, by starting phi on the
  exponential wall on the right. This would produce an inflationary
  universe, in which the squashing ran down to the round seven sphere.
  The scalar field that represents the size of the seven sphere, has a
  potential that looks unstable. However, if one takes into account the
  back reaction of the scalar field on the four form, the effective
  potential becomes stable. This looks good, but the potentials are too
  steep to give enough inflation. Maybe if we can include the dynamical
  effects of symmetry breaking, we can get something more reasonable.
  The aim is to find a description of the origin of the universe, on the
  basis of fundamental theory.
  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, about the initial
  instanton. However, there is an important difference between our
  instanton, and previous proposals for open inflation. They have all
  assumed false vacuum potentials, and have used the Coleman De Lucia
  instanton, which is non singular. However, our instanton has a
  singularity at the south pole. There has been a lot of discussion of
  this singularity. In the Lorentzian analytically continued spacetime,
  the singularity is time like and naked. People have worried about this
  singularity, because it seemed to make the spacetime non predictable.
  Anything could come out of the singularity. However, 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. Support for this boundary condition, has come
  from the work of Garriga, who has shown that in some cases at least,
  the singularity in the instanton, is just an artefact of Kaluza Klein
  reduction from higher dimensional spacetimes. In these situations,
  perturbations would obey this Dirichelet boundary condition.
  When one analytically continues to Lorentzian spacetime, the
  Dirichelet boundary condition, implies that perturbations reflect at
  the time like singularity. This has a significant effect on the two
  point correlation function of the perturbations. I show preliminary
  calculations that Neil and a student have made, for the case of omega
  equals zero point three.
  The first shows the two point correlation function of the microwave
  background, as a function of angle, for our instanton, and for false
  vacuum open inflation. The difference, which is plotted on a magnified
  scale, is like a step function at 30 degrees, the angle subtended by
  the curvature radius, on the surface of last scattering. The next
  graph, shows the power spectrum of this correlation function. You see
  it has small oscillations, that come from the Fourier transform, of
  the step function.
  The present observations of the microwave fluctuations, are 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.

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