Text 531, 132 rader
Skriven 2004-10-26 13:48:00 av Michael Ragland (1:278/230)
Ärende: (Part2) Hawking's Anthrop
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***** Continued From Previous Message *****
y 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.

"It's uncertain whether intelligence has any long term survival value.
Bacteria do quite well without it."

Stephen Hawking
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