is not clear why the universe should start out inating in the rst place.
The instantons I have described predict that the universe starts out in an inating, de Sitter like state. Thus they solve the rst problem, the fact that the universe is isotropic. However, there are diculties with the other three problems. According to the no boundary proposal, the a-priori probability of an instanton, is to the minus the Euclidean action. But if the Reechi scalar is positive, e
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, favours 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, . If the form of is given, this in turn means a shorter period of V
V
ination. Thus the universe may not achieve the number of -foldings, needed to ensure matter +
e
is near to one now.
9
In the case of the open Lorentzian analytical continuation considered here, the no boundary a-priori probabilities would be heavily weighted towards matter + = 0. 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 ne tuning problems of the hot big bang. These are: the amplitude of the uctuations 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 uctuations of the metric, which freeze in amplitude when their co-moving wavelength leaves the horizon during ination.
Thus, the spectrum of the tensor perturbation will be roughly one over the horizon size, in Planck units. Longer co-moving wavelengths, will leave the horizon earlier during ination. Thus the spectrum of the tensor perturbations, at the time they re-enter the horizon, will slowly increase with wave length, up to a maximum of one over the size of the instanton.
Amplitude of perturbations when they
come into the visible universe
1
of
size
instanton
e
litud
Amp
Time
Time when Ω < 1 (16) The time at which the maximum amplitude re-enters the horizon, is also the time at which begins to drop below one. There are two competing e ects. One is the a-priori probability from the no boundary proposal, which wants to make the instantons large. The other is the probability of the formation of galaxies. This requires sucient ination to keep omega near to one, and a sucient amplitude of the uctuations. Both these favour small instanton sizes. Where the balance occurs depends on whether we weight with the density of galaxies per unit proper volume, or by the total number of galaxies. If we weight with the present proper density of galaxies, the probability distribution for , would be sharply peaked at about = 10 . ;3 Predictions for Weighting with proper density of galaxies, = 0 001 : Weighting with total number of galaxies, = 1 (17) This is the minimum value, that would give one galaxy in the observable universe, and clearly does 10 not agree with observation. On the other hand, one might think that one should weight with a factor proportional to the total number of galaxies in the universe. In this case, one would multiply the probability by a factor n, where is the number of -foldings during ination. This would lead to ;3 e n e the prediction that = 1, which seems to be consistent with observation, as I shall discuss. So far I haven't taken into account the anthropic requirement, that the cosmological constant is very small now. Eleven dimensional supergravity contains a three form gauge eld, with a four form eld 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 y apart.