The Anthropic Principle, is usually said to have weak and strong versions. According to the strong Anthropic Principle, there are millions of di erent universes, each with di erent 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 e ective couplings are supposed to
vary with position, and intelligent life occurs only in those regions in which the couplings have the right values. Even those who reject the Strong Anthropic Principle, would accept some Weak Anthropic arguments. For instance, the reason stars are roughly half way through their evolution, is that life could not have developed before stars, or have continued when they burnt out.
When one goes to quantum cosmology however, and uses the no boundary proposal, the distinction between the Weak and Strong Anthropic Principles disappears. The di erent physical constants are
just di erent moduli of the internal space, in the compactication of M theory, or eleven dimensional supergravity. All possible moduli will occur in the path integral over compact metrics. By contrast, if the path integral was over non compact metrics, one would have to specify the values of the moduli at innity. Each set of moduli at innity would dene a di erent super selection sector of the theory,
and there would be no summation over sectors. It would then be just an accident that the moduli at innity have those particular values, like four uncompactied dimensions, that allow intelligent life.
Thus it seems that the Anthropic Principle really requires the no boundary proposal, and vice versa.
One can make the Anthropic Principle precise, by using Bayes statistics.
Bayesian Statistics
(matter
)
P
j
Gal
axy
/
(
matter )
(matter )
(6)
P
Gal
axy
j
P
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 ne 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 constants near what we observe. There may be some other region of moduli space that allows some di erent 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.
4
Euclidean Four Sphere
=
+
( + sin )
1
2
2
2
2
2
ds
d
H sin2 H d
d
South
North
Pole
Pole
(7)
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 denite 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.
Analytical Continuation to a Closed Universe
Analytically continue = equator +
it
Equator
σ = 0
=
+
( + sin )
1
2
2
2
2
2
ds
;dt
H cosh2 Ht d
d
(8)
One can analytically continue the coordinate, , as equator + . One obtains a Lorentzian metric,
it
which is a closed Friedmann solution, with a scale factor that goes like cosh( ). So this is a closed H
t
universe, that starts at the Euclidean instanton, and expands exponentially.
5
Analytical contination of the
four sphere to an open universe
Anayltically continue = , =
(9)
it
i
=
+ (
) (
+ sinh )
2
1
2
2
2
2
2
ds
;dt
H sinh Ht d
d
However, one can analytically continue the four sphere in another way. Dene = , and = .
t
i
i
This gives an open Friedmann universe, with a scale factor like
( ).
sinh
H
t
Penrose diagram of an open analytical continuation