(10)
Thus one can get an apparently spatially innite 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 inate for ever. This would mean they would never form galaxies. A perfectly round four sphere has a lower action, and hence a higher a-priori probability 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 at.
This means there are equal opportunities for dimensions. All dimensions, in the compact Euclidean geometry, start out with curvatures of the same order. But in the Lorentzian analytical continuation, some dimensions can remain small, while others inate and become large. However, equal opportunities for dimensions might allow more than four to inate. 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 instantons. These are solutions of the Euclidean eld equations. So we need to study deformed four spheres in the e ective theory obtained by dimensional reduction of eleven
6
dimensional supergravity, to four dimensions. These Kaluza Klein theories contain various scalar elds, that come from the three index eld, and the moduli of the internal space. For simplicity, I will describe only the single scalar eld case.
Energy Momentum Tensor
=
+ ( )]
1
(11)
T
;
g
V
2
The scalar eld, , will have a potential, ( ). In regions where the gradients of are small,
V
the energy momentum tensor will act like a cosmological constant, = 8
, where is Newton’s
GV
G
constant in four dimensions. Thus it will curve the Euclidean metric, like a four sphere.
However, if the eld is not at a stationary point of , it can not have zero gradient everywhere.
V
This means that the solution can not have O(5) symmetry, like the round four sphere. The most it can have is O(4) symmetry. In other words, the solution is a deformed four sphere.
O(4) Instantons
=
+ ( )( + sin )
2
2
2
2
2
2
ds
d
b
d
d
b
φ
σ = 0
σ
σ
max
max
(12)
One can write the metric of an O(4) instanton, in terms of a function, ( ). Here is the radius b
b
of a three sphere of constant distance, , from the north pole of the instanton. If the instanton were
a perfectly round four sphere, would be a sine function of . It would have one zero at the north b
pole, and a second at the south pole, which would also be a regular point of the geometry. However, if the scalar eld at the north pole is not at a stationary point of the potential, it will vary 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 di erent. The scalar eld
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, and holds for any polynomial potential, and for any exponential potential, with an exponent, , less then 2. The scale factor, , will go to zero at the a